c4029-ch04

7/18/2019 C4029-Ch04

http://slidepdf.com/reader/full/c4029-ch04 1/104

4

B o u n d a r y p r o b l e m s w i t h o n e c l o s e d b o u n d a r y

4 . 1 L a p l a c e a n d P o i s s o n e q u a t i o n s

I t i s w e l l k n o w n t h a t a v e c t o r e l d

~

K h a v i n g a v a n i s h i n g c u r l

~

K = 0 c a n b e

r e p r e s e n t e d b y a p o t e n t i a l f u n c t i o n U

K

x

( x y z ) =

@ U

@ x

K

y

( x y z ) =

@ U

@ y

K

z

( x y z ) =

@ U

@ z

: ( 4 . 1 . 1 )

S e t t i n g u p t h e d i v e r g e n c e o f s u c h a e l d r e s u l t s i n

@ K

x

@ x

+

@ K

y

@ y

+

@ K

z

@ z

= d i v

~

K =

@

@ x

@ U

@ x

+

@

@ y

@ U

@ y

+

@

@ z

@ U

@ z

= U : ( 4 . 1 . 2 )

T h e r e a r e m a n y p r a c t i c a l e x a m p l e s i n p h y s i c s a n d e n g i n e e r i n g f o r t h e L a p l a c e

e q u a t i o n U = 0 . I n h y d r o d y n a m i c s , i n p r o b l e m s o f h e a t c o n d u c t i o n , i n e l e c -

t r o m a g n e t i s m a n d i n m a n y o t h e r e l d s t h e L a p l a c e e q u a t i o n o r i t s i n h o m o -

g e n e o u s c o u n t e r p a r t , t h e P o i s s o n e q u a t i o n , a p p e a r s a s

U = ( x y ) : ( 4 . 1 . 3 )

L e t u s n o w s o l v e a s i m p l e t w o - d i m e n s i o n a l b o u n d a r y v a l u e p r o b l e m f o r t h e

L a p l a c e e q u a t i o n . T h e c l o s e d d o m a i n s h o u l d b e a r e c t a n g l e o f d i m e n s i o n s a

a n d b , s e e F i g u r e 4 . 1 .

-

x

a

a

g

1

( y ) g

2

( y )

f

1

( x )

f

2

( x )

b b

6

y

F i g u r e 4 . 1

R e c t a n g u l a r b o u n d a r y

1 3 7 © 2003 by CRC Press LLC

7/18/2019 C4029-Ch04


1 3 8 B o u n d a r y p r o b l e m s w i t h o n e c l o s e d b o u n d a r y

T h e s e i n h o m o g e n e o u s b o u n d a r y c o n d i t i o n s a r e d e s c r i b e d b y

U ( x y = 0 ) = f

1

( x ) 0 x a ( 4 . 1 . 4 )

U ( x y = b ) = f

2

( x ) 0 x a ( 4 . 1 . 5 )

U ( x = 0 y ) = g

1

( y ) 0 y b ( 4 . 1 . 6 )

U ( x = a y ) = g

2

( y ) 0

y

b : ( 4 . 1 . 7 )

S o t h a t t h i s p r o b l e m i s w e l l d e n e d , t h e f u n c t i o n s m u s t b e c o n t i n u o u s a t

t h e c o r n e r s , b u t i t i s n o t n e c e s s a r y t h a t t h e b o u n d a r y v a l u e f u n c t i o n s s a t i s f y

U = 0 . L e t b e t h e v a l u e s o f t h e b o u n d a r y f u n c t i o n s a t t h e c o r n e r ,

t h e n t h e y h a v e t o s a t i s f y

f

1

( 0 ) = g

1

( 0 ) = f

2

( a ) = g

2

( b ) =

f

1

( a ) = g

2

( 0 ) = f

2

( 0 ) = g

1

( b ) = :

( 4 . 1 . 8 )

U s i n g t h e u s u a l s e t u p U ( x y ) = X ( x ) Y ( y ) f o r l i n e a r p a r t i a l d i e r e n t i a l

e q u a t i o n s w i t h c o n s t a n t c o e c i e n t s o n e g e t s

X

0 0

( x ) + p

2

X ( x ) = 0 ( 4 . 1 . 9 )

Y

0 0

( y ) ; p

2

Y ( y ) = 0 ( 4 . 1 . 1 0 )

w h e r e p i s t h e s e p a r a t i o n c o n s t a n t . T h e g e n e r a l s o l u t i o n s o f t h e s e e q u a t i o n s

r e a d

X ( x ) =

1

X

m = 1

A

m

s i n ( p

m

x + C

m

) ( 4 . 1 . 1 1 )

Y ( y ) =

1

X

n = 1

B

n

s i n h ( p

n

y + C

n

) ( 4 . 1 . 1 2 )

b u t o t h e r s o l u t i o n s l i k e h a r m o n i c p o l y n o m i a l s ( s e c t i o n 2 . 3 ) o r f u n c t i o n s o f a

c o m p l e x v a r i a b l e l i k e i n ( 3 . 3 . 1 ) e x i s t , t o o . T h e p a r t i a l a m p l i t u d e s A

m

B

n

,

t h e s e p a r a t i o n c o n s t a n t s p

m

a n d t h e c o n s t a n t s C

m

C

n

a r e s t i l l u n k n o w n .

T h e y h a v e t o b e d e t e r m i n e d b y t h e b o u n d a r y c o n d i t i o n s . S i n c e u = 0

i s l i n e a r , t h e s u m o v e r p a r t i c u l a r s o l u t i o n s i s a g a i n a s o l u t i o n ( s u p e r p o s i t i o n

p r i n c i p l e ) . S i n c e t h e p a r t i a l d i e r e n t i a l e q u a t i o n u = 0 i s h o m o g e n e o u s e a c h

p a r t i c u l a r s o l u t i o n m a y b e m u l t i p l i e d b y a f a c t o r ( p a r t i a l a m p l i t u d e ) . T h e

p h a s e c o n s t a n t s C

m

m a y b e u s e d t o n d a s o l u t i o n w i t h t w o f u n c t i o n s

A

m

s i n ( p

m

x + C

m

) = A

m

s i n p

m

x c o s C

m

+ A

m

c o s p

m

x s i n C

m

( 4 . 1 . 1 3 )

a n d a n a l o g o u s l y f o r s i n h .

T h e s o l u t i o n o f U ( x y ) = 0 c a n n o w b e w r i t t e n a s

© 2003 by CRC Press LLC

7/18/2019 C4029-Ch04


L a p l a c e a n d P o i s s o n e q u a t i o n s 1 3 9

U ( x y ) =

1

X

k = 0

D

k

s i n ( p

k

x + C

k

) s i n h

p

k

y +

~

C

k

: ( 4 . 1 . 1 4 )

T o s i m p l i f y c a l c u l a t i o n s w e c h o o s e

U ( 0 y ) = g

1

( y ) = f

1

( 0 ) = = 0 ( 4 . 1 . 1 5 )

U ( a y ) = g

2

( y ) = f

1

( a ) = = 0 ( 4 . 1 . 1 6 )

U ( x b ) = f

2

( x ) = g

1

( b ) = = 0 ( 4 . 1 . 1 7 )

b u t

U ( x 0 ) = f

1

( x ) 6= 0 ( 4 . 1 . 1 8 )

a n d , f o r c o n t i n u i t y , f

2

( a ) = g

2

( b ) = = 0 m u s t b e v a l i d . T o s a t i s f y t h e

b o u n d a r y c o n d i t i o n ( 4 . 1 . 1 5 ) b y t h e s o l u t i o n ( 4 . 1 . 1 4 ) , t h e e x p r e s s i o n

U ( 0 y ) =

1

X

k = 0

D

k

s i n ( C

k

) s i n h

p

k

y +

~

C

k

= 0 ( 4 . 1 . 1 9 )

m u s t b e v a l i d . T h i s c a n b e s a t i s e d b y C

k

= 0 f o r a l l k . T h e b o u n d a r y

c o n d i t i o n ( 4 . 1 . 1 6 ) d e m a n d s

U ( a y ) =

1

X

k = 0

D

k

s i n ( p

k

a ) s i n h

p

k

y +

~

C

k

= 0 : ( 4 . 1 . 2 0 )

F r o m t h i s c o n d i t i o n t h e s e p a r a t i o n c o n s t a n t s p

k

w i l l i m m e d i a t e l y b e d e t e r -

m i n e d :

p

k

a = k p

k

= k = a : k = 0 1 2 : : : : ( 4 . 1 . 2 1 )

L i k e i n m a n y b o u n d a r y v a l u e p r o b l e m s t h e s e p a r a t i o n c o n s t a n t s a r e d e t e r -

m i n e d b y t h e b o u n d a r y c o n d i t i o n s . N e x t w e c o n s i d e r t h e b o u n d a r y c o n d i t i o n

( 4 . 1 . 1 7 )

U ( x b ) =

1

X

k = 0

D

k

s i n

k x

a

s i n h

k b

a

+

~

C

k

= 0 ( 4 . 1 . 2 2 )

w h i c h c a n b e s a t i s e d b y

k b

a

+

~

C

k

= 0

~

C

k

= ;

k b

a

: ( 4 . 1 . 2 3 )

F i n a l l y c o n d i t i o n ( 4 . 1 . 1 8 ) h a s t o b e t a k e n i n t o a c c o u n t .

U ( x 0 ) = f

1

( x ) =

1

X

k = 1

D

k

s i n

k x

a

s i n h

;

k b

a

: ( 4 . 1 . 2 4 )


7/18/2019 C4029-Ch04



O b s e r v e t h a t n o w k = 0 i s n o t a l l o w e d a n d t h a t t h e c o n t i n u i t y c o n d i t i o n

( 4 . 1 . 1 5 ) f

1

( 0 ) = 0 i s s a t i s e d a u t o m a t i c a l l y .

A s o n e s e e s , ( 4 . 1 . 2 4 ) r e p r e s e n t s a F o u r i e r s e r i e s e x p a n s i o n o f t h e g i v e n

f u n c t i o n f

1

( x ) . T o n d t h e F o u r i e r e x p a n s i o n c o e c i e n t s w e m u l t i p l y

( 4 . 1 . 2 4 ) b y s i n ( m x = a ) m 6= k a n d i n t e g r a t e o v e r x

a

Z

0

f

1

( x ) s i n

m x

a

d x =

a

Z

0

X

k = 1

D

k

s i n

k x

a

s i n

m x

a

s i n h

;

k b

a

d

x

: ( 4 . 1 . 2 5 )

I n s e c t i o n 2 . 3 w e d i s c u s s e d t h e o r t h o g o n a l i t y o f s o m e f u n c t i o n s , c o m p a r e

( 2 . 3 . 1 7 ) . T h e a t t r i b u t e o f o r t h o g o n a l i t y i s v e r y i m p o r t a n t . I t a l l o w s t o e x -

p a n d g i v e n f u n c t i o n s l i k e f

1

( x ) a n d t o e x p r e s s t h e m b y a s e r i e s a c c o r d i n g t o

o r t h o g o n a l f u n c t i o n s . I n ( 4 . 1 . 2 5 ) o n e h a s

a

Z

b

s i n

k x

a

s i n

m x

a

d x = 0 f o r k 6= m : ( 4 . 1 . 2 6 )

T h i s i s e a s y t o p r o v e . T h e s i m p l e t r a n s f o r m a t i o n x = a = m o d i e s ( 4 . 1 . 2 6 )

i n t o

a

Z

0

s i n k s i n m d =

a

0 f o r k

6= m

= 2 f o r k = m

m k = 1 2 : : : : : ( 4 . 1 . 2 7 )

T h u s w e g e t

a

Z

0

f

1

( x ) s i n

k x

a

d x =

a

2

D

k

s i n h ( ; k b = a ) ( 4 . 1 . 2 8 )

w h i c h g i v e s t h e F o u r i e r e x p a n s i o n c o e c i e n t s D

k

. T h e s o l u t i o n o f o u r

b o u n d a r y v a l u e p r o b l e m d e n e d b y ( 4 . 1 . 1 5 ) t o ( 4 . 1 . 1 8 ) i s t h e n g i v e n b y

U ( x y ) =

1

X

k = 1

2 s i n h ( k b = a ; k y = a )

a s i n h ( k b = a )

a

Z

0

f

1

( ) s i n ( k = a ) d s i n ( k x = a ) : ( 4 . 1 . 2 9 )

B u t w h a t a b o u t t h e m o r e g e n e r a l p r o b l e m ( 4 . 1 . 4 ) t o ( 4 . 1 . 7 ) ? W e w i l l c o m e

b a c k t o i t i n p r o b l e m 2 o f t h i s s e c t i o n .

T h e b o u n d a r y p r o b l e m w e j u s t s o l v e d i s i n h o m o g e n e o u s b e c a u s e f

1

( x ) 6= 0 .

W h y c o u l d w e n o t h o m o g e n i z e t h e p r o b l e m a s w e l e a r n t i n s e c t i o n 1 . 2 ? T h e r e

w e h a v e s h o w n t h a t a n i n h o m o g e n e o u s p r o b l e m c o n s i s t i n g o f a n h o m o g e n e o u s

e q u a t i o n t o g e t h e r w i t h a n i n h o m o g e n e o u s c o n d i t i o n m a y b e c o n v e r t e d i n t o a n

i n h o m o g e n e o u s e q u a t i o n o f t h e t y p e ( 4 . 1 . 3 ) w i t h a n h o m o g e n e o u s b o u n d a r y

c o n d i t i o n .

T h e r e a r e t w o m e t h o d s t o s o l v e l i n e a r i n h o m o g e n e o u s p a r t i a l d i e r e n t i a l

e q u a t i o n s :


7/18/2019 C4029-Ch04



1 . S e a r c h a f u n c t i o n U s a t i s f y i n g t h e h o m o g e n e o u s b o u n d a r y c o n d i t i o n s

a n d d e t e r m i n e f r e e p a r a m e t e r s l i k e e x p a n s i o n c o e c i e n t s w i t h i n U i n

s u c h a w a y t h a t U s a t i s e s t h e i n h o m o g e n e o u s p a r t i a l d i e r e n t i a l e q u a -

t i o n s . T h i s d e l i v e r s a p a r t i c u l a r s o l u t i o n o f t h e i n h o m o g e n e o u s e q u a t i o n

t h a t h o w e v e r d o e s n o t s a t i s f y t h e h o m o g e n e o u s e q u a t i o n .

2 . F i r s t , s o l v e t h e m a t c h i n g h o m o g e n e o u s p a r t i a l d i e r e n t i a l e q u a t i o n a n d

e x p a n d t h e i n h o m o g e n e o u s t e r m a c c o r d i n g t o t h e s o l u t i o n s o f t h e

h o m o g e n e o u s e q u a t i o n .

W e s t a r t w i t h m e t h o d 1 . T h e d i e r e n t i a l e q u a t i o n ( 4 . 1 . 3 ) i s i n h o m o g e n e o u s

a n d t h e b o u n d a r y c o n d i t i o n s s h o u l d b e h o m o g e n e o u s . W i t h r e g a r d t o o u r

e x p e r i e n c e w e s e t u p

U ( x y ) =

1

X

= 1

c

( y ) s i n

x

a

( 4 . 1 . 3 0 )

w h e r e c

( y ) a r e t h e e x p a n s i o n c o e c i e n t s . N o w ( 4 . 1 . 3 0 ) s h o u l d r s t s a t i s f y

t h e b o u n d a r y c o n d i t i o n s . W e m u l t i p l y ( 4 . 1 . 3 0 ) b y s i n ( x = a ) a n d i n t e g r a t e

o v e r x . D u e t o ( 4 . 1 . 2 7 ) w e t h e n o b t a i n

a

Z

0

s i n

x

a

s i n

x

a

d x =

a

2

( 4 . 1 . 3 1 )

a n d

c

( y ) =

2

a

a

Z

0

U ( x y ) s i n

x

a

d x : ( 4 . 1 . 3 2 )

D u e t o t h e h o m o g e n e o u s b o u n d a r y c o n d i t i o n s

U ( 0 y ) = U ( a y ) = U ( x 0 ) = U ( x b ) = 0 ( 4 . 1 . 3 3 )

t h e c

m u s t s a t i s f y c

( 0 ) = c

( b ) = 0 f o r a l l . W e n o w s e t u p

c

( y ) =

1

X

= 1

s i n

y

b

( 4 . 1 . 3 4 )

s o t h a t

=

2

b

b

Z

0

c

( y ) s i n

y

b

d y ( 4 . 1 . 3 5 )

a n d t h e e x p r e s s i o n s a t i s f y i n g t h e h o m o g e n e o u s b o u n d a r y c o n d i t i o n s r e a d s

U ( x y ) =

1

X

= 1

1

X

= 1

s i n

x

a

s i n

y

b

( 4 . 1 . 3 6 )

w h e r e n o w

=

4

a b

a

Z

0

b

Z

0

U ( x y ) s i n

x

a

s i n

y

b

d y d x ( 4 . 1 . 3 7 )


7/18/2019 C4029-Ch04



w h i c h a r e s t i l l u n k n o w n ! ( 4 . 1 . 3 6 ) s a t i s e s t h e b o u n d a r y c o n d i t i o n s , b u t U

m u s t s a t i s f y t h e i n h o m o g e n e o u s e q u a t i o n t o o . T o n d a p a r t i c u l a r s o l u t i o n

o f t h e i n h o m o g e n e o u s p a r t i a l d i e r e n t i a l e q u a t i o n e x p a n d t h e i n h o m o g e n e o u s

t e r m w i t h r e s p e c t t o t h e s o l u t i o n s o f t h e m a t c h i n g h o m o g e n e o u s e q u a t i o n .

B u t w a i t , w e d o n o t y e t h a v e t h e s e s o l u t i o n s . S o w e j u s t h a v e t o w r i t e d o w n

a s e t u p c o m p a t i b l e w i t h t h e b o u n d a r y c o n d i t i o n s :

( x y ) =

1

X

= 1

1

X

= 1

d

s i n

x

a

s i n

y

b

( 4 . 1 . 3 8 )

w h e r e

d

=

4

a b

a

Z

0

b

Z

0

( x y ) s i n

y

b

s i n

x

a

d y d x : ( 4 . 1 . 3 9 )

B u t f r o m w h e r e d o w e g e t ( x y ) ? W e h a v e t o i n s e r t ( 4 . 1 . 3 6 ) , w h i c h c o n -

t a i n s t h e h o m o g e n e o u s b o u n d a r y c o n d i t i o n s , i n t o t h e i n h o m o g e n e o u s P o i s s o n

e q u a t i o n ( 4 . 1 . 3 )

( x y ) = U

x x

+ U

y y

=

1

X

= 1

1

X

= 1

s i n

x

a

s i n

y

b

;

2

2

a

2

;

2

2

b

2

=

1

X

= 1

1

X

= 1

d

s i n

x

a

s i n

y

b

: ( 4 . 1 . 4 0 )

N o w w e c a n r e a d o t h e

f r o m ( 4 . 1 . 4 0 )

=

; a

2

b

2

2

( a

2

2

+ b

2

2

)

d

( 4 . 1 . 4 1 )

a n d t h e b o u n d a r y p r o b l e m ( 4 . 1 . 3 ) , ( 4 . 1 . 3 3 ) i s s o l v e d a n d h a s t h e s o l u t i o n

U ( x y ) =

1

X

= 1

1

X

= 1

; a

2

b

2

d

2

( a

2

2

+ b

2

n u

2

)

s i n

x

a

s i n

y

b

: ( 4 . 1 . 4 2 )

B u t r e m e m b e r , t h i s o n l y i s a p a r t i c u l a r s o l u t i o n o f ( 4 . 1 . 3 ) a n d d o e s n o t s o l v e

t h e h o m o g e n e o u s e q u a t i o n .

N o w l e t u s l o o k h o w m e t h o d 2 w o r k s . W e r s t h a v e t o s o l v e t h e m a t c h i n g

h o m o g e n e o u s e q u a t i o n U = 0 a n d t h e n e x p a n d w i t h r e s p e c t t o t h e f u n c -

t i o n s s o l v i n g U = 0 . W e d e m o n s t r a t e t h e p r o c e d u r e o n a t h r e e - d i m e n s i o n a l

s p h e r i c a l p r o b l e m . W e u s e ( 3 . 1 . 7 4 ) a n d t h e o r d i n a r y d i e r e n t i a l e q u a t i o n s

( 3 . 1 . 7 6 ) - ( 3 . 1 . 7 8 ) . T h e s o l u t i o n s t o ( 3 . 1 . 7 7 ) h a d b e e n g i v e n i n s e c t i o n 2 . 3 ,

0 0

+ m

2

= 0 i s s o l v e d b y c o s m ' a n d s i n m ' a n d M a t h e m a t i c a s o l v e s

R

0 0

( r ) +

2

r

R

0

( r ) ;

l ( l + 1 )

r

2

R ( r ) = 0 ( 4 . 1 . 4 3 )


7/18/2019 C4029-Ch04



w h i c h i s N O T a B e s s e l e q u a t i o n ( 2 . 2 . 4 8 ) b u t i t i s o f t h e t y p e ( 2 . 2 . 6 5 ) t h a t

i s a n E u l e r e q u a t i o n o f t y p e ( 2 . 2 . 3 3 ) w i t h x

0

= 0 .

DSolve[R0 0 [r]+2*R 0 [r]/r-(l*(l+1))*R[r]/rˆ2==0,R[r],r]

y i e l d s

R r ] ! r

( ; 1 ; l )

C 1 ] + r

l

C 2 ]

T h u s , t h e s o l u t i o n o f t h e m a t c h i n g h o m o g e n e o u s e q u a t i o n U = 0 r e a d s i n

s p h e r i c a l c o o r d i n a t e s

U ( r # ' ) =

1

X

l = 0

1

X

m = 0

;

a

m

l

r

l

+ b

m

l

r

; l ; 1

N P

m

l

( c o s # ) e

i m '

: ( 4 . 1 . 4 4 )

N i s a n o r m a l i z i n g f a c t o r a n d a

m

l

a n d b

m

l

a r e t h e p a r t i a l a m p l i t u d e s . E x -

p a n d i n g t h e i n h o m o g e n e o u s t e r m o f t h e i n h o m o g e n e o u s e q u a t i o n u =

g i v e s

( r # ' ) =

X

l m

l m

( r ) P

m

l

( c o s # ) e

i m '

: ( 4 . 1 . 4 5 )

M u l t i p l i c a t i o n b y P

; m

l

( c o s # ) e x p ( ; i m ' ) , i n t e g r a t i o n a n d u s i n g ( 2 . 3 . 1 9 ) a n d

d x =

; s i n # d #

R R

s i n # d # d ' = 4 r e s u l t s i n

l m

( r ) =

Z Z

( r # ' ) P

; m

l

( c o s # ) e

; i m '

s i n # d # d '

2 l + 1

( ; 1 )

m

8

: ( 4 . 1 . 4 6 )

I f o n e i n s e r t s t h e e x p r e s s i o n

U =

1

X

l = 0

1

X

m = l

U

l m

( r ) P

m

l

( c o s # ) e

i m '

( 4 . 1 . 4 7 )

i n t o t h e i n h o m o g e n e o u s e q u a t i o n w h i c h w e w r i t e n o w U = ; 4 a s u s u a l

i n e l e c t r o m a g n e t i s m a n d i f o n e m u l t i p l i e s b y t h e o r t h o g o n a l f u n c t i o n s a n d

i n t e g r a t e s , o n e g e t s

1

r

2

d

d r

r

2

d U

l m

( r )

d r

;

l ( l + 1 )

r

2

U

l m

( r ) = ; ( ; 1 )

m

( 2 l + 1 )

l m

( r ) : ( 4 . 1 . 4 8 )

T h i s i s a n i n h o m o g e n e o u s o r d i n a r y d i e r e n t i a l e q u a t i o n . T h e m a t c h i n g h o -

m o g e n e o u s e q u a t i o n h a s a s o l u t i o n o b t a i n e d b y t h e a n a l o g o u s c o m m a n d a s

g i v e n a b o v e a n d r e a d s l i k e t h e s o l u t i o n R[r] g i v e n a b o v e . T h e W r o n s k i a n

d e t e r m i n a n t ( 2 . 1 . 1 2 ) n o w t a k e s t h e f o r m

W = r

l

( ; l ; 1 ) r

; l ; 2

; l r

l ; 1

r

; l ; 1

= ; ( 2 l + 1 ) r

; 2

: ( 4 . 1 . 4 9 )

I n s e r t i n g W a n d ( 4 . 1 . 4 4 ) i n t o ( 2 . 1 . 1 4 ) t h e n a p a r t i c u l a r s o l u t i o n o f ( 4 . 1 . 4 8 ) i s

o b t a i n e d :

U

l m

( r ) = ( ; 1 )

m

r

l

1

Z

r

l m

( r

0

) r

0 ; l + 1

d r

0

+ r

; l ; 1

r

Z

0

l m

( r

0

) r

0 l + 2

d r

0

!

: ( 4 . 1 . 5 0 )


7/18/2019 C4029-Ch04



S i n c e t h e

l m

( r ) a r e u n k n o w n , M a t h e m a t i c a c a n g i v e o n l y a f o r m a l s o l u t i o n

o f t h e i n h o m o g e n e o u s e q u a t i o n ( 4 . 1 . 4 8 ) . T h e c o m m a n d

DSolve[R0 0 [r]+2*R0 [r]/r-(l*(l+1))*R[r]/rˆ2==B[r],R[r],r]

g i v e s

R r ] ! r

( ; 1 ; l )

C 1 ] + r

l

C 2 ] +

1

1 + 2 l

r

; 1 ; l

;

Z

R

K $ 1 1 8

K $ 1 1 7

2 + l

B K $ 1 1 7 ] d K $ 1 1 7 + r

1 + 2 l

Z

r

K $ 1 3 1

K $ 1 3 0

1 ; l

B K $ 1 3 0 ] d K $ 1 3 0

:

U s i n g a d d i t i o n f o r m u l a s 2 . 1 ] , 2 . 9 ] f o r t h e s p h e r i c a l f u n c t i o n s a n d t h e f o r m u l a

c o s = c o s # c o s #

0

+ s i n # s i n #

0

c o s ( ' ; '

0

)

o n e c a n m o d i f y ( 4 . 1 . 5 0 ) t o r e a d

U ( r # ' ) =

1

Z

0

Z

0

2

Z

0

r

0 2

( r

0

#

0

'

0

)

p

r

2

+ r

0 2

; 2 r r

0

c o s

d '

0

s i n #

0

d #

0

d r

0

w h e r e i s g i v e n b y ( 4 . 1 . 4 5 ) . I n s e r i e s e x p a n s i o n s o f t h e t y p e

p

l m

=

Z Z

r

l

P

m

l

( c o s # ) e

+ i m '

( r # ' ) s i n # d # d '

t h e e x p a n s i o n c o e c i e n t s p

l m

a r e c a l l e d m u l t i p o l e m o m e n t s . T h e r e a r e m a n y

p h y s i c a l a n d e n g i n e e r i n g p r o b l e m s i n w h i c h s u c h e x p a n s i o n s a p p e a r : b u r d e n o f

t h e o r g a n i s m b y - r a y s d u e t o t h e u r a n i u m c o n t e n t i n b r i c k s o r t h e d i s t r i b u t i o n

o f e l e c t r i c c h a r g e s o n a b o d y .

W e t r e a t e d a n i n h o m o g e n e o u s p r o b l e m t h a t h a d b e e n i n h o m o g e n e o u s b e -

c a u s e t h e d i e r e n t i a l e q u a t i o n ( 4 . 1 . 3 ) h a s b e e n i n h o m o g e n e o u s , b u t t h e b o u n d -

a r y c o n d i t i o n s ( 4 . 1 . 3 3 ) w e r e h o m o g e n e o u s . W e p r e s e n t e d t w o m e t h o d s t o s o l v e

s u c h a p r o b l e m . B u t a p r o b l e m o f a n h o m o g e n e o u s e q u a t i o n w i t h i n h o m o g e -

n e o u s b o u n d a r y c o n d i t i o n s i s i n h o m o g e n e o u s t o o ! D u e t o t h e t h e o r e m t h a t t h e

g e n e r a l s o l u t i o n U o f a n i n h o m o g e n e o u s l i n e a r e q u a t i o n U = f ( x y ) c o n s i s t s

o f t h e s u p e r p o s i t i o n o f t h e g e n e r a l s o l u t i o n W o f t h e h o m o g e n e o u s e q u a t i o n

p l u s a p a r t i c u l a r s o l u t i o n V o f t h e i n h o m o g e n e o u s e q u a t i o n w e c a n e x e c u t e

t h e f o l l o w i n g m e t h o d . T o s o l v e U = f ( x y ) o n e m a y w r i t e U = W + V ,

s o t h a t U = W + V = f ( x y ) , w h e r e V = f ( x y ) a n d W = 0 .

S i n c e w e d e m a n d h o m o g e n e o u s b o u n d a r y c o n d i t i o n s f o r U ( b o u n d a r y ) = 0 =

V ( b o u n d a r y ) + W ( b o u n d a r y ) , o n e h a s V ( b o u n d a r y ) = ; W ( b o u n d a r y ) . N o w

t h e f u n c t i o n V ( x y ) s h o u l d s a t i s f y t h e i n h o m o g e n e o u s b o u n d a r y c o n d i t i o n s


7/18/2019 C4029-Ch04



f o r W a n d a t t h e s a m e t i m e i t m u s t b e s u c h t h a t V = f ( x y ) . I t i s q u i t e

d i c u l t t o c r e a t e s u c h a f u n c t i o n V .

W e s t a r t w i t h t h e b o u n d a r y c o n d i t i o n s ( 4 . 1 . 1 5 ) t o ( 4 . 1 . 1 8 ) f o r V f o r t h e

r e c t a n g l e a s s h o w n i n F i g u r e 4 . 1 i n t h e s p e c i a l i z e d f o r m

V ( 0 y ) = V ( a y ) = V ( x b ) = 0 V ( x 0 ) = 1 : ( 4 . 1 . 5 1 )

A f u n c t i o n s a t i s f y i n g t h e s e c o n d i t i o n s h a s t o b e d i s c o v e r e d . A n e x a m p l e w o u l d

b e

V ( x y ) =

x

x + y

+

x ; a

x ; y ; a

; 2 +

x

a

+

( a + 2 b ) y

b ( a + b )

;

2 x y

a b

+

a ; x

a

2 y + a

y + a

;

( a + 2 b ) y

b ( a + b )

+

y

y + a

+

a y

b ( a + b )

a

a + b

+

x

a

;

x

x + b

;

x ; a

x ; a ; b

: ( 4 . 1 . 5 2 )

I f o n e c a l c u l a t e s f ( x y ) = V , t h e n U = f ( x y ) i s t h e i n h o m o g e n e o u s

e q u a t i o n f o r U . I t s s o l u t i o n s a t i s e s t h e h o m o g e n e o u s b o u n d a r y c o n d i t i o n

U ( b o u n d a r y ) = 0 .

A n o t h e r m e t h o d t o s o l v e t h e i n h o m o g e n e o u s P o i s s o n e q u a t i o n i s t h e u s e

o f t h e G r e e n f u n c t i o n , c o m p a r e ( 2 . 4 . 2 9 ) . T h e G r e e n f u n c t i o n f o r U =

a n d f o r t h e r e c t a n g l e m a y b e d e r i v e d a s f o l l o w s : i n s e r t i n g ( 4 . 1 . 3 0 ) , w h i c h

s a t i s e s t h e b o u n d a r y h o m o g e n e o u s c o n d i t i o n s , U ( 0 y ) = U ( a y ) = 0 i n t o

t h e P o i s s o n e q u a t i o n y i e l d s

U =

1

X

= 1

s i n

x

a

c

0 0

( y ) ;

2

2

a

2

c

( y )

= ( x y ) : ( 4 . 1 . 5 3 )

T h i s d e m o n s t r a t e s t h a t t h e e x p r e s s i o n i n b r a c k e t s p l a y s t h e r o l e o f F o u r i e r

e x p a n s i o n c o e c i e n t s f o r ( x y ) s o t h a t

c

0 0

( y ) ;

2

2

a

2

c

( y ) =

2

a

a

Z

0

( x y ) s i n

x

a

d x = g ( y ) : ( 4 . 1 . 5 4 )

T h e n e w f u n c t i o n g ( y ) i s t h e i n h o m o g e n e o u s t e r m o f t h e d i e r e n t i a l e q u a t i o n

f o r c

( y )

c

0 0

( y ) ;

2

2

a

2

c

( y ) = g

( y ) : ( 4 . 1 . 5 5 )

D u e t o t h e b o u n d a r y c o n d i t i o n s ( 4 . 1 . 3 3 ) t h e c

h a v e t o s a t i s f y

c

( y ) = U ( 0 y ) c

( 0 ) = c

( b ) = 0 = 1 2 : : : : ( 4 . 1 . 5 6 )


7/18/2019 C4029-Ch04



T h e n , a c c o r d i n g t o s e c t i o n 2 . 4 t h e s o l u t i o n h a s t h e f o r m

c

( y ) =

b

Z

0

G

( y ) g

( ) d ( 4 . 1 . 5 7 )

w h e r e n o w t h e G r e e n f u n c t i o n f o r t h e r e c t a n g l e c a n b e r e a d o

G

( y ) =

8

>

>

>

<

>

>

>

:

;

a

s i n h ( = a )

s i n h ( b = a )

s i n h

( b ; y )

a

0 y b

;

a

s i n h ( y = a )

s i n h ( b = a )

s i n h

( b ; )

a

0 y b

: ( 4 . 1 . 5 8 )

P l e a s e o b s e r v e o u r n o t a t i o n : ( x y ) i s t h e i n h o m o g e n e o u s t e r m i n t h e P o i s -

s o n e q u a t i o n d e t e r m i n e d b y t h e a c t u a l p h y s i c a l p r o b l e m , w h e r e a s f ( x y ) i s

t h e i n h o m o g e n e o u s t e r m c r e a t e d b y t h e h o m o g e n i z a t i o n o f t h e i n h o m o g e n e o u s

b o u n d a r y c o n d i t i o n s b e l o n g i n g t o t h e h o m o g e n e o u s d i e r e n t i a l e q u a t i o n .

T h e G r e e n f u n c t i o n f o r a r e c t a n g l e w i t h h o m o g e n e o u s b o u n d a r y c o n d i -

t i o n s , n a m e l y

G ( x y ) =

2

a

1

X

= 1

G

( y ) s i n

x

a

s i n

a

( 4 . 1 . 5 9 )

e n a b l e s o n e t o w r i t e t h e s o l u t i o n o f ( 4 . 1 . 3 ) , ( 4 . 1 . 3 3 ) i n t h e f o r m

U ( x y ) =

1

X

= 1

c

( y ) s i n

x

a

=

a

Z

0

b

Z

0

G ( x y ) f ( ) d d : ( 4 . 1 . 6 0 )

U p t o n o w w e h a v e d i s c u s s e d s i m p l e r e g u l a r r e c t a n g l e s . B u t h o w t o s o l v e

p r o b l e m s l i k e a s s h o w n i n F i g u r e s 4 . 2 a n d 4 . 3 ?

- -

x x

6 6

y y

F i g u r e 4 . 2 F i g u r e 4 . 3

R e c t a n g l e w i t h i n c i s i o n R e c t a n g l e w i t h e x c i s i o n


7/18/2019 C4029-Ch04



N e w m e t h o d s h o w e v e r a r e n e c e s s a r y t o s o l v e s u c h p r o b l e m s w i t h c o r n e r s o r

h o l e s ( a d m i s s i o n o f s i n g u l a r i t i e s , v a r i a t i o n a l m e t h o d s e t c . ) .

B o u n d a r y p r o b l e m s f o r a c i r c l e a r e e a s i e r t h a n t h e s i t u a t i o n s m e n t i o n e d .

A c c o r d i n g t o p r o b l e m 4 o f s e c t i o n 3 . 1 , t h e L a p l a c e e q u a t i o n i n c y l i n d r i c a l

c o o r d i n a t e s m a y b e s e p a r a t e d i n t o t w o o r d i n a r y d i e r e n t i a l e q u a t i o n s

R

0 0

( r ) +

1

r

R

0

( r ) ;

2

R

( r )

r

2

= 0 ( 4 . 1 . 6 1 )

( w e u s e d b = 0 , s i n c e o u r p r o b l e m i s i n d e p e n d e n t o n z a n d a !

2

) a n d

( ! )

0 0

( ' ) +

2

( ' ) = 0 : ( 4 . 1 . 6 2 )

T h e i n d e x i n d i c a t e s t h a t w e e x p e c t a s e t o f s o l u t i o n s s i n c e t h e o r i g i n a l

p a r t i a l d i e r e n t i a l e q u a t i o n i s l i n e a r . ( 4 . 1 . 6 1 ) i s a n E u l e r e q u a t i o n o f t y p e

( 2 . 2 . 3 3 ) h a v i n g t h e s o l u t i o n R ( r ) r

r

;

, s o t h a t

= = 1 2 : : :

f o r a s o l u t i o n v a l i d a r o u n d t h e w h o l e c i r c l e ( 0

'

2 ) . T h u s t h e g e n e r a l

s o l u t i o n o f t h e L a p l a c e e q u a t i o n r e a d s

U ( r ' ) =

a

0

2

+

1

X

= 1

c

r

( a

c o s ' + b

s i n ' ) : ( 4 . 1 . 6 3 )

T h e s o l u t i o n r

;

i s s i n g u l a r a t r = 0 a n d h a d b e e n e x c l u d e d . W e d e m a n d

U ( r = 0 ' ) = r e g u l a r , n o t s i n g u l a r . I f t h e b o u n d a r y c o n d i t i o n a l o n g t h e

c i r c u l a r l i n e d o e s n o t c o v e r t h e w h o l e c i r c u m f e r e n c e t h e n t h e

a r e n o l o n g e r

i n t e g e r s .

N o w w e n e e d b o u n d a r y c o n d i t i o n s o n t h e c i r c u l a r c i r c u m f e r e n c e l i k e

U ( r = R ' ) = f ( ' ) : ( 4 . 1 . 6 4 )

U s i n g t h i s c o n d i t i o n , ( 4 . 1 . 6 3 ) r e a d s

U ( R ' ) =

a

0

2

+

1

X

= 0

c

R

( a

c o s ' + b

s i n ' ) = f ( ' ) ( 4 . 1 . 6 5 )

a n d t h e F o u r i e r - c o e c i e n t s a r e

a

b

=

1

2

Z

0

f ( ' )

c o s '

s i n '

d ' a

0

=

1

2

Z

0

f ( ' ) d ' : ( 4 . 1 . 6 6 )

T o s e e t h e c o n v e r g e n c e o f t h e s u m

P

w e c a n d e m a n d c

R

= 1 c

= R

;

,

s o t h a t

U ( r ' ) =

a

0

2

+

1

X

= 1

r

R

( a

c o s ' + b

s i n ' ) : ( 4 . 1 . 6 7 )


7/18/2019 C4029-Ch04



U s i n g t h e a d d i t i o n t h e o r e m f o r s i n a n d c o s a n d ( 4 . 1 . 6 6 ) w e c a n m o d i f y ( 4 . 1 . 6 7 )

t o r e a d

U ( r ' ) =

1

2

2

4

2

Z

0

f ( ) d + 2

1

X

= 1

r

R

2

Z

0

f ( ) ( c o s ' c o s + s i n ' s i n ) d

3

5

=

1

2

2

Z

0

f ( )

"

1 + 2

1

X

= 1

r

R

c o s ( ' ; )

#

d : ( 4 . 1 . 6 8 )

T a k i n g a d v a n t a g e o f t h e f o r m u l a f o r t h e i n n i t e g e o m e t r i c s e r i e s w e w r i t e

1

1 ; e

i '

=

1

X

= 0

e

i '

=

1

X

= 0

( c o s ' + i s i n ' ) ( 4 . 1 . 6 9 )

w h e r e = r = R a n d

1

X

= 0

c o s ' = R e

1

1 ; e

i '

=

1 ; c o s '

1 ; 2 c o s ' +

2

: ( 4 . 1 . 7 0 )

C o m p a r i n g w i t h ( 2 . 3 . 9 ) o n e r e c o g n i z e s t h e l a s t r h s t e r m a s t h e g e n e r a t i n g

f u n c t i o n o f t h e p o l y n o m i a l s

. U s i n g

1 + 2

1

X

= 1

c o s ' =

1 ;

2

1 ; 2 c o s ' +

2

( 4 . 1 . 7 1 )

o n e g e t s t h e s o - c a l l e d P o i s s o n i n t e g r a l f o r a c i r c l e

U ( r ' ) =

1

2

2

Z

0

R

2

; r

2

R

2

; 2 R r c o s ( ' ; ) + r

2

f ( ) d ( 4 . 1 . 7 2 )

f r o m w h i c h t h e G r e e n f u n c t i o n f o r t h e c i r c l e c a n b e r e a d o .

P o i s s o n i n t e g r a l s a l s o e x i s t i n t h r e e d i m e n s i o n s f o r e l e c t r o s t a t i c o r g r a v -

i t a t i o n a l p r o b l e m s , s i n c e b o t h p o t e n t i a l s s a t i s f y t h e P o i s s o n e q u a t i o n . L e t

o r r

0

#

0

'

0

b e t h e c o o r d i n a t e s o f t h e s o u r c e p o i n t , w h e r e a s m a l l c h a r g e

e l e m e n t d i s l o c a t e d a n d x y z o r r # ' t h e c o o r d i n a t e s o f t h e e l d p o i n t ,

t h e n t h e i r r e l a t i v e d i s t a n c e d i s g i v e n b y

d =

p

( x ; )

2

+ ( y ; )

2

+ ( z ; )

2

o r

p

r

2

+ R

2

; 2 r R c o s ( ) ( 4 . 1 . 7 3 )


7/18/2019 C4029-Ch04


7/18/2019 C4029-Ch04



N o w t h e p o t e n t i a l U i n t h e e l d p o i n t o r i g i n a t i n g f r o m a c h a r g e d i s t r i b u t i o n

d i s a p p a r e n t l y g i v e n b y

U ( x y z ) =

Z

d

d =

Z Z Z

( )

p

( x ; )

2

+ ( y ; )

2

+ ( z ; )

2

d d d ( 4 . 1 . 8 2 )

w h i c h s a t i s e s U = .

B y t h e w a y , o n e c a n s h o w t h a t 1 = d , w h e r e d i s g i v e n b y ( 4 . 1 . 8 1 ) i s n o t h i n g

e l s e t h a n t h e G r e e n f u n c t i o n o f a p o i n t c h a r g e .

I f t h e s u r f a c e p o t e n t i a l U ( F ) i s g i v e n o n t h e s u r f a c e F o f a s p h e r e t h e n t h e

p o t e n t i a l i n t h e d i s t a n c e r f r o m t h e o r i g i n i s g i v e n b y t h e P o i s s o n i n t e g r a l

f o r t h e s p h e r e

U ( r ) = R

3

1 ;

r

2

R

2

2

Z

0

Z

0

U

F

( # ' ) s i n #

( R

2

+ r

2

; 2 r R c o s )

3 = 2

d # d ' : ( 4 . 1 . 8 3 )

T h i s i s t h e s o l u t i o n o f a D i r i c h l e t p r o b l e m . I n m a n y o t h e r e l e c t r o s t a t i c

p r o b l e m s , a s w e l l a s i n s t a t i o n a r y t h e r m a l p r o b l e m s , t h e m e t h o d s d e s c r i b e d

h e r e a r e u s e d .

U p t o n o w , w e h a v e t r e a t e d b o u n d a r y v a l u e p r o b l e m s o f a r e c t a n g l e , o f

a s p h e r e a n d o f a c i r c l e . N o w , t o n i s h t h i s s e c t i o n , w e d i s c u s s c y l i n d r i c a l

p r o b l e m s . F o r a n i n n i t e l y l o n g c y l i n d e r t h e p o t e n t i a l U ( r ' ) d e p e n d s o n

p o l a r c o o r d i n a t e s i n t h e x y p l a n e . S u c h p r o b l e m s w i l l b e t r e a t e d i n s e c t i o n

4 . 2 . I f t h e c y l i n d r i c a l p r o b l e m o f t h e L a p l a c e e q u a t i o n i s a x i a l l y s y m m e t r i c ,

t h e n t h e t w o - d i m e n s i o n a l p r o b l e m i s d e s c r i b e d b y a p o t e n t i a l U ( r z ) s a t i s f y i n g

@

2

U

@ r

2

+

1

r

@ U

@ r

+

@

2

U

@ z

2

= 0 : ( 4 . 1 . 8 4 )

S e p a r a t i o n i n t o t w o o r d i n a r y d i e r e n t i a l e q u a t i o n s a n d t h e b o u n d a r y c o n d i -

t i o n s

U ( R 0 ) = U

0

U ( R z ) = U

0

f o r 0 < z < l U ( R l ) = 0 ( 4 . 1 . 8 5 )

w h e r e R a n d l a r e r a d i u s a n d l e n g t h o f t h e c y l i n d e r r e s p e c t i v e l y , r e s u l t i n

U ( r z ) = U

0

"

1 ; 2

1

X

n = 1

J

0

(

n

r = R ) s i n h (

n

z = R )

n

J

1

(

n

) s i n h (

n

l = R )

#

( 4 . 1 . 8 6 )

w h e r e

n

a r e t h e p o s i t i v e r o o t s o f J

0

( ) = 0 a n d w h e r e

R

Z

0

r J

0

(

n

r = a ) = R

2

J

1

(

n

) =

n

:

J

0

a n d J

1

a r e B e s s e l f u n c t i o n s .


7/18/2019 C4029-Ch04


P r o b l e m s

1 . C a l c u l a t e d i v g r a d U i n c a r t e s i a n a n d s p h e r i c a l c o o r d i n a t e s .

2 . S o l v e t h e b o u n d a r y p r o b l e m d e n e d b y U = 0 a n d t h e i n h o m o g e n e o u s

c o n d i t i o n s ( 4 . 1 . 4 ) t o ( 4 . 1 . 7 ) . T h e r e s u l t s h o u l d b e

U

1

( x y ) =

2

a

1

X

k = 1

(

s i n h ( k y = a )

s i n h ( k b = a )

a

Z

0

f

2

( ) s i n ( k = a ) d s i n ( k x = a )

+

s i n h ( ( b ; y ) k = a )

s i n h ( k b = a )

a

Z

0

f

1

( ) s i n ( k = a ) d s i n ( k x = a )

)

+

2

b

1

X

k = 1

(

s i n h ( k x = b )

s i n h ( k a = b )

b

Z

0

g

2

( ) s i n ( k = b ) d s i n ( k y = b )

+

s i n h ( ( a ; x ) k = b )

s i n h ( k a = b )

b

Z

0

g

1

( ) s i n ( k = b ) d

s i n ( k y = b )

)

:

I s t h i s s o l u t i o n c o n t i n u o u s o n a c o r n e r w h e r e U 6= 0 ? ( N o ) I f t h e

b o u n d a r y c o n d i t i o n s a r e s i m p l i e d t o ( 4 . 1 . 8 ) t h e n t h e s o l u t i o n w i l l b e

U

2

( x y ) = U

1

( x y ) + U

0

( x y ) , w h e r e

U

0

( x y ) = c o s

y

2 b

s i n h ( a = 2 b )

s i n h

x

2 b

+

s i n h ( a = 2 b )

s i n h

2 b

( a ; x )

+ s i n

y

2 b

s i n h ( a = 2 b )

s i n h

x

2 b

+

s i n h ( a = 2 b )

s i n h

2 b

( a ; x )

a n d w h e r e t h e e x p a n s i o n c o e c i e n t s g i v e n b y t h e i n t e g r a l s n o w r e a d

a

Z

0

( f

2

( ) ; U

0

( b ) ) s i n ( k = a ) d

a

Z

0

( f

1

( ) ; U

0

( 0 ) ) s i n ( k = a ) d

b

Z

0

( g

2

( ) ; U

0

( a ) ) s i n ( k = b ) d

b

Z

0

( g

1

( ) ; U

0

( 0 ) ) s i n ( k = b ) d :

3 . S o l v e U

x x

+ U

y y

=

; 2 f o r t h e r e c t a n g l e d e n e d b y 0

x

a

; b = 2

y b = 2 w i t h t h e h o m o g e n e o u s b o u n d a r y c o n d i t i o n U ( b o u n d a r y ) = 0 .

U s e U = V + W V ( 0 y ) = 0 V ( a y ) = 0 V ( x y ) = A x

2

+ B x + C . T h e

f u n c t i o n W w i l l b e d e n e d b y W = 0 a n d t h e b o u n d a r y c o n d i t i o n s


7/18/2019 C4029-Ch04



W ( 0 ) = 0 W ( a y ) = 0 W ( x ; b = 2 ) = ; V ( x ) W ( x b = 2 ) = ; V ( x ) .

T h e s o l u t i o n s h o u l d b e

U ( x y ) = x ( a ; x ) ;

8 a

2

3

X

n

c o s h ( 2 n + 1 ) y = a ] s i n ( 2 n + 1 ) x = a ]

( 2 n + 1 )

3

c o s h ( 2 n + 1 ) b = 2 a ]

:

B y t h e w a y , t h e s i m i l a r e q u a t i o n U

x x

+ U

y y

= ; 4 h a s t h e s i m p l e s o l u t i o n

U ( x y ) = a

2

; ( x

2

+ y

2

) :

4 . S o l v e U

x x

+ U

y y

= ; x y f o r a c i r c u l a r d o m a i n . T h e c i r c l e h a s t h e c e n t e r a t

x = 0 y = 0 , t h e r a d i u s i s a a n d t h e h o m o g e n e o u s b o u n d a r y c o n d i t i o n

i s g i v e n b y U ( r = a ' ) = 0 . A g a i n o n e s h o u l d u s e U = V + W =

; ( 1 = 1 2 ) x y ( x

2

+ y

2

) + W W = 0 W ( r = a ' ) =

; V ( r = a ' ) a n d

t h e s o l u t i o n i s

U ( r ' ) =

;

r

3

2 4

s i n 2 ' +

a

4

4 8

+

Z

;

s i n 2 t

a

2

; r

2

a

2

; 2 a r c o s ( t ; ' ) + r

2

d t :

5 . S o l v e t h e b o u n d a r y v a l u e p r o b l e m U = 0 U ( 0 y ) = A U ( a y ) = A y

f o r 0

x

a 0

y

b @ u ( x y = 0 ) = @ y = 0 @ u ( x y = b ) = @ y = 0 :

T h e s o l u t i o n s h o u l d b e

U ( x y ) = A + A ( b ; 2 ) x = 2 a

;

4 A b

2

1

X

k = 0

1

( 2 k + 1 )

2

s i n h ( 2 k + 1 ) x = b ]

s i n h ( 2 k + 1 ) a = b ]

c o s

( 2 k + 1 ) y

b

:

6 . A s s u m e a c y l i n d e r o f h e i g h t l d e n e d b y i t s r a d i u s R 0 r R 0

z

l . S o l v e t h e s t e a d y h e a t c o n d u c t i o n p r o b l e m . D u e t o a s s u m e d

s y m m e t r y , @ = @ ' = 0 a n d t h e b o u n d a r y c o n d i t i o n s f o r t h e t e m p e r a t u r e

d i s t r i b u t i o n T ( r z )

T ( r 0 ) = f

0

( r ) T ( r l ) = f

1

( r ) T ( R z ) = ' ( z ) ( 4 . 1 . 8 7 )

t h e s o l u t i o n r e a d s

T ( r z ) =

2

l

1

X

n = 1

1

I

0

( n R = l )

(

I

0

n r

l

l

Z

0

' ( ) s i n

n

l

d

+

n

l

R

Z

0

(

; 1 )

n

f

1

( )

; f

0

( ) ] G

n

( r p ) d

)

s i n

n z

l

w h e r e t h e G r e e n f u n c t i o n G i s d e n e d b y


7/18/2019 C4029-Ch04



G

n

( r ) =

8

>

>

>

>

>

>

<

>

>

>

>

>

>

:

K

0

( n a = R ) I

0

( n r = l )

; I

0

( n a = l ) K

o

( n r = l ) ] I

0

( n = l ) r

K

0

( n a = l ) I

0

( n = l )

; I

0

( n a = l ) K

0

( n = l ) ] I

0

( n r = l )

r :

I

0

( x ) = J ( i x ) K

0

( x ) = Y

0

( i x ) a r e m o d i e d B e s s e l f u n c t i o n s , s e e ( 2 . 2 . 5 1 ) ,

a n d p r o b l e m 5 i n s e c t i o n 2 . 2 . S p e c i a l i z a t i o n o f t h e b o u n d a r y c o n d i t i o n s

( 4 . 1 . 8 7 ) f

0

( r ) = 0 f

1

( r ) = 0 ' ( z ) = T

0

= c o n s t y i e l d s

T ( r z ) =

4 T

0

1

X

n = 0

I

0

( 2 n + 1 ] r = l ) s i n ( 2 n + 1 ] z = l )

I

0

( 2 n + 1 ] R = l ) ( 2 n + 1 )

o r

T ( r z ) = 2 T

0

1

X

n = 1

J

0

(

n

r = R ) s i n h (

n

z = R )

n

J

1

(

n

) s i n h (

n

l = R )

i f f

0

( r ) = 0 f

1

( r ) = T

0

' ( z ) = 0 .

7 . A p l a n e c i r c u l a r d i s k o f r a d i u s R c a r r i e s a s u r f a c e c h a r g e Q t h a t i s

d i s t r i b u t e d a c c o r d i n g t o = e = ( 4 R

p

R

2

; r

2

) . C a l c u l a t e t h e p o t e n t i a l

U ( P ) i n t h e e l d p o i n t P , w h i c h i s l o c a t e d v e r t i c a l l y a b o v e t h e c e n t e r

o f t h e d i s k i n a d i s t a n c e p . S o l u t i o n :

U ( P ) = ;

Q

4 R

a r c s i n

; r

2

+ R

2

; p

2

p

R

4

+ p

4

; R

2

p

2

f o r 0 r R : ( 4 . 1 . 8 8 )

8 . A d i o d e c o n s i s t i n g o f t w o e l e c t r o d e s s i t u a t e d o n t h e x a x i s i n t h e d i s t a n c e

d f r o m e a c h o t h e r e m i t s a n e l e c t r o n c u r r e n t j =

p

2 e U ( x ) = m a t t h e

r s t e l e c t r o d e s i t u a t e d a t x = 0 . T h e p o t e n t i a l d i s t r i b u t i o n b e t w e e n

t h e e l e c t r o d e s i s t h e n d e s c r i b e d b y d

2

U = d x

2

= ; 4 a n d t h e b o u n d a r y

c o n d i t i o n U ( a ) = U

a

. S o l v e t h e o n e - d i m e n s i o n a l h o m o g e n e o u s e q u a t i o n

U

0 0

= c o n s t U

; 1 = 2

w i t h t h e s e t u p U = c o n s t x

n

r e s u l t i n g i n U ( x ) =

c o n s t j

2 = 3

x

4 = 3

. A f t e r i n s e r t i o n o f t h e b o u n d a r y c o n d i t i o n o n e o b t a i n s

t h e L a n g m u i r l a w o f p l a s m a p h y s i c s

j =

U

3 = 2

a

p

2 e

9 a

2

p

m

: ( 4 . 1 . 8 9 )

H e r e m a n d e a r e m a s s a n d c h a r g e o f a n e l e c t r o n , r e s p e c t i v e l y .

9 . S o l v e

1

r

@

@ r

r

@ U

@ r

+

@

2

U

@ z

2

+

1

r

2

@

2

U

@ '

2

= 0 : ( 4 . 1 . 9 0 )


7/18/2019 C4029-Ch04



S o l u t i o n : ( p a n d q a r e s e p a r a t i o n c o n s t a n t s )

U ( r z ' ) = ( A J

n

( q r ) + B N

n

( q r ) )

;

C e

q z

+ D e

; q z

( E s i n p ' + F c o s p ' ) ( 4 . 1 . 9 1 )

U ( r z ' ) = ( A J

n

( i q r ) + B N

n

( i q r ) ) ( C s i n q z + D c o s q z )

( E s i n p ' + F c o s p ' ) : ( 4 . 1 . 9 2 )

1 0 . A s s u m e a g i v e n p o t e n t i a l U ( x y z ) . T h e e l d

~

K b e l o n g i n g t o t h i s p o -

t e n t i a l i s g i v e n b y

~

K = r U : ( 4 . 1 . 9 3 )

C a l c u l a t e t h e d i e r e n t i a l e q u a t i o n s f o r t h e e l d l i n e s a t t a c h e d t o t h e

e l d v e c t o r

~

K . T h e y r e a d

d x

@ U = @ x

=

d y

@ U = @ y

=

d z

@ U = @ z

: ( 4 . 1 . 9 4 )

M a t h e m a t i c a h e l p s t o v i s u a l i z e t h e e l d l i n e s 2 ]

PlotVectorField3D[

{y/(3*z),-x/(2*z),1.},

{x,-1.,1.},{y, -1.,1.},{z,1.,3.}] ( 4 . 1 . 9 5 )

F i g u r e 4 . 4

V i s u a l i z a t i o n o f t h e e l d g i v e n b y ( 4 . 1 . 9 5 )


7/18/2019 C4029-Ch04



1 1 . A N e u m a n n b o u n d a r y c o n d i t i o n m a y r e a d i n s p h e r i c a l c o o r d i n a t e s

@ U

@ r

r = R

= f ( #

0

'

0

) ( 4 . 1 . 9 6 )

w h e r e n o w t h e b o u n d a r y v a l u e s a r e e x p a n d e d

f ( #

0

'

0

) =

1

X

n = 1

2 n + 1

4

2

Z

0

Z

0

f ( # ' ) P

n

( c o s ) s i n # d # d ' ( 4 . 1 . 9 7 )

w h e r e c o s i s g i v e n b y ( 4 . 1 . 7 9 ) . F o r r < R t h e s o l u t i o n i s

U ( r # ' ) =

1

X

n = 1

R

n

r

R

n

Y

n

( # ' ) ( 4 . 1 . 9 8 )

a n d f o r r > R

U ( r # ' ) = ; R

1

X

n = 0

1

n + 1

R

r

n + 1

Y

n

( # ' ) : ( 4 . 1 . 9 9 )

H e r e

Y

n

( # ' ) = P

n

( c o s # ) e x p ( i n ' ) ( 4 . 1 . 1 0 0 )

a r e c a l l e d s p h e r i c a l f u n c t i o n s .

1 2 . C a l c u l a t e t h e s u r f a c e t e m p e r a t u r e d i s t r i b u t i o n T ( r # ) o f a s p h e r e o f

r a d i u s R i n w h i c h t h e r e i s a c o n t i n u o u s h e a t p r o d u c t i o n q = c o n s t a n d

w h i c h l o s e s t h e e q u a l a m o u n t o f e n e r g y b y c o n d u c t i o n 4 . 8 ] . T h i s i n d u c e s

a b o u n d a r y c o n d i t i o n

@ T ( R # )

@ r

+ T ( R # ) = f ( # ) : ( 4 . 1 . 1 0 1 )

f ( # ) i s a r b i t r a r y , i s t h e h e a t t r a n s f e r c o e c i e n t w h i c h a p p e a r s i n t h e

N e w t o n c o o l i n g d o w n l a w

r T + ( T ; T

0

) = 0 : ( 4 . 1 . 1 0 2 )

T h e s o l u t i o n o f T = ; q i s g i v e n b y

T ( r # ) =

q

6

( R

2

; r

2

) +

q R

3

+

2

Z

0

f ( # ) s i n # d #

+

R

2

1

X

n = 1

2 n + 1

R = + n

r

R

n

P

n

( c o s # )

Z

0

f ( # ) P

n

( c o s # ) s i n # d # : ( 4 . 1 . 1 0 3 )

T h i s s o l u t i o n c a n a l s o b e f o u n d b y h o m o g e n i z a t i o n o f t h e b o u n d a r y

c o n d i t i o n .


7/18/2019 C4029-Ch04


4 . 2 C o n f o r m a l m a p p i n g i n t w o a n d t h r e e d i m e n s i o n s

T h e m e t h o d o f c o n f o r m a l m a p p i n g i s a c o n s e q u e n c e o f t h e C a u c h y - R i e m a n n

e q u a t i o n s . T h e s e e q u a t i o n s a r e b a s i c i n t h e t h e o r y o f f u n c t i o n s o f a c o m p l e x

v a r i a b l e z = x + i y . A f u n c t i o n f ( z ) = u ( x y ) + i v ( x y ) i s c a l l e d a n a l y t i c a t

t h e p o i n t z

0

= x

0

+ i y

0

i n t h e c o m p l e x G a u s s p l a n e , i f i t i s p o s s i b l e t o n d

a d e r i v a t i v e , s o t h a t t h e f u n c t i o n m a y b e d e v e l o p e d i n t o a T a y l o r s e r i e s . I f

f

0

( z

0

) d o e s n o t e x i s t t h e n t h e p o i n t z

0

i s c a l l e d s i n g u l a r ( s i n g u l a r i t y ) . T h e

d i e r e n t i a b i l i t y a t z

0

m u s t b e i n d e p e n d e n t f r o m t h e d i r e c t i o n f r o m w h i c h t h e

p o i n t z

0

i s a p p r o a c h e d , s e e F i g u r e 4 . 5 .

-

x

6

i y

-

x ! 0

y = 0

6

x = 0

y ! 0

F i g u r e 4 . 5

A p p r o a c h i n g a p o i n t z

0

i n t h e c o m p l e x p l a n e

C o m p l e x d e r i v a t i o n i s d e n e d b y

l i m

z ! 0

f

z

= l i m

x ! 0

u

x

+ i

v

x

=

@ u

@ x

+ i

@ v

@ x

( 4 . 2 . 1 )

a n d

l i m

z ! 0

f

z

= l i m

y ! 0

; i

u

y

+

v

y

= ; i

@ u

@ y

+

@ v

@ y

: ( 4 . 2 . 2 )

F o r i n d e p e n d e n c e f r o m d i r e c t i o n t h e C a u c h y - R i e m a n n e q u a t i o n s m u s t b e

v a l i d :

@ u

@ x

=

@ v

@ y

@ u

@ y

=

;

@ v

@ x

: ( 4 . 2 . 3 )

A s a n e x a m p l e = f ( z ) = z

2

= u ( x y ) + i v ( x y ) u ( x y ) = x

2

; y

2

v = 2 x y

s a t i s e s ( 4 . 2 . 3 ) a n d i s t h e r e f o r e a n a l y t i c , b u t f ( z ) = z

= x ; i v u = x v = ; y

i s n o t a n a l y t i c . D e r i v a t i o n o f ( 4 . 2 . 3 ) r e s u l t s i n

@

2

u

@ x

2

=

@

2

v

@ x @ y

@

2

u

@ y

2

= ;

@

2

v

@ x @ y

o r u = 0 v = 0 : ( 4 . 2 . 4 )


7/18/2019 C4029-Ch04


C o n f o r m a l m a p p i n g i n t w o a n d t h r e e d i m e n s i o n s 1 5 7

S o l u t i o n s o f t h e t w o L a p l a c e e q u a t i o n s ( 4 . 2 . 4 ) m a y b e w r i t t e n u ( x y ) =

c o n s t v ( x y ) = c o n s t . A n y c o m p l e x n u m b e r c a n b e t r a n s l a t e d i n t o a p o i n t

i n t h e ( x y ) p l a n e a n d v i c e v e r s a . I f t h i s i s d o n e f o r a s e t o f p o i n t s , o n e s p e a k s

o f m a p p i n g o f t h e x y p l a n e o n t o t h e u v p l a n e . M a t h e m a t i c a i s o f g r e a t h e l p

i n p r o d u c i n g s u c h p l o t s 4 ] . I t o e r s t h e c o m m a n d s

<<ComplexMapPlot.m; ComplexMapPlot[Sin[z],z,

RectangularGrid[{-1.,1.},{-1.,1.}]] ( 4 . 2 . 5 )

W h e n y o u s t a r t M a t h e m a t i c a , d o n ' t f o r g e t t o p r e s s s h i f t a n d e n t e r t o g e t h e r

w i t h y o u r r s t c o m m a n d . I n ( 4 . 2 . 5 ) t h e r e c t a n g u l a r g r i d i n t h e u v p l a n e w i l l

b e p l o t t e d o n t o t h e z ( x , y ) p l a n e . T h e m a p p i n g i s d o n e b y t h e f u n c t i o n z ] =

S i n z ] . T h e c o o r d i n a t e s i n b r a c k e t s d e t e r m i n e t h e l e f t u p p e r a n d l e f t l o w e r c o r -

n e r o f t h e g r i d . T h e n u m b e r o f h o r i z o n t a l l i n e s f r o m ; 1 t o + 1 a n d o f v e r t i c a l

l i n e s f r o m ; 1 t o + 1 h a s t h e d e f a u l t v a l u e ( 1 4 ) o f t h e p a c k a g e s . I f t h e r e a d e r

h a s n o a c c e s s t o t h e p a c k a g e ComplexMapPlot, t h e c o m m a n d ComplexMap

m a y h e l p . I t i s c o n t a i n e d i n A d d o n s / S t a n d a r d P a c k a g e s / G r a p h i c s .

T h e c o m m a n d

echo $Packages( 4 . 2 . 6 )

i n f o r m s y o u w h i c h p a c k a g e s h a v e b e e n l o a d e d . O n e m a y a l s o f o r c e t h e l o a d i n g

o f a p a c k a g e b y t h e M a t h e m a t i c a c o m m a n d

AppendTo[$Path, "/usr/local/mathematica..."] ( 4 . 2 . 7 )

d o w n t o t h e d i r e c t o r y , w h e r e t h e p a c k a g e i s l o c a t e d . A n o t h e r w a y i s g i v e n b y

t h e c o m m a n d

<<Graphics

ComplexMap

CartesianMap[Identity,{0,1.},{0,1.}]

w h i c h c r e a t e s t h e r e c t a n g u l a r g r i d i n t h e u v p l a n e , s e e F i g u r e 4 . 6

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

F i g u r e 4 . 6

R e c t a n g u l a r g r i d i n t h e u v p l a n e


7/18/2019 C4029-Ch04



T o s e e t h e m a p p i n g o f t h i s g r i d i n t o t h e x y p l a n e b y t h e f u n c t i o n z

2

, w e r s t

e x p a n d z

2

= ( x + i y )

2

= x

2

+ 2 i x y + y

2

, s o t h a t u ( x y ) = x

2

; y

2

v ( x y ) =

2 x y . N o w t h e c o m m a n d

RC=ContourPlot[u[x,y],{x,0,1.},{y,0,1.},

ContourShading->False]

y i e l d s

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

F i g u r e 4 . 7

P l o t o f u ( x y ) i n t h e x y p l a n e

N e x t w e p l o t v ( x y ) . T h e c o m m a n d 4 . 2 . 8 )

RV=ContourPlot[v[x,y],{x,0,1.},{y,0,1.},

ContourShading->False]

( w h a t h a p p e n s i f y o u d r o p , ContourShading->False? ) a n d F i g u r e 4 . 8

h a s b e e n c r e a t e d .

N o w w e w o u l d l i k e t o p u t t h e s e t w o p l o t s t o g e t h e r . T h i s i s d o n e v e r y s i m p l y

b y t h e c o m m a n d

Show[RC,RV]( 4 . 2 . 9 )

w h i c h r e s u l t s i n F i g u r e 4 . 9 .

I n ( 4 . 2 . 9 ) w e l e a r n e d a n e w M a t h e m a t i c a c o m m a n d h o w t o c o m b i n e t w o

F i g u r e s . I f o n e w a n t s t o s u p p r e s s t h e s h o w i n g o f a p i c t u r e o n e c a n w r i t e

Plot[...DisplayFunction->Identity]

T o c o m b i n e s e v e r a l s u c h p l o t s a n d t o s h o w t h e m t o g e t h e r o n e u s e s

Show[Pl1,Pl2...

DisplayFunction->$DisplayFunction].


7/18/2019 C4029-Ch04


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

F i g u r e 4 . 8

P l o t o f u ( x y ) i n t h e x y p l a n e

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

F i g u r e 4 . 9

C o m b i n e d p l o t z ( x y )


7/18/2019 C4029-Ch04



B u t w o u l d i t b e p o s s i b l e t o c r e a t e t h e l a s t g u r e b y a d i r e c t c o m m a n d ?

CartesianMap h a s w o r k e d t o p r o d u c e t h e g r i d o f F i g u r e 4 . 6 . T h e i d e a

t o m a p z

2

i n t o t h e c a r t e s i a n p l a n e u v w o u l d n o t s o l v e i t , w e h a v e t o u s e t h e

i n v e r s e f u n c t i o n

p

z . W e t y p e

gun[z_]=Sqrt[z];

CartesianMap[gun,{0,1.},{0,1.}] ( 4 . 2 . 1 0 )

0.2 0.4 0.6 0.8 1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

F i g u r e 4 . 1 0

C a r t e s i a n M a p o f

p

z

T h e d i s a d v a n t a g e o f c o n f o r m a l m a p p i n g i s t h e f a c t , t h a t t h e s a t i s f a c t i o n o f

g i v e n b o u n d a r y c o n d i t i o n s c a n n o t b e m e t : t h e f u n c t i o n ( z ) d e t e r m i n e s t h e

b o u n d a r y c o n d i t i o n . W e n e e d a m e t h o d t o n d t h e m a p p i n g f u n c t i o n a s a

c o n s e q u e n c e o f t h e g i v e n b o u n d a r y c o n d i t i o n s ! F o r b o u n d a r i e s c o n s i s t i n g o f

s t r a i g h t l i n e s t h e S c h w a r z - C h r i s t o f f e l t r a n s f o r m a t i o n i s h e l p f u l . I t m a p s

t h e r e a l a x i s o f t h e ( u v ) p l a n e i n t o a n y a r b i t r a r y p o l y g o n i n t h e z ( x y )

p l a n e , t h e u p p e r h a l f ( v > 0 ) o f t h e p l a n e i n t o t h e i n t e r i o r o f t h e p o l y g o n .

T h e i n v e r s e t r a n s f o r m a t i o n m a p s t h e z ( x y ) p l a n e p o l y g o n i n t o t h e u p p e r h a l f

o f t h e p l a n e . T o u n d e r s t a n d t h e s i t u a t i o n , w e c o n s i d e r t h e e x p r e s s i o n z

0

( )

d z

d

= A (

;

1

)

;

1

: ( 4 . 2 . 1 1 )

A i s a c o m p l e x c o n s t a n t a n d

1

i s r e a l , t h e p o i n t a o n t h e r e a l u a x i s . A g i v e n

c o m p l e x n u m b e r z = x + i y c a n b e r e p r e s e n t e d b y z = r e

i #

i n p o l a r c o o r d i n a t e s

r # . H e r e r =

p

x

2

+ y

2

i s t h e m a g n i t u d e o f z a n d # = a r c t a n ( y = x ) i s c a l l e d

a r g u m e n t a r g . N o w

a r g

d z

d

=

a r g A

;

1

a r g A

f o r

<

1

>

1

:

( 4 . 2 . 1 2 )

M o v i n g a l o n g t h e r e a l u a x i s , a r g ( ;

1

) = f o r <

1

a n d a r g ( ;

1

) = 0

f o r >

1

. T h u s d u r i n g m o v i n g a l o n g t h e u a x i s t h e a r g u m e n t o f d z = d


7/18/2019 C4029-Ch04



d i s c o n t i n u o u s l y j u m p s b y t h e a m o u n t

1

w h e n p a s s e s

1

. B a s e d o n t h e s e

f a c t s , t h e s e t u p f o r t h e t r a n s f o r m a t i o n i s g i v e n b y

d z

d

= A ( ;

1

)

;

1

( ;

2

)

;

2

: : : ( ;

n

)

;

n

: ( 4 . 2 . 1 3 )

F o r a c l o s e d p o l y g o n t h e c o n d i t i o n

n

X

i = 1

i

= 2 ( 4 . 2 . 1 4 )

m u s t b e v a l i d . I n t e g r a t i o n o f ( 4 . 2 . 1 3 ) y i e l d s

z = A

Z

( ;

1

)

;

1

( ;

2

)

;

2

: : : ( ;

n

)

;

n

d + B : ( 4 . 2 . 1 5 )

T h e c o m p l e x i n t e g r a t i o n c o n s t a n t s a l l o w t o r o t a t e ( A ) a n d t o t r a n s l a t e ( B )

t h e p o l y g o n i n t h e z p l a n e , r e s p e c t i v e l y .

W e h a v e d i s c u s s e d b o u n d a r y v a l u e p r o b l e m s o n r e c t a n g l e s i n e a r l i e r s e c -

t i o n s . W h a t d o e s n o w c o n f o r m a l m a p p i n g o e r ? L e t u s c o n s i d e r t h e m a p p i n g

o n t o a r e c t a n g l e i n t h e z p l a n e . W e c h o o s e

1

= ; 1 = k

2

= ; 1

3

= 1

4

=

1 = k 0 k 1 : ( 4 . 2 . 1 5 ) y i e l d s

z = z

0

+ A

Z

1

p

( 1 ;

2

) ( 1 ; k

2

2

)

d = z

0

+ A s n

; 1

( k ) : ( 4 . 2 . 1 6 )

T h e s o l u t i o n i s g i v e n b y t h e M a t h e m a t i c a c o m m a n d

Integrate[((1-xˆ2)*(1-kˆ2*xˆ2))ˆ(-1./2.),x]( 4 . 2 . 1 7 )

T h i s g i v e s t h e h y p e r g e o m e t r i c f u n c t i o n o f t w o v a r i a b l e s

( x ( 1 ; x

2

)

0 5

( 1 ; k

2

x

2

)

0 5

A p p e l l F 1 1 = 2 0 : 5 0 : 5 3 = 2 x

2

k

2

x

2

] )

( ( 1

; x

2

) ( 1

; k

2

x

2

) )

0 5

:

H o w ? R e m e m b e r : b e t w e e n 1 a n d 1 . , t h e r e i s a d i e r e n c e . R e p e a t

Integrate[((1-xˆ2)*(1-kˆ2*xˆ2))ˆ(-1/2),x] ( 4 . 2 . 1 8 )

T h i s y i e l d s

p

1 ; x

2

p

1 ; k

2

x

2

E l l i p t i c F A r c S i n x ] k

2

]

p

( 1 ; x

2

) ( 1 ; k

2

x

2

)

: ( 4 . 2 . 1 9 )

B u t c o u l d w e c a n c e l t h e s q u a r e r o o t s ?

MM=%%

Cancel[MM]

N o , t h i s d o e s n o t w o r k . T h e M a t h e m a t i c a b o o k e x p l a i n s t h a t c a n c e l w o r k s

f o r p o l y n o m i a l s b u t n o t f o r r o o t s .

W h a t d o w e k n o w a b o u t EllipticF? L e t ' s a s k M a t h e m a t i c a .

?EllipticF g i v e s

E l l i p t i c F p h i , m ] . T h i s i s t h e e l l i p t i c i n t e g r a l o f t h e r s t k i n d t h a t h a s b e e n

d i s c u s s e d i n ( 2 . 7 . 1 5 ) . C a n w e g o t o t h e


7/18/2019 C4029-Ch04



?JacobiSN[x,k]

T r y

InverseFunction[ArcSin]( 4 . 2 . 2 0 )

T h i s r e s u l t s i n S i n . O . K . N o w :

InverseFunction[EllipticF '

,m]]g i v e s

I n v e r s e F u n c t i o n E l l i p t i c F ' , m ] ]

A p p a r e n t l y t h i s i s t o o d i c u l t . T e s t t o p l o t t h e s e r e s u l t s u s i n g

CartesianMap

s e e m s t o f a i l t o o .

I f t h e b o u n d a r i e s a r e n o l o n g e r d e s c r i b a b l e b y s t r a i g h t l i n e s , t h e c h o i c e o f

s i n g u l a r i t i e s m a y h e l p . L e t u s i n v e s t i g a t e

( z ) =

q

2

l n

z + a

z ; a

: ( 4 . 2 . 2 1 )

T h i s f u n c t i o n d e s c r i b e s a d o u b l e s o u r c e ( e l e c t r i c c h a r g e s e t c . ) o f e q u a l i n -

t e n s i t y q , w h i c h a r e s i t u a t e d a t a a n d ; a . M a t h e m a t i c a h e l p s t o c l a r i f y t h e

s i t u a t i o n . L e t u s l e a r n b y t r i a l a n d e r r o r . W e t y p e

< < Graphics ComplexMap

a n d d o n o t f o r g e t t o h i t t h e t w o k e y s S h i f t a n d E n t e r t o g e t h e r , i f t h i s i s t h e

r s t c o m m a n d g i v e n i n t h e M a t h e m a t i c a w i n d o w . W e c o n t i n u e b y

echo $Packages

fz[z_]=Log[(z+1)/(z-1)];

CartesianMap[fz,{-1.,1.},{-1.,1.}]

D o n o t w r i t e

fz[z]b u t w r i t e o n l y

fzi n t h e l a s t c o m m a n d . O t h e r w i s e t h e

c a r t e s i a n m a p p i n g d o e s n o t w o r k . T h e c o m m a n d g i v e n a b o v e r e s u l t s i n a p l o t

s h o w i n g ( 4 . 2 . 2 1 ) . W e r e m e m b e r : w h e n w e w a n t t o p l o t ( 4 . 2 . 2 1 ) w e h a v e t o

u s e t h e i n v e r s e f u n c t i o n o f ( 4 . 2 . 2 1 ) . P e r h a p s M a t h e m a t i c a m a y h e l p ? T y p e

InverseFunction[fz[z]]b u t t h i s d o e s n o t h e l p . W e t r y

InverseFunction[fz] a n d w e g e t f z

; 1

S o w e n d t h e i n v e r s e f u n c t i o n o u r s e l v e s a n d w e t y p e

gz[z_]=(1 + Exp[z])/(Exp[z]-1);

CartesianMap[gz,{-Pi,Pi},{-Pi,Pi}]

w h i c h p r o d u c e s F i g u r e 4 . 1 1 . I n t h i s p i c t u r e w e s e e t h a t t h e l i n e s u ( x y ) =

c o n s t a n d v ( x y ) = c o n s t c r o s s e a c h o t h e r i n a n a n g l e o f 9 0

: t h e m a p p i n g

i s o r t h o g o n a l a n d i t p r e s e r v e s t h e a n g l e s . S u c h a n a l y t i c t r a n s f o r m a t i o n s a r e

c a l l e d c o n f o r m a l . T a k i n g t h e l i m i t a ! 0 q ! 1 a q ! R

2

o n e g e t s t h e

p o t e n t i a l o f a d i p o l e

( z ) =

R

2

z

: ( 4 . 2 . 2 2 )


7/18/2019 C4029-Ch04



T h i s e x p r e s s i o n r e a l i z e s a t r a n s f o r m a t i o n b y r e c i p r o c a l r a d i i i n t h e t w o - d i m e n -

s i o n a l z p l a n e : t h e d o m a i n o u t s i d e t h e c i r c l e w i t h r a d i u s R i s m a p p e d o n t o

t h e i n t e r i o r d o m a i n w i t h i n t h e c i r c l e r e p r e s e n t i n g t h e c u t o f a n i n n i t e l o n g

c y l i n d e r s t a n d i n g v e r t i c a l l y o n t h e z p l a n e . A d d i n g t h e t e r m V

0

z r e p r e s e n t i n g

a o w c o m i n g f r o m l e f t ( x < 0 ) p a r a l l e l t o t h e x a x i s , o n e o b t a i n s

( z ) = V

0

z + V

0

R

2

= z = V

0

( x + i y ) +

V

0

R

2

( x ; i y )

x

2

+ y

2

: ( 4 . 2 . 2 3 )

-4 -2 2 4

-2

-1

1

2

F i g u r e 4 . 1 1

C o n f o r m a l m a p p i n g o f ( 4 . 2 . 2 1 )

E q u a t i o n ( 4 . 2 . 2 3 ) d e s c r i b e s t h e e l d o f o w a r o u n d a c i r c u l a r c y l i n d e r . I n

t h e t w o p o i n t s x = R y = 0 w e n d s t a g n a t i o n p o i n t s , w h e r e t h e s t r e a m i n g

v e l o c i t y i s z e r o , s e e F i g u r e 4 . 1 2 . T h e b o u n d a r y c o n d i t i o n t h a t t h e v e l o c i t y

c o m p o n e n t v e r t i c a l t o a r i g i d w a l l x

2

+ y

2

= R

2

m u s t v a n i s h , i s s a t i s e d .

T o s e e t h i s w e p a y a t t e n t i o n t o t h e f a c t t h a t u ( x y ) = c o n s t d e s c r i b e s t h e

p o t e n t i a l o f t h e o w a n d t h e s t r e a m l i n e s a r e g i v e n b y v ( x y ) = c o n s t . R e a d i n g

o f r o m ( 4 . 2 . 2 3 ) w e g e t

v ( x y ) = V

0

y ;

V

0

R

2

y

x

2

+ y

2

= 0 u ( x y ) = V

0

x +

V

0

R

2

x

x

2

+ y

2

:

T h e s e f u n c t i o n s m a y b e p l o t t e d u s i n g t h e c o m m a n d ( 4 . 2 . 8 )

ContourPlot

o r b y c o n f o r m a l m a p p i n g u s i n g

CartesianMap

F o r t h i s g o a l w e c a l c u l a t e t h e i n v e r s e f u n c t i o n t o ( 4 . 2 . 2 3 ) .

I t r e a d s

giza1[z_]=z/(2*V0)+Sqrt[zˆ2/(4*V0ˆ2)-Rˆ2]; ( 4 . 2 . 2 4 )

I n o r d e r t o p l o t i t , w e t y p e

<<Graphics

ComplexMap

echo $Packages

t o r e c e i v e


7/18/2019 C4029-Ch04



G r a p h i c s C o m p l e x M a p e c h o ,

U t i l i t i e s F i l t e r O p t i o n s e c h o , G l o b a l e c h o , S y s t e m e c h o

a n d t h e n

Clear[R,V0,P1];R=1.;V0=1.;

P1=CartesianMap[giza1, {-6.,6.},{-6.,6.}]

E x e c u t i n g t h e l a s t c o m m a n d o n e r e c e i v e s a s t r a n g e p i c t u r e , j u s t t h e r i g h t s i d e

( x > 0 ) o f F i g u r e 4 . 1 2 . T h i s i s u n d e r s t a n d a b l e , b e c a u s e r o o t h a s t w o s i g n s .

W e t h u s t y p e

Clear[R,V0,giza2,P2];

R=1.;V0=1.;

giza2[z_]=z/(2*V0)-Sqrt[zˆ2/(4*V0ˆ2)-Rˆ2];

P2=CartesianMap[giza2,{-6.,6.},{-6.,6.}]

t o r e c e i v e t h e l e f t s i d e ( x < 0 ) o f F i g u r e 4 . 1 2 . T h i s i s n o w t h e t i m e t o c o m b i n e

t h e t w o p l o t s .

W e t y p e

Clear[PP];

PP=Show[P1,P2,PlotRange->All]

a n d n a l l y w e g e t F i g u r e 4 . 1 2 .

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

F i g u r e 4 . 1 2

F l o w a r o u n d a c y l i n d e r . M a p o f ( 4 . 2 . 2 3 )


7/18/2019 C4029-Ch04



T h r e e - d i m e n s i o n a l p r o b l e m s ( x y z ) e x h i b i t i n g a x i a l s y m m e t r y a r o u n d t h e z

a x i s , s e e F i g u r e 4 . 1 3 , m a y a l s o b e s o l v e d u s i n g c o n f o r m a l m a p p i n g . T o d o

t h i s a c o m p l e x p l a n e E ( z ) i s i n t r o d u c e d t h a t i s l o c a t e d p e r p e n d i c u l a r t o

t h e x y p l a n e o f a t h r e e - d i m e n s i o n a l c o o r d i n a t e s y s t e m x y z . L e t b e t h e

a n g l e b e t w e e n t h e x a x i s a n d t h e n e w a x i s . T o c r e a t e F i g u r e 4 . 1 3 w e g i v e

t h e c o m m a n d s

Clear[P1, G1]; P1 = {Text["x", {8., 0}],

Text["z", {0., 8.}], Text["xi", {6.5, 2.0}];

Text["E", {2., 2.5}], Text["alpha", {2.5, 0.45}],

Text["y", {4.8, 3.9}], Line[{{0, 0}, {8.0, 0.},

{0, 0}, {0, 8.0}, {0, 0}, {6.5, 2.0}}],

Line[{{0., 4.0}, {4.0, 5.4}, {4.0, 5.4},

{4.0, 1.2}}], Line[{{0., 0.}, {4.8, 3.9}}]};G1 = Graphics[P1];Show[G1]

x

z

xiE

alpha

y

F i g u r e 4 . 1 3

C o n f o r m a l m a p p i n g i n t h r e e - d i m e n s i o n a l s p a c e

L e t u s a s s u m e t h a t ' ( z i ) i s a s o l u t i o n o f '

z z

+ '

= 0 w i t h i n t h e p l a n e

E ( z ) . T h e n r o t a t i o n a r o u n d t h e z a x i s , d e s c r i b e d b y t h e a n g l e m a y b e

e e c t u a t e d b y t h e t r a n s f o r m a t i o n = x c o s + y s i n . T h i s d e l i v e r s a t h r e e -

d i m e n s i o n a l s o l u t i o n ' ( z

i x c o s + y s i n ] ) = ' ( x y z ) , w h i c h s o l v e s

'

x x

+ '

y y

+ '

z z

= 0 f o r i n s t a n c e ' ( x y z ) = ( z + i )

2

= z

2

; x

2

c o s

2

;

y

2

s i n

2

; 2 x y c o s s i n + i 2 z . S i n c e t h i s e x p r e s s i o n i s v a l i d f o r a r b i t r a r y

, w e m a y i n t e g r a t e o v e r r e s u l t i n g i n

' ( x y z ) =

2

Z

0

F ( z i ) f ( ) d =

2

Z

0

F ( z i x c o s + y s i n ] ) f ( ) d : ( 4 . 2 . 2 5 )

T h e t h r e e - d i m e n s i o n a l b o u n d a r y c o n d i t i o n s m a y b e s a t i s e d b y t h e a p p r o p r i -

a t e c h o i c e o f f ( ) 1 . 2 ] .


7/18/2019 C4029-Ch04


P r o b l e m s

1 . T h e f u n c t i o n ( 4 . 2 . 2 3 ) s h o u l d r e c e i v e a n a d d i t i o n a l t e r m r e a d i n g

( z ) = V

0

z + V

0

R

2

= z + ( i ; = 2 ) l n z : ( 4 . 2 . 2 6 )

T h e n e w t e r m d e s c r i b e s c i r c u l a t i o n a r o u n d t h e c y l i n d e r t h a t m e a n s ; =

H

~ w d ~s 6= 0 , b u t c u r l ~ w = 0 f o r t h e v e l o c i t y e l d ~ w o f t h e o w p a r a l l e l

t o t h e x a x i s . I t h a s b e e n s h o w n t h a t t h i s c i r c u l a t i o n p r o d u c e s a f o r c e

p e r p e n d i c u l a r t o t h e d i r e c t i o n o f t h e f r e e s t r e a m v e l o c i t y ~ w . T h i s f o r c e i s

c a l l e d l i f t . A c c o r d i n g t o t h e K u t t a - J o u k o w s k y f o r m u l a t h e l i f t f o r c e

i n t h e y d i r e c t i o n i s g i v e n b y

K

y

= ; w

x

( x = 1 y = 0 ) : ( 4 . 2 . 2 7 )

H e r e i s t h e d e n s i t y o f a i r . I n v e s t i g a t e a n d p l o t t h e e q u a t i o n ( 4 . 2 . 2 6 ) .

A J o u k o w s k i a e r o f o i l w i t h o u t c i r c u l a t i o n ( ; = 0 ) m a y b e d e s c r i b e d b y

ComplexMapPlot[z+1/z,z,{Circle[{0.25,0},1.25]}]

c o m p a r e 4 ] .

2 . I n v e s t i g a t e t h e s o - c a l l e d K a r m a n - T r e f f z p r o l e s d e s c r i b e d b y

=

; l a

+ l a

=

z ; a

z + a

k

:

H e r e a a n d l a r e g e o m e t r i c p a r a m e t e r s a n d ( 2 ; k ) d e s c r i b e s t h e a n g l e

b e t w e e n t h e u p p e r a n d l o w e r p a r t o f t h e a i r f o i l a t i t s e n d .

3 . T w o - d i m e n s i o n a l , t i m e - i n d e p e n d e n t h e a t c o n d u c t i o n p r o b l e m s s a t i s f y

t h e e q u a t i o n T ( x y ) = 0 , s o t h a t c o n f o r m a l m a p p i n g c a n b e u s e d . L e t

u s c o n s i d e r a t u b e o f r a d i u s R a n d t e m p e r a t u r e T

0

t h a t i s b u r i e d i n a

d e p t h h b e l o w t h e s u r f a c e o f t h e e a r t h ( x = 0 ) e x h i b i t i n g a t e m p e r a t u r e

T ( x = 0 y ) = 0 4 . 8 ] . T h i s i s a p r a c t i c a l p r o b l e m f o r a p l u m b e r t o a v o i d

t h e f r e e z i n g o f w a t e r t u b e s i n c o l d w i n t e r t i m e s . S h o w t h a t t h e p r o b l e m

c a n b e s o l v e d b y

=

z ; c

z + c

c =

p

h

2

; R

2

( 4 . 2 . 2 8 )

w h i c h m a p s t h e u p p e r h a l f p l a n e o n - t o t h e c i r c u l a r t u b e c r o s s s e c t i o n .

T h e s o l u t i o n r e a d s

T ( x y ) =

T

0

l n ( h + c ) = R ]

l n

(

p

( x

2

; c

2

+ y

2

)

2

+ 4 c

2

y

2

( x ; c )

2

+ y

2

)

: ( 4 . 2 . 2 9 )

4 . S o l v e p r o b l e m 3 b y u s i n g a s e p a r a t i o n s e t u p T ( x y ) = X ( x ) Y ( y ) .


7/18/2019 C4029-Ch04



5 . I f y o u h a v e t h e p a c k a g e 4 ] i n v e s t i g a t e a n d p l o t

ComplexMapPlot[Table[z+zˆn,{n,1,20}],z,

{Circle[{0,0},1]},PlotPoints->100];

6 . I n v e s t i g a t e a n d p l o t 2 ]

PolarMap[Identity,{0,1.},{0, 2. Pi}]

7 . T y p e

<<Graphics

PlotField

PlotVectorField[{Sin[x],Cos[y]},{x,0,Pi},{y,0,Pi}]

s e e 2 ]

8 . T y p e t h e f o l l o w i n g c o m m a n d s :

Clear[psi];v=1.;

psi[x_,y_]=-v*y+v*y/(xˆ2+yˆ2)

C1=ContourPlot[psi[x,y],{x,-4.,4.},{y,-4.,4.},

ContourShading->False,

ContourSmoothing->2,

PlotPoints->80,Contours->{0},

DisplayFunction->Identity];

C2=ContourPlot[psi[x,y],{x,-4.,4.},{y,-4.,4.},

ContourShading->False,

ContourSmoothing->2,

PlotPoints->80,Contours->30,

DisplayFunction->Identity];

Show[ContourGraphics[C1],ContourGraphics[C2],

DisplayFunction->$DisplayFunction]

T r y t o u n d e r s t a n d t h e c o m m a n d s a n d w h a t h a p p e n s . C o m p a r e t h e p l o t

w i t h F i g u r e 4 . 1 2 .

9 . T h e t e m p e r a t u r e o f h o t w a t e r p r o d u c e d b y a s o l a r c o l l e c t o r d e p e n d s o n

t h e b o u n d a r y c o n d i t i o n s t a k i n g i n t o a c c o u n t i r r a d i a t i o n , h e a t t r a n s f e r

a n d h e a t c o n d u c t i o n . A s o l a r h e a t c o l l e c t o r c o n s i s t i n g o f t w o s h e e t s o f

t h i c k n e s s d = 0 : 0 0 5 m , e x t e n d e d 2 a i n t h e x d i r e c t i o n a n d i n n i t e i n t h e

y d i r e c t i o n , i s s u b j e c t e d t o a s o l a r i r r a d i a t i o n I = 7 0 0 W / m

2

a n d s h o u l d

d e l i v e r T

R

= 3 0

C . F o r a h e a t t r a n s f e r c o e c i e n t = 4 W / m

2

K a n d

a h e a t c o n d u c t i v i t y = 4 0 0 W / m K c a l c u l a t e t h e t e m p e r a t u r e T

m a x

o f t h e w a t e r h e a t e d b y t h e c o l l e c t o r f o r a n e n v i r o n m e n t t e m p e r a t u r e

T

0

= 1 0

C . A s s u m e

T

0 0

( x ) ; ( T ; T

0

) = d + I = d = 0 T ( x = 0 ) = T ( 2 a ) = T

R

( d T = d x )

x = 0

=

0

T ( x ) = T

0

+ I = + ( T

R

; T

0

) ; I = ( ) c o s h

p

= d ( a ; x ) = c o s h

p

= d .


7/18/2019 C4029-Ch04


4 . 3 D ' A l e m b e r t w a v e e q u a t i o n a n d s t r i n g v i b r a t i o n s

E l a s t i c w a v e s i n b o u n d e d b o d i e s a r e r e e c t e d a t t h e b o d y s u r f a c e . T h e s u -

p e r p o s i t i o n o f t h e i n c i d e n t w a v e a n d t h e w a v e r e e c t e d b y t h e w a l l c r e a t e s a

s t a n d i n g w a v e . I t s w a v e l e n g t h d e p e n d s o n t h e d i m e n s i o n s o f t h e b o d y a n d

t h u s o n t h e b o u n d a r y c o n d i t i o n s . A s t r i n g i s a o n e - d i m e n s i o n a l c y l i n d r i c a l

b o d y o f v e r y s m a l l d i a m e t e r , s o t h a t t h e e x u r a l s t i n e s s m a y b e n e g l e c t e d .

T h e t h e o r y o f e l a s t i c i t y 1 . 2 ] d e r i v e s f o r l o n g i t u d i n a l o s c i l l a t i o n s u ( x t ) ( i n t h e

x d i r e c t i o n a l o n g t h e c y l i n d e r a x i s ) t h e o n e - d i m e n s i o n a l d ' A l e m b e r t w a v e

e q u a t i o n

@

2

u ( x t )

@ t

2

=

E

@

2

u ( x t )

@ x

2

( 4 . 3 . 1 )

a n d f o r t h e l a t e r a l ( t r a n s v e r s e ) o s c i l l a t i o n s v ( x t ) t h e e q u a t i o n

@

2

v ( x t )

@ t

2

=

p

@

2

v ( x t )

@ x

2

: ( 4 . 3 . 2 )

H e r e i s t h e ( c o n s t a n t ) l i n e d e n s i t y , E t h e m o d u l u s o f e l a s t i c i t y a n d p i s t h e

t e n s i o n a p p l i e d o n t h e s t r i n g . T h e s e e q u a t i o n s a r e b a s e d o n t h e a s s u m p t i o n o f

t h e v a l i d i t y o f t h e l i n e a r H o o k e l a w f o r s m a l l d e e c t i o n s . F o r l a r g e d e e c t i o n s

t h e n o n l i n e a r C a r r i e r e q u a t i o n ( 3 . 4 . 5 5 ) h a s t o b e u s e d . U s u a l l y t h e w a v e

p h a s e s p e e d s c

l

=

p

E = a n d c

t

=

p

p = a r e i n t r o d u c e d . U s i n g c f o r c

l

a n d

c

t

a s w e l l a s w i n s t e a d o f u a n d v r e s p e c t i v e l y , w e m a y w r i t e

1

c

2

@

2

w

@ t

2

=

@

2

w

@ x

2

: ( 4 . 3 . 3 )

T h i s e q u a t i o n h a s t h e s o l u t i o n ( 1 . 2 . 2 ) o r

w ( x t ) = f ( x + c t ) + g ( x

; c t ) : ( 4 . 3 . 4 )

T h i s g e n e r a l s o l u t i o n o f t h e p a r t i a l d i e r e n t i a l e q u a t i o n o f s e c o n d o r d e r i n x

h a s t o b e a d a p t e d t o t h e t w o b o u n d a r y c o n d i t i o n s o f a s t r i n g c l a m p e d o n b o t h

e n d s ( x = 0 x = l )

w ( 0 t ) = 0 w ( l t ) = 0 : ( 4 . 3 . 5 )

S i n c e t h e r e i s a l s o t h e s e c o n d i n d e p e n d e n t v a r i a b l e t i n t h e p a r t i a l e q u a t i o n

o f s e c o n d o r d e r w i t h r e s p e c t t o t , w e h a v e t o c h o o s e t w o i n i t i a l c o n d i t i o n s :

w ( x 0 ) = h ( x ) w

t

( x 0 ) = H ( x ) ( 4 . 3 . 6 )

w h e r e h a n d H a r e a r b i t r a r y b u t d i e r e n t i a b l e f u n c t i o n s . I n s e r t i n g t = 0 i n t o

( 4 . 3 . 4 ) o n e o b t a i n s

w ( x 0 ) = f ( x ) + g ( x ) = h ( x ) ( 4 . 3 . 7 )


7/18/2019 C4029-Ch04


D ' A l e m b e r t w a v e e q u a t i o n a n d s t r i n g v i b r a t i o n s 1 6 9

a n d

w

t

( x 0 ) = c ( f

0

( x ) ; g

0

( x ) ) = H ( x ) : ( 4 . 3 . 8 )

I n t e g r a t i o n o f t h i s e q u a t i o n a n d a d d i n g ( 4 . 3 . 7 ) r e s u l t s i n

f ( x ) =

1

2

h ( x ) +

1

2 c

Z

H ( x ) d x ( 4 . 3 . 9 )

w h i l e s u b t r a c t i o n g i v e s

g ( x ) =

1

2

h ( x )

;

1

2 c

Z

H ( x ) d x : ( 4 . 3 . 1 0 )

N o w t h e a r g u m e n t r e p l a c e m e n t ( s e e ( 3 . 3 . 1 1 ) ) x ! x c t d e l i v e r s t h e s o l u t i o n

w ( x t ) =

1

2

2

4

h ( x + c t ) + h ( x ; c t ) +

1

c

x + c t

Z

x ; c t

H ( x ) d x

3

5

( 4 . 3 . 1 1 )

s a t i s f y i n g t h e i n i t i a l c o n d i t i o n s ( 4 . 3 . 6 ) . T o t a k e i n t o a c c o u n t t h e b o u n d a r y

c o n d i t i o n s ( 4 . 3 . 5 ) w e i n s e r t x = 0 g i v i n g

2 w ( 0 t ) = h ( c t ) + h ( ; c t ) +

1

c

+ c t

Z

; c t

H ( x ) d x = 0 : ( 4 . 3 . 1 2 )

T o s a t i s f y ( 4 . 3 . 1 2 ) w e n o w h a v e t o m a k e a s s u m p t i o n s c o n c e r n i n g t h e a r b i t r a r y

f u n c t i o n s h a n d H : t h e y m u s t b e o d d ( a s y m m e t r i c ) , h ( ; c t ) = ; h ( c t ) H ( ; x )

= ; H ( x ) . B u t t h i s s a t i s e s o n l y t h e r s t b o u n d a r y c o n d i t i o n ( 4 . 3 . 5 ) . I n o r d e r

t o s a t i s f y w ( l t ) = 0 w e u s e t h e S t e f a n t r i c k t o e x p a n d t h e s o l u t i o n f u n c t i o n

o v e r t h e b o u n d a r i e s x < 0 x > l a n d h ( l ; x ) = ; h ( l + x ) H ( l ; x ) =

; H ( l + x ) .

A n o t h e r m e t h o d t o s o l v e ( 4 . 3 . 3 ) i s s e p a r a t i o n u s i n g w ( x t ) = X ( x )

T ( t ) ,

g i v i n g

T ( t ) = A s i n ! t + B c o s ! t ( 4 . 3 . 1 3 )

X ( x ) = C s i n

!

c

x + D c o s

!

c

x ( 4 . 3 . 1 4 )

w h e r e !

2

i s t h e s e p a r a t i o n c o n s t a n t . T o s a t i s f y ( 4 . 3 . 5 ) w e o b t a i n D = 0 , a n d

t h e a s y m m e t r i c e i g e n f u n c t i o n

X ( x ) = C s i n

n x

l

n = 1 2 : : : ( 4 . 3 . 1 5 )

T h e n t h e e i g e n v a l u e i s g i v e n b y

!

n

=

n c

l

( 4 . 3 . 1 6 )


7/18/2019 C4029-Ch04



a n d w i t h C = 1 t h e s o l u t i o n t o ( 4 . 3 . 3 ) r e a d s

w ( x t ) =

1

X

n = 0

( A

n

s i n !

n

t + B

n

c o s !

n

t ) s i n

n x

l

: ( 4 . 3 . 1 7 )

T o s a t i s f y t h e i n i t i a l c o n d i t i o n s w e h a v e t o e x p a n d h ( x ) H ( x ) w i t h r e s p e c t t o

t h e e i g e n f u n c t i o n s :

t = 0 w ( x 0 ) = h ( x ) =

1

X

n = 0

B

n

s i n

n x

l

( 4 . 3 . 1 8 )

t = 0 w

t

( x 0 ) = H ( x ) =

1

X

n = 0

A

n

!

n

s i n

n x

l

: ( 4 . 3 . 1 9 )

F o r g i v e n i n i t i a l d e e c t i o n d e n e d b y h ( x ) a n d H ( x ) , t h e s t r i n g o s c i l l a t i o n s

a r e d e s c r i b e d b y ( 4 . 3 . 1 7 ) . B u t h o w d o e s t h e s o l u t i o n r e a d i f a n e x t e r i o r f o r c e

K ( x t ) e x c i t e s t h e s t r i n g c o n t i n u o u s l y ? A p p a r e n t l y ( 4 . 3 . 3 ) h a s t o b e m o d i e d :

@

2

w

@ x

2

;

1

c

2

@

2

w

@ t

2

= K ( x t ) = K

0

( x ) s i n ! t =

X

n

K

n

s i n

n x

l

s i n ! t ( 4 . 3 . 2 0 )

w h e r e

K

n

=

2

l

l

Z

0

K

0

( x ) s i n

n x

l

d x : ( 4 . 3 . 2 1 )

S i n c e w e k n o w t h a t t h e s o l u t i o n o f ( 4 . 3 . 2 0 ) i s g i v e n a s a s u m o f t h e g e n e r a l

s o l u t i o n s o f t h e h o m o g e n o u s e q u a t i o n ( 4 . 3 . 3 ) a n d a p a r t i c u l a r s o l u t i o n o f t h e

i n h o m o g e n e o u s e q u a t i o n ( 4 . 3 . 2 0 ) , w e m a y w r i t e f o r t h e l a t t e r U ( x ) s i n ! t ,

w h e r e ! i s t h e f r e q u e n c y o f t h e e x t e r i o r f o r c e . W e t h u s h a v e

U

0 0

( x ) +

!

2

c

2

U ( x ) = K

0

( x ) ( 4 . 3 . 2 2 )

X

n

s i n

n x

l

; A

n

2

n

2

l

2

+

!

2

c

2

A

n

+ K

n

= 0 o r

A

n

=

c

2

K

n

!

2

n

; !

2

: ( 4 . 3 . 2 3 )

R e s o n a n c e a n d v e r y l a r g e o r i n n i t e a m p l i t u d e s A

n

o c c u r f o r ! !

n

, i f

t h e f r e q u e n c y ! o f t h e e x t e r i o r f o r c e i s n e a r l y ( o r e x a c t l y ) e q u a l t o o n e o f

t h e e i g e n f r e q u e n c i e s !

n

o f t h e s t r i n g . I f a m a r c h i n g f o r m a t i o n o f s o l d i e r s

c r o s s e s a b r i d g e , o n e o f t h e e i g e n f r e q u e n c i e s o f t h e b r i d g e m a y b e e x c i t e d

a n d t h e b r i d g e c o u l d c o l l a p s e . T o m a r c h i n s t e p w h e n c r o s s i n g a b r i d g e i s

s t r i c t l y f o r b i d d e n f o r m a t h e m a t i c a l r e a s o n s . E v e n a n a r m y h a s t o a c c e p t


7/18/2019 C4029-Ch04



p h y s i c a l f a c t s . B u t w h a t a b o u t d a m p i n g t h e o s c i l l a t i o n s ? W e w i l l d i s c u s s t h i s

p o s s i b i l i t y i n t h e p r o b l e m s a t t a c h e d t o t h i s s e c t i o n . O n t h e o t h e r h a n d , a r e

t h e r e t i m e - i n d e p e n d e n t s o l u t i o n s o f ( 4 . 3 . 3 ) ? S u r e , i n n e w d e n o t a t i o n w ( x ) !

G ( x ) o n e h a s G

0 0

( x ) = 0 a n d

G ( x ) = a x + b : ( 4 . 3 . 2 4 )

I n o r d e r t o b e a b l e t o s a t i s f y t h e b o u n d a r y c o n d i t i o n s ( 4 . 3 . 5 ) , w e h a v e t o

a c c e p t a d i s c o n t i n u o u s s o l u t i o n

G ( x ) =

G

0

x = f o r 0 x

G

0

( x ; l ) = ( ; l ) f o r x l :

( 4 . 3 . 2 5 )

T h i s s t r a n g e s o l u t i o n d e s c r i b e s a s t r i n g t h a t i s c o n t i n u o u s l y p u l l e d o u t a t

x = b y a f o r c e o f s t r e n g t h G

o

. T h i s p o i n t f o r c e h a s a l l a t t r i b u t e s o f a

G r e e n f u n c t i o n

1 . s y m m e t r y G ( x ) = G ( x ) ,

2 . s a t i s e s t h e p e r t i n e n t h o m o g e n e o u s e q u a t i o n ,

3 . i t s r s t d e r i v a t i v e s a t i s e s t h e d i s c o n t i n u i t y c o n d i t i o n ( 2 . 4 . 3 0 ) a n d G

0 0

=

( x ; ) ( D i r a c D e l t a ) .

B u t i f G ( x ) d e s c r i b e s a l o c a l p o i n t f o r c e a t , t h e n a d i s t r i b u t e d f o r c e o f

s t r e n g t h f ( ) a n d i n t e g r a t i o n c o u l d c o n s t r u c t a n o t h e r s o l u t i o n b y

w ( x ) =

l

Z

0

G ( x ) f ( ) d : ( 4 . 3 . 2 6 )

F o r f ( x ) = !

2

w ( x ) = c

2

, ( 4 . 3 . 2 6 ) c r e a t e s t h e i n t e g r a l e q u a t i o n f o r t h e s t r i n g

w ( x ) =

!

2

c

2

l

Z

0

G ( x ) w ( ) d : ( 4 . 3 . 2 7 )

H e r e t h e G r e e n f u n c t i o n i s t h e k e r n e l o f t h e i n t e g r a l e q u a t i o n , w h i c h c o n t a i n s

t h e b o u n d a r y c o n d i t i o n s w i t h i n t h e i n t e g r a t i o n l i m i t s . T h e i n t e g r a l e q u a t i o n

h a s t h e a d v a n t a g e t h a t g e n e r a l t h e o r e m s m a y b e d e d u c e d q u i t e e a s i l y . L e t

= !

2

= c

2

= !

2

= c

2

b e t h e e i g e n v a l u e s b e l o n g i n g t o w

( x ) a n d w

( x )

r e s p e c t i v e l y , t h e n o n e m a y w r i t e d o w n t h e i d e n t i t y

l

Z

0

w

( ) w

( ) d =

l

Z

0

w

( ) d

l

Z

0

G ( x ) w

( x ) d x : ( 4 . 3 . 2 8 )

E x c h a n g i n g a n d , w r i t i n g a s e c o n d i d e n t i t y a n d s u b t r a c t i n g t h e m o n e

o b t a i n s

(

;

)

l

Z

0

w

( ) w

( ) d = 0 ( 4 . 3 . 2 9 )


7/18/2019 C4029-Ch04



w h i c h s t a t e s t h a t t h e e i g e n f u n c t i o n s a r e o r t h o g o n a l . I t i s a l s o p o s s i b l e t o p r o v e

t h a t t h e e i g e n v a l u e s a r e a l w a y s r e a l o r t h a t t h e k e r n e l m a y b e e x p a n d e d a s a

b i l i n e a r f o r m o f t h e e i g e n f u n c t i o n s :

G ( x ) =

X

w

( x ) w

( )

=

2

l

X

s i n

x

l

s i n

l

2

2

= l

2

: ( 4 . 3 . 3 0 )

N o w t h e q u e s t i o n a r i s e s i f a s i m i l a r s o l u t i o n l i k e ( 4 . 3 . 2 6 ) e x i s t s f o r t h e t i m e -

d e p e n d e n t s o l u t i o n , t o o ? W e w i l l d i s c u s s t h i s q u e s t i o n i n p r o b l e m 5 .

W h a t h a p p e n s w i t h o u r s o l u t i o n s i n t h e l i m i t l

! 1 ? O n e m a y e x p e c t a

c o n t i n u o u s s p e c t r u m o f e i g e n v a l u e s a s w i l l b e s h o w n w h e n t r e a t i n g m e m b r a n e s

i n t h e n e x t s e c t i o n .

P r o b l e m s

1 . I n v e s t i g a t e t h e d a m p i n g o f r e s o n a n c e o s c i l l a t i o n s . T h e p e r t i n e n t e q u a -

t i o n i s a m o d i c a t i o n o f ( 4 . 3 . 3 ) a n d m a y r e a d

1

c

2

@

2

w

2

@ t

2

=

@

2

w

@ x

2

; k

@ w

@ t

: ( 4 . 3 . 3 1 )

A s e t u p

w ( x t ) = e x p ( ; a t ) c o s ! t s i n ( x = l ) ( 4 . 3 . 3 2 )

s a t i s e s t h e b o u n d a r y c o n d i t i o n s a t x = 0 a n d x = l a n d s e e m s t o b e

u s e f u l . L e t u s s e e i f M a t h e m a t i c a c o u l d h e l p .

T y p e

Clear[w];w[x,t]=Exp[-a*t]*Cos[!

*t]*

Sin[Pi*x/l]

Expand[(D[w[x,t],{t,2}]/cˆ2-D[w[x,t],{x,2}]+

k*D[w[x,t],t])/w[x,t]]

T h e r e s u l t d o e s n o t h e l p v e r y m u c h . I t r e a d s

; k a +

a

2

c

2

;

!

2

c

2

; k ! T a n t ! ] +

2 a ! T a n t ! ]

c

2

+

2

l

2

N o w t r y a n o t h e r w a y

S1=k*D[w[x,t],t];

S2=D[w[x,t],{t,2}]/cˆ2;

S3=-D[w[x,t],{x,2}]

w h i c h r e p r e s e n t s e x a c t l y t h e p a r t i a l d i e r e n t i a l e q u a t i o n ( 4 . 3 . 3 1 ) . T h e n

g i v e

Simplify[S1+S2+S3]


7/18/2019 C4029-Ch04



t o r e c e i v e

e

; a t

S i n

x

l

]

( (

a

2

l

2

; a c

2

k l

2

+ c

2

2

; l

2

!

2

)

C o s t ! ] +

(

2 a ; c

2

k

)

l

2

! S i n t ! ]

)

c

2

l

2

F i n d t h a t e x p r e s s i o n f o r a t h a t s a t i s e s t h e e q u a t i o n ( 4 . 3 . 3 1 ) .

2 . T h e e q u a t i o n d e s c r i b i n g d a m p e d o s c i l l a t i o n s e x c i t e d b y a n e x t e r i o r p e -

r i o d i c f o r c e r e a d s

1

c

2

@

2

w

@ t

2

=

@

2

w

@ x

2

; k

@ w

@ t

+ f ( x ) c o s ! t : ( 4 . 3 . 3 3 )

T r y a s o l u t i o n u s i n g t h e s e t u p

wd[x,t]=g[x]*Cos[!

*t]+h[x]*Sin[!

*t];

Simplify[D[wd[x,t],{t,2}]/cˆ2-D[wd[x,t],{x,2}]+k*D[wd[x,t],t]-f[x]*Cos[! *t]]

D o e s t h e r e s u l t o e r t w o o r d i n a r y d i e r e n t i a l e q u a t i o n s f o r g ( x ) a n d

h ( x ) ?

3 . P r o v e t h a t ( 4 . 3 . 2 6 ) s o l v e s w

0 0

( x ) + f ( x ) = 0 . R e c a l l : G

0 0

= ( x ; ) .

4 . A s s u m e a n i n h o m o g e n e o u s s t r i n g w i t h l o c a l l y v a r y i n g t e n s i o n p ( x ) a n d

l i n e d e n s i t y ( x ) s o t h a t t h e w a v e e q u a t i o n r e a d s

@

@ x

p ( x )

@ u

@ x

= ( x )

@

2

u

@ t

2

: ( 4 . 3 . 3 4 )

T r y a s e p a r a t i o n s e t u p 3 . 1 3 ] u ( x t ) = T ( t ) X ( x ) a n d u s e t h e b o u n d a r y

c o n d i t i o n s :

u ( 0 t ) +

@ u ( 0 t )

@ x

= 0

u ( l t ) +

@ u ( l t )

@ x

= 0 ( 4 . 3 . 3 5 )

t o g e t h e r w i t h t h e i n i t i a l c o n d i t i o n s

u ( x 0 ) = '

0

( x )

@ u ( x 0 )

@ t

= '

1

( x ) 0 x l : ( 4 . 3 . 3 6 )

T h e r e s u l t s h o u l d r e a d

T

n

( t ) = A

n

c o s t + B

n

s i n t

u ( x t ) =

X

n

( A

n

c o s

n

t + B

n

s i n

n

t ) X

n

( x )

w h e r e

n

a r e t h e s e p a r a t i o n c o n s t a n t s

u ( x 0 ) = '

0

( x ) =

P

n

A

n

X

n

( x )

@ u ( x 0 ) = @ t = '

1

( x ) =

P

n

n

B

n

X

n

( x )


7/18/2019 C4029-Ch04



A

n

=

Z

l

0

( x ) '

0

( x ) X

n

( x ) d x

B

n

= ( 1 =

n

)

Z

l

0

( x ) '

1

( x ) X

n

( x ) d x :

5 . A s t r i n g o f l e n g t h l i s c l a m p e d o n b o t h e n d s a n d a t t i m e t = 0 i s p l u c k e d

a t x = s u c h t h a t u ( 0 ) = G

0

. T h e r e i s n o i n i t i a l v e l o c i t y , u

t

( 0 ) = 0 .

S h o w t h a t t h e s o l u t i o n o f t h e s t r i n g e q u a t i o n i s g i v e n b y

u ( x t ) =

2 G

0

2

l

2

( l ; )

1

X

n = 1

1

n

2

s i n

n

l

s i n

n x

l

c o s

n c

l

t : ( 4 . 3 . 3 7 )

6 . A s t r i n g o f l e n g t h 2 l i s c l a m p e d a t x = ; l a n d x = + l . A t t i m e t = 0

i t i s p l u c k e d a t x = 0 s o t h a t u ( x 0 ) = 0 u

t

( x 0 ) = v

0

, j x j < " a n d

u

t

( x 0 ) = 0 , " <

jx

j l . A f t e r t a k i n g t h e l i m i t "

! 0 t h e s o l u t i o n

s h o u l d b e 4 . 8 ]

u ( x t ) =

4 v

0

c

1

X

n = 0

1

2 n + 1

c o s

2 n + 1

2 l

x

s i n

2 n + 1

2 l

c t

: ( 4 . 3 . 3 8 )

4 . 4 H e l m h o l t z e q u a t i o n a n d m e m b r a n e v i b r a t i o n s

I n p h y s i c s a n d e n g i n e e r i n g a m e m b r a n e i s d e n e d a s a t w o - d i m e n s i o n a l p l a n e ,

v e r y t h i n b o d y , n e a r l y a s k i n , w i t h n e g l i g i b l e b e n d i n g r i g i t y . F o r s u c h a b o d y

t h e t h e o r y o f e l a s t i c i t y d e r i v e s r s t a t w o - d i m e n s i o n a l w a v e e q u a t i o n t h a t

b e c o m e s a t w o - d i m e n s i o n a l H e l m h o l t z e q u a t i o n a f t e r s e p a r a t i o n o f t h e t i m e

d e p e n d e n t t e r m s i n ! t . I t r e a d s

w ( x y ) + !

2

= c

2

w ( x y ) = 0 : ( 4 . 4 . 1 )

F o r a c l a m p e d r e c t a n g u l a r m e m b r a n e a b t h e b o u n d a r y c o n d i t i o n s r e a d

w ( x = a = 2 y ) = 0

w ( x y = b = 2 ) = 0

( 4 . 4 . 2 )

a n d t h e s o l u t i o n o f t h e w a v e e q u a t i o n i s t h e n

u ( x y t ) =

X

m n

A

m n

c o s

n x

a

c o s

m y

b

s i n ( !

m n

t +

m n

) ( 4 . 4 . 3 )

w h e r e n m = 1 3 : : : : T h e p h a s e s p e e d c o f t h e t r a n s v e r s a l o s c i l l a t i o n u ( x y t )

i s g i v e n b y

p

p = a n d t h e e i g e n f r e q u e n c i e s , d e t e r m i n e d b y ( 4 . 4 . 2 ) , a r e

!

2

m n

= c

2

n

2

2

a

2

+

m

2

2

b

2

: ( 4 . 4 . 4 )


7/18/2019 C4029-Ch04


H e l m h o l t z e q u a t i o n a n d m e m b r a n e v i b r a t i o n s 1 7 5

T h e p a r t i a l a m p l i t u d e s A

m n

h a v e t o b e d e t e r m i n e d f r o m t h e i n i t i a l c o n d i -

t i o n s u ( x y 0 ) = g ( x y ) a n d u

t

( x y 0 ) = 0 . F o r a d i s l o c a t e d r e c t a n g l e ( l e f t

l o w e r c o r n e r i s s i t u a t e d i n t h e o r i g i n x = 0 y = 0 w i t h m o d i e d b o u n d a r y

c o n d i t i o n s ) o n e o b t a i n s 4 . 8 ]

u ( x y t ) =

1

X

m n = 0

A

m n

c o s

2 n + 1

2 a

x

c o s

2 m + 1

2 b

y

c o s ( !

m n

t )

A

m n

=

4

a b

a

Z

0

b

Z

0

g ( x y ) c o s

2 n + 1

2 a

x

c o s

2 m + 1

2 b

y

d x d y

!

2

m n

= c

2

2

"

2 n + 1

2 a

2

+

2 m + 1

2 b

2

#

:

I n t h e c a s e o f i n h o m o g e n e o u s b o u n d a r y c o n d i t i o n s o n e m a y e i t h e r h o m o g e n i z e

t h e c o n d i t i o n s o r o n e r s t s o l v e s t h e b o u n d a r y p r o b l e m s f o r ( 4 . 4 . 1 ) a n d a

s u p e r p o s i t i o n o f t h e p a r t i c u l a r s o l u t i o n s w i t h u n k n o w n a m p l i t u d e s i s u s e d t o

s o l v e t h e i n i t i a l v a l u e p r o b l e m .

I f t h e r e c t a n g l e i s s p e c i a l i z e d i n t o a s q u a r e ( a = b ) e q u a t i o n ( 4 . 4 . 4 ) t a k e s

t h e f o r m

!

2

m n

= c

2

2

( n

2

+ m

2

) = a

2

: ( 4 . 4 . 5 )

A p p a r e n t l y t h e s e e i g e n v a l u e s a r e m u l t i p l e t w o e i g e n f u n c t i o n s b e l o n g t o t h e

s a m e e i g e n v a l u e ( d e g e n e r a t e e i g e n v a l u e p r o b l e m ) . L i n e s ( x ) c h a r a c t e r i z e d b y

u ( x ) = 0 a r e c a l l e d n o d a l l i n e s . A l o n g s u c h n o d a l l i n e s a m e m b r a n e m a y

b e c u t i n s u b d o m a i n s . T h e s e s m a l l e r s u b d o m a i n s a r e a s s o c i a t e d w i t h h i g h e r

f r e q u e n c i e s . T h e l o w e s t e i g e n v a l u e i s a s s o c i a t e d w i t h t h e u n d i v i d e d w h o l e

d o m a i n ( C o u r a n t ' s t h e o r e m ) .

D e g e n e r a t e e i g e n v a l u e p r o b l e m s o e r t h e p o s s i b i l i t y t o n d n o d a l l i n e s t h a t

a r e n o t i d e n t i c a l w i t h c o o r d i n a t e c u r v e s . T h u s t h e t w o d e g e n e r a t e e i g e n f u n c -

t i o n s o f a s q u a r e m e m b r a n e

w ( x y ) = s i n

2 x

a

s i n

y

a

+ s i n

x

a

s i n

2 y

a

= 2 s i n

x

a

s i n

y

a

c o s

x

a

+ c o s

y

a

= 0 ( 4 . 4 . 6 )

h a v e n o t o n l y t h e n o d a l l i n e s s i n ( x = a ) = 0 s i n ( y = a ) = 0 b u t a l s o

2 c o s

2 a

( x + y ) c o s

2 a

( x ; y ) = 0 o r x + y = a x ; y = a :

T h u s x + y = a a n d x

; y = a a r e n o d a l l i n e s , t o o . T h e s e s t r a i g h t l i n e s a r e

t h e d i a g o n a l s o f t h e s q u a r e , w h i c h i s n o w c u t i n t o f o u r t r i a n g l e s .

W e h a v e s e e n t h a t t h e l o w e s t e i g e n v a l u e i s i m p o r t a n t , b u t w h i c h m e m b r a n e

h a s t h e l o w e s t e i g e n v a l u e ? C a n y o u h e a r t h e s h a p e o f a m e m b r a n e ? T h e r e


7/18/2019 C4029-Ch04



h a s b e e n a 7 5 - y e a r - o l d d i s c u s s i o n . F i r s t , i n 1 9 2 3 , i t w a s p r o v e n 4 . 1 ] t h a t f o r

m e m b r a n e s o f v a r i o u s s h a p e s , b u t t h e s a m e a r e a , s u r f a c e d e n s i t y a n d t e n s i o n ,

t h e c i r c u l a r m e m b r a n e h a s t h e l o w e s t e i g e n v a l u e ( F a b e r t h e o r e m ) . I n 1 9 6 6

K a c 4 . 2 ] a s k e d i f o n e c a n h e a r t h e s h a p e o f a d r u m a n d f o u n d t h e a n s w e r

N O , b u t s h o w e d t h a t o n e c a n h e a r t h e c o n n e c t i v i t y o f t h e d r u m . O n t h e o t h e r

h a n d , i n 1 9 8 9 , G o r d o n 4 . 3 ] s t a t e d t h a t t h e r e a r e i s o s p e c t r a l m a n i f o l d s a n d

t h a t i t i s n o t p o s s i b l e t o \ h e a r t h e s h a p e " .

A n y w a y , w e n o w i n v e s t i g a t e t h e e i g e n f r e q u e n c i e s o f a c i r c u l a r m e m b r a n e .

W e u s e t h e r e s u l t s o f p r o b l e m 4 o f s e c t i o n 3 . 1 f o r s o l v i n g ( 4 . 4 . 1 ) i n c y l i n d r i c a l

c o o r d i n a t e s :

w ( r ' ) =

X

p

A

p

J

p

!

p

c

r

a

p

s i n p ' + b

p

c o s p ' ] : ( 4 . 4 . 7 )

I n c l u s i o n o f t h e t i m e - d e p e n d e n t t e r m y i e l d s t h e s o l u t i o n o f t h e w a v e e q u a t i o n

w ( r ' t ) =

X

p

A

p

J

p

!

p

c

r

c o s p ' s i n ( ! t +

p

) : ( 4 . 4 . 8 )

B o t h s o l u t i o n s s a t i s f y t h e h o m o g e n e o u s b o u n d a r y c o n d i t i o n s w ( r = R ' ) =

0 , w h e r e R i s t h e r a d i u s o f t h e c i r c u l a r m e m b r a n e , s e e ( 2 . 4 . 4 5 ) . T h e u n -

k n o w n p a r t i a l a m p l i t u d e s m a y b e u s e d t o s a t i s f y i n i t i a l c o n d i t i o n s o f t h e

t i m e - d e p e n d e n t w a v e e q u a t i o n . I f f o r i n s t a n c e w ( r ' t = 0 ) = f ( r ' ) a n d

w

t

( r ' 0 ) = h ( r ' ) a r e g i v e n , t h e n a l s o t h e a m p l i t u d e s a

p

b

p

i n t h e t i m e -

d e p e n d e n t t e r m

P

p

( a

p

c o s !

p

t + b

p

s i n !

p

t ) c a n b e u s e d t o s a t i s f y t h e i n i -

t i a l c o n d i t i o n . I f i n h o m o g e n e o u s b o u n d a r y c o n d i t i o n s l i k e g ( r ' ) a r e g i v e n ,

t h e n i t m a y b e u s e f u l t o s t a r t w i t h w ( r ' t ) = v ( r ' ) e x p ( i ! t ) . L e t v

p

( r ' )

t h e e i g e n f u n c t i o n s o f t h e H e l m h o l t z e q u a t i o n , t h e n w ( r ' t ) = v

p

( r ' ) +

P

p

( a

p

c o s !

p

t + b

p

s i n !

p

t ) v

p

( r ' ) h a s t o s a t i s f y t h e i n i t i a l c o n d i t i o n s a n d

o n e h a s t o d e t e r m i n e a a n d b s o t h a t f o r t = 0 w ( r ' 0 ) ! v

p

( r ' ) v a n i s h e s .

T h e b e h a v i o u r o f t h e s o l u t i o n ( 4 . 4 . 7 ) i s v e r y i n t e r e s t i n g f o r r ! R . I n

t h i s c a s e t h e e q u a t i o n ( 2 . 4 . 4 3 ) d e s c r i b i n g B e s s e l f u n c t i o n b e c o m e s y

0 0

+ y =

0 . T h i s m e a n s t h a t t h e B e s s e l f u n c t i o n s b e h a v e l i k e s i n x o r c o s x f o r

e x t r e m e l y l a r g e x . T h i s r e c a l l s t h e r e s u l t o f p r o b l e m 3 o f s e c t i o n 2 . 5 . L e t u s

c o n s i d e r t h e i n n i t e c i r c u l a r m e m b r a n e . I n t h e B e s s e l e q u a t i o n ( 2 . 4 . 4 3 ) w e

n e g l e c t t h e t e r m r

; 2

b y a s s u m i n g p = 0 , b u t w e k e e p t h e r

; 1

t e r m . T h u s ,

t h e e q u a t i o n r e a d s

u

0 0

( r ) +

1

r

u

0

( r ) +

!

2

c

2

u ( r ) = 0 : ( 4 . 4 . 9 )

U s i n g !

2

= c

2

= k

2

, M a t h e m a t i c a s o l v e s t h i s e q u a t i o n :

DSolve[u0 0 [r]+u0 [r]/r+kˆ2*u[r]==0,u[r],r]( 4 . 4 . 1 0 )

r e s u l t s i n

u ! B e s s e l J 0 k r ] C 1 ] + B e s s e l Y 0 k r ] C 2 ]

T h e N e u m a n n f u n c t i o n ( B e s s e l f u n c t i o n o f t h e s e c o n d k i n d , a l s o d e n o t e d

b y N ) h a s a s i n g u l a r i t y a t r = 0 a n d h a s t o b e e x c l u d e d h e r e . F o r t h e c i r c u l a r

m e m b r a n e b o u n d e d b y a n i t e r a d i u s R t h e e i g e n v a l u e s k a r e d e t e r m i n e d b y

t h e z e r o s o f J

0

( k R ) , s e e ( 2 . 4 . 4 5 ) . N o w f o r R ! 1 , t h e a r g u m e n t k R t e n d s t o


7/18/2019 C4029-Ch04



1 , a n d w e o b t a i n a n i n n i t e c o n t i n u o u s s p e c t r u m o f e i g e n v a l u e s . W e e x p e c t

t o r e p l a c e a s u m

P

J

0

( k R ) o v e r t h e e i g e n v a l u e s b y a n i n t e g r a l . W e t h u s s e t

u p

u ( r ) =

1

Z

0

A ( k ) J

0

( k r ) d k : ( 4 . 4 . 1 1 )

T o s o l v e t h e f u l l p r o b l e m i n c l u d i n g t h e t i m e d e p e n d e n c e a n d a s s u m i n g s y m -

m e t r y @ = @ ' = 0 w e w r i t e

u ( r t ) =

1

Z

0

A ( k ) J

0

( k r ) c o s k c t d k ( 4 . 4 . 1 2 )

a n d g i v e i n i t i a l c o n d i t i o n s

u ( r t = 0 ) = u

0

( r )

@ u ( r t = 0 )

@ t

= 0 0 < r 1 : ( 4 . 4 . 1 3 )

T h i s d e s c r i b e s a n i n i t i a l p l u c k i n g o f t h e m e m b r a n e w i t h v e l o c i t y z e r o d i s -

t r i b u t e d o v e r t h e r a d i u s . F o r t = 0 t h e s o l u t i o n ( 4 . 4 . 1 2 ) s a t i s e s ( 4 . 4 . 1 3 ) i f

u

0

( r ) =

1

Z

0

A ( k ) J

0

( r k ) d k : ( 4 . 4 . 1 4 )

I n o r d e r t o d e t e r m i n e t h e e x p a n s i o n c o e c i e n t s A ( k ) w e w r i t e ( k

; > )

1

r

d

d r

r

d

d r

+

2

!

J

0

( r ) = 0 J

0

( r )

1

r

d

d r

r

d

d r

+

2

!

J

0

( r ) = 0 ( ; J

0

( r ) ) ( 4 . 4 . 1 5 )

d

d r

r

J

0

( r )

d J

0

( r )

d r

; J

0

( r )

d J

0

( r )

d r

=

;

2

;

2

J

0

( r ) J

0

( r ) r :

T h i s d e m o n s t r a t e s t h a t t h e r h s t e r m m a y b e w r i t t e n a s a d i e r e n t i a l . M u l t i -

p l i c a t i o n o f ( 4 . 4 . 1 4 ) b y r J

0

( r ) a n d i n t e g r a t i o n r e s u l t s i n

1

Z

0

u

0

( r ) r J

0

( r ) d r =

1

Z

0

1

Z

0

A ( ) J

0

( r ) d r J

0

( r ) d r : ( 4 . 4 . 1 6 )

U s i n g ( 4 . 4 . 1 5 ) w e c a n w r i t e

1

Z

0

u

0

( r ) r J

0

( r ) d r =

l i m

R ! 1

1

Z

0

A ( )

R

2

;

2

J

0

( R )

d J

0

( R )

d R

; J

0

( R )

d J

0

( R )

d R

d : ( 4 . 4 . 1 7 )


7/18/2019 C4029-Ch04



T o e v a l u a t e t h e B e s s e l f u n c t i o n s f o r l a r g e a r g u m e n t s R , w e u s e a n a s y m p -

t o t i c e x p a n s i o n 1 . 2 ] , 2 . 1 ]

J

p

( x ) =

1

Z

0

c o s ( x s i n '

; p ' ) d ' ( 4 . 4 . 1 8 )

w h i c h w e w r i t e i n t h e f o r m

J

p

( R ) =

p

2

c o s R ; ( p + 1 = 2 ) = 2 ]

p

R

( 4 . 4 . 1 9 )

s o t h a t f o r p = 0 ( 4 . 4 . 1 7 ) n o w r e a d s

1

Z

0

u

0

( r ) r J

0

( r ) d r = l i m

R ! 1

1

1

Z

0

A ( )

(

2

;

2

)

p

s i n ( ; ) R

;

;

c o s ( + ) R

+

d ( 4 . 4 . 2 0 )

( w e u s e d t h e a d d i t i o n t h e o r e m o f t r i g o n o m e t r i c f u n c t i o n s ) . T h e r h s t e r m

d e s c r i b e s a r e s o n a n c e a t = . U s i n g t h e D i r i c h l e t i n t e g r a l

1

Z

0

s i n z

z

d z =

2

( 4 . 4 . 2 1 )

o n e o b t a i n s

A ( ) =

1

Z

0

u

0

( r ) r J

0

( r ) d r ( 4 . 4 . 2 2 )

c o m p a r e ( 4 . 4 . 1 4 ) a n d u s e i t :

u

0

( r ) =

1

Z

0

1

Z

0

u

0

( r

0

) r

0

J

0

( r

0

) J

0

( r ) d r

0

d : ( 4 . 4 . 2 3 )

I n s e r t i n g ( 4 . 4 . 2 2 ) i n ( 4 . 4 . 1 2 ) , o n e o b t a i n s t h e s o l u t i o n f o r t h e i n n i t e m e m -

b r a n e

u ( r t ) =

1

Z

0

1

Z

0

u

0

( r

0

) r

0

J

0

( r

0

) d r

0

J

0

( r ) c o s c t d : ( 4 . 4 . 2 4 )

W a v e s i n a v e r y l a r g e l a k e o r s u r f a c e w a v e s d u e t o t h e s u r f a c e t e n s i o n o f a

l i q u i d l a y e r c a n b e d e s c r i b e d b y ( 4 . 4 . 2 4 ) .

S o l u t i o n s o f t h e H e l m h o l t z e q u a t i o n f o r a b o u n d a r y n o t d e s c r i b a b l e b y

c o o r d i n a t e s y s t e m s i n w h i c h t h e p a r t i a l d i e r e n t i a l e q u a t i o n i s n o t e a s i l y s e p -

a r a b l e , c a n o n l y b e f o u n d b y o t h e r m e t h o d s l i k e n u m e r i c a l i n t e g r a t i o n , c o l l o -

c a t i o n m e t h o d s o r v a r i a t i o n a l m e t h o d s , c o m p a r e t h e s e c t i o n s o f c h a p t e r 5 .


7/18/2019 C4029-Ch04



T h e H e l m h o l t z e q u a t i o n p l a y s a n i m p o r t a n t r o l e i n e l e c t r o m a g n e t i s m .

A p p l y i n g t h e o p e r a t o r c u r l o n c u r l

~

E = ; @

~

B = @ t a n d t h e o p e r a t o r

0

@ = @ t o n

c u r l

~

H = @ " "

0

~

E = @ t +

~

j o n t h e M a x w e l l e q u a t i o n s o n e o b t a i n s t h e e l e c t r o -

m a g n e t i c w a v e e q u a t i o n s :

~

E ;

1

c

2

@

2

~

E

@ t

2

= g r a d

" "

0

+

0

@

~

j

@ t

( 4 . 4 . 2 5 )

~

B

;

1

c

2

@

2

~

B

@ t

2

=

;

0

c u r l

~

j : ( 4 . 4 . 2 6 )

H e r e "

0

= 0 : 8 8 5 4 1 9 1 0

; 1 1

A s / V m

0

= 1 : 2 5 6 6 3 7 1 0

; 6

V s / A m a n d d i v ( " "

0

~

E )

= d i v

~

H = 0 h a v e b e e n u s e d . c =

p

1 =

0

" "

0

i s t h e v e l o c i t y o f l i g h t w i t h i n a

m e d i u m p o s s e s s i n g t h e m a t e r i a l p a r a m e t e r s " a n d . F o r a t i m e d e p e n d e n c e

e x p ( i ! t ) a n d a d o m a i n f r e e o f c h a r g e s a n d c u r r e n t s

~

j , t h e e q u a t i o n s

r e s u l t i n a v e c t o r H e l m h o l t z e q u a t i o n ( 3 . 1 . 8 0 ) . I t i s n o t d i c u l t t o s o l v e

t h e e q u a t i o n s i n c a r t e s i a n c o o r d i n a t e s , s e e ( 3 . 1 . 8 1 ) - ( 3 . 1 . 8 3 ) , b u t a l l o t h e r

c o o r d i n a t e s p r e s e n t d i c u l t i e s , c o m p a r e ( 3 . 1 . 8 4 ) . U s i n g t h e t r i c k (

~

K )

l

=

K

l

h e l p s . A s a n e x a m p l e w e m a y u s e c a r t e s i a n c o m p o n e n t s a s f u n c t i o n s o f

s p h e r i c a l c o o r d i n a t e s

~

K ( r # ' ) = K

x

( r # ' )

~e

x

+ K

y

( r # ' ) ~e

y

+ K

z

( r # ' ) ~e

z

@

2

K

x

@ r

2

+

2

r

@ K

x

@ r

+

@

2

K

x

r

2

@ #

2

+

c o t #

r

2

@ K

x

@ #

+

1

r

2

s i n

2

#

@

2

K

x

@ '

2

+ k

2

K

x

= 0 ( 4 . 4 . 2 7 )

( a n d a n a l o g o u s e q u a t i o n s f o r K

y

K

z

) . A s u s u a l , k = ! = c . I n e n g i n e e r i n g

p r o b l e m s c y l i n d r i c a l g e o m e t r y i s m o r e i m p o r t a n t . O n e i s i n t e r e s t e d i n t h e

s o l u t i o n s o f ( 4 . 4 . 2 5 ) a n d ( 4 . 4 . 2 6 ) e i t h e r f o r c l o s e d h o l l o w b o x e s ( r e s o n a t o r s )

o r i n i n n i t e l y l o n g h o l l o w c y l i n d e r s ( w a v e g u i d e s ) . F o r a z - i n d e p e n d e n c e

e x p ( i z ) t h e M a x w e l l e q u a t i o n s r e a d i n g e n e r a l c y l i n d r i c a l c o o r d i n a t e s

q

1

q

2

( q

3

= z ) :

1

p

g

2 2

@ B

z

@ q

2

; i B

2

=

i !

c

2

E

1

( 4 . 4 . 2 8 )

i B

1

;

1

p

g

1 1

@ B

z

@ q

1

=

i !

c

2

E

2

( 4 . 4 . 2 9 )

1

p

g

1 1

@ B

2

@ q

1

;

1

p

g

2 2

@ B

1

@ q

2

=

i !

c

2

E

z

( 4 . 4 . 3 0 )

1

p

g

2 2

@ E

z

@ q

2

; i E

2

= ; i ! B

1

( 4 . 4 . 3 1 )

i E

1

;

1

p

g

1 1

@ E

z

@ q

1

= ; i ! B

2

( 4 . 4 . 3 2 )

1

p

g

1 1

@ E

2

@ q

1

;

1

p

g

2 2

@ E

1

@ q

2

= ; i ! B

z

: ( 4 . 4 . 3 3 )


7/18/2019 C4029-Ch04



E x p r e s s i n g E

2

( q

1

q

2

) b y B

1

( q

1

q

2

) a n d B

z

( q

1

q

2

) e t c . , a n d i n s e r t i n g i n t o

( 4 . 4 . 3 1 ) , e t c . , r e s u l t s i n t h e F o c k e q u a t i o n s :

B

1

( q

1

q

2

) =

i

!

2

;

2

c

2

!

p

g

2 2

@ E

z

@ q

2

+

c

2

p

g

1 1

@ B

z

@ q

1

B

2

( q

1

q

2

) =

i

!

2

;

2

c

2

;

!

p

g

1 1

@ E

z

@ q

1

+

c

2

p

g

2 2

@ B

z

@ q

2

( 4 . 4 . 3 4 )

E

1

( q

1

q

2

) =

; i c

2

!

2

;

2

c

2

!

p

g

2 2

@ B

z

@ q

2

;

p

g

1 1

@ E

z

@ q

1

E

2

( q

1

q

2

) =

i c

2

!

2

;

2

c

2

!

p

g

1 1

@ B

z

@ q

1

+

p

g

2 2

@ E

z

@ q

2

( 4 . 4 . 3 5 )

w h e r e E

z

a n d B

z

m a y b e c a l c u l a t e d f r o m ( E )

z

= E

z

( B )

z

= B

z

a n d

t h e b o u n d a r y c o n d i t i o n s o n a m e t a l l i c w a l l E

t

( r = R ' ) = E

x

= 0 B

n

= 0 .

F o r a c i r c u l a r c y l i n d e r t h e t a n g e n t i a l c o m p o n e n t i n t h e z d i r e c t i o n i s t h e n

g i v e n b y

E

z

( r z ' t ) = E

0

z

e x p ( i ! t

; i z )

c o s m '

J

m

p

k

2

;

2

r

: ( 4 . 4 . 3 6 )

T h e v a n i s h i n g o f t h i s e x p r e s s i o n f o r r = R y i e l d s

p

k

2

;

2

R = j

m n

s e e ( 2 . 4 . 4 5 ) . I t t h u s f o l l o w s f r o m t h e l o w e s t e i g e n v a l u e 2 : 4 0 4 8 t h a t t h e r e a r e

n o p r o p a g a t i n g w a v e s w i t h f r e q u e n c i e s

! < c

r

2 : 4 0 4 8

2

R

2

+

2

: ( 4 . 4 . 3 7 )

T h i s i s t h e s o - c a l l e d c u t - o f r e q u e n c y .

A c c o r d i n g t o t h e F o c k e q u a t i o n s t h e r e a r e t w o t y p e s o f w a v e s i n a w a v e -

g u i d e :

1 . t r a n s v e r s e e l e c t r i c w a v e ( T E w a v e ) , E

z

( q

1

q

2

z ) = 0 B

n

( b o u n d a r y , z ) =

0 B

1

B

2

E

1

a n d E

2

a r e d e s c r i b e d b y B

z

f r o m t h e F o c k e q u a t i o n s ,

2 . t r a n s v e r s e m a g n e t i c w a v e ( T M w a v e ) , B

z

( q

1

q

2

z ) = 0 E

t

( b o u n d a r y , z ) =

0 E

1

E

2

B

1

B

2

a r e d e s c r i b e d b y E

z

. F o r a w a v e - g u i d e w i t h r e c t a n g u l a r

c r o s s s e c t i o n ( a

b ) t h e r e s u l t s a r e


7/18/2019 C4029-Ch04



T M T E

E

x

= ; f i

m

a

c o s

m x

a

s i n

n y

b

E

x

= f i k

n

b

c o s

m x

a

s i n

n y

b

E

y

= ; f i

n

b

s i n

m x

a

c o s

n y

b

E

y

= ; f i k

m

a

s i n

m x

a

c o s

n y

b

E

z

= f

2

m

2

a

2

+

n

2

b

2

s i n

m x

a

s i n

n y

b

E

z

= 0

B

x

= f i k

n

b

s i n

m x

a

c o s

n y

b

B

x

= f i

m

a

s i n

m x

a

c o s

n y

b

B

y

= ; f i k

m

a

c o s

m x

a

s i n

n y

b

B

y

= f i

n

b

c o s

m x

a

s i n

n y

b

B

z

= 0 B

z

= f

2

m

2

a

2

+

n

2

b

2

c o s

m x

a

c o s

n y

b

( 4 . 4 . 3 8 )

w h e r e

f = e x p ( i ! t ; i z ) k = ! = c

2

= k

2

;

m

2

2

a

2

;

n

2

2

b

2

:

T h e f a c t o r i = e x p ( i = 2 ) i n d i c a t e s a r e l a t i v e p h a s e s h i f t . F o r a c i r c u l a r c r o s s

s e c t i o n o f t h e w a v e g u i d e o n e g e t s f o r t h e T M w a v e

E

r

= ; i f

j

m n

R

c o s m ' J

0

m

j

m n

r

R

B

r

= ; i k f

m

r

s i n m ' J

m

j

m n

r

R

E

'

= i f

m

r

s i n m ' J

m

j

m n

r

R

B

'

= ; i k f

j

m n

R

c o s m ' J

0

m

j

m n

r

R

E

z

=

j

2

m n

R

2

f c o s m ' J

m

j

m n

r

R

B

z

= 0 ( 4 : 4 : 3 9 )

w h e r e

2

= k

2

; j

2

m n

= R

2

: A g a i n j

m n

r e p r e s e n t s t h e n - t h r o o t o f t h e B e s s e l

f u n c t i o n J

m

. U s i n g t h e H e r t z v e c t o r

~

Z o n e m a y i n v e s t i g a t e t h e e m i s s i o n o f

r a d i a t i o n f r o m c e l l u l a r p h o n e s . I t i s e a s y t o p r o v e t h a t t h e s e t u p

~

E = ;

1

c

2

@

2

~

Z

@ t

2

+ g r a d d i v

~

Z

~

B =

1

c

2

c u r l

@

~

Z

@ t

( 4 . 4 . 4 0 )

s a t i s e s c u r l

~

E = ; @

~

B = @ t . T h e e q u a t i o n c u r l

~

B =

0

= @ " "

0

~

E = @ t a n d i n t e g r a -

t i o n w i t h r e s p e c t t o t i m e y i e l d

~

Z =

1

c

2

@

2

~

Z

@ t

2

= ; k

2

~

Z : ( 4 . 4 . 4 1 )


7/18/2019 C4029-Ch04



I n t h e d e r i v a t i o n o f t h i s e q u a t i o n ( 3 . 1 . 8 0 ) h a s b e e n u s e d . I n s p h e r i c a l c o o r -

d i n a t e s ( 4 . 4 . 4 1 ) h a s t h e c o u p l e d f o r m o f ( 3 . 1 . 8 4 ) . I n c a r t e s i a n c o o r d i n a t e s i t

w o u l d h a v e t h e f o r m ( 4 . 4 . 2 7 ) a n d c o u l d b e s o l v e d a n a l o g o u s l y t o ( 3 . 1 . 7 4 ) . F o r

a H e r t z d i p o l e a e r i a l o f a c e l l u l a r p h o n e w e m a k e s i m p l i c a t i o n s

@

@ '

= 0 Z

'

= 0 Z

#

= 0 Z e x p ( i ! t ) ( 4 . 4 . 4 2 )

a n d t h e s e t u p

Z

r

= r w ( r # ) : ( 4 . 4 . 4 3 )

T h e s o l u t i o n o f ( 4 . 4 . 2 7 ) f o r K

x

! Z

r

r e a d s

Z

r

( r ' ) = r w ( r # ) = P

l

( c o s # ) R

l

( r ) ( 4 . 4 . 4 4 )

w h e r e

R

0 0

l

+

2

r

R

0

l

+

k

2

;

l ( l + 1 )

r

2

R

l

= 0 ( 4 . 4 . 4 5 )

c o m p a r e ( 2 . 2 . 6 5 ) a n d ( 3 . 1 . 7 6 ) . T h e s e f u n c t i o n s R

l

a r e c a l l e d s p h e r i c a l B e s s e l

f u n c t i o n s o r B e s s e l f u n c t i o n s o f f r a c t i o n a l o r d e r

R

l

( r ) =

1

p

r

J

l + 1 = 2

( k r ) : ( 4 . 4 . 4 6 )

T h e y a r e s o l u t i o n s o f ( 2 . 2 . 6 5 ) a n d o f t h e t y p e d i s c u s s e d i n p r o b l e m 3 o f s e c t i o n

2 . 5 . T h e J f u n c t i o n s d e s c r i b e s t a n d i n g w a v e s . I n o u r p r o b l e m w e a r e i n t e r -

e s t e d i n p r o p a g a t i n g w a v e s . B e s i d e s t h e s t a n d i n g w a v e s o l u t i o n s o f ( 2 . 2 . 6 5 )

a n d o f ( 2 . 2 . 4 8 ) f o r B e s s e l f u n c t i o n s o f i n t e g e r o r d e r , t h e r e e x i s t p r o p a g a t i n g

s o l u t i o n s , t h e H a n k e l f u n c t i o n s ( B e s s e l f u n c t i o n s o f t h i r d k i n d ) . T h e y a r e

d e n e d b y

H

( 1 )

( z ) = J

( z ) + i Y

( z )

H

( 1 )

( z ) = J

( z ) ; i Y

( z ) : ( 4 . 4 . 4 7 )

F o r f r a c t i o n a l o r d e r t h e y d e g e n e r a t e i n t o t h e R

l

w h e r e a s t h e J

l + 1 = 2

a r e r e l a t e d

t o s t a n d i n g w a v e s l i k e s i n a n d c o s , t h e R

l

a r e r e l a t e d t o p r o p a g a t i n g w a v e s

e x p ( i k r ) . T h e R

l

( r ) c a n b e d e s c r i b e d b y t h e R o d r i g u e z f o r m u l a s

R

l

( r ) = r

l

1

r

d

d r

l

e x p ( i k r )

r

( 4 . 4 . 4 8 )

s o t h a t

R

0

( r ) =

e x p ( ; i k r )

r

R

1

( r ) = R

0

0

( r ) =

e x p ( ; i k r )

r

i k ;

1

r

R

2

( r ) = R

0

1

( r ) ;

1

r

R

1

( r ) =

e x p ( ; i k r )

r

; k

2

;

3 i k

r

+

3

r

2

( 4 . 4 . 4 9 )


7/18/2019 C4029-Ch04



w i t h t h e a d d i t i o n t h e o r e m s

R

0

l

( r ) = R

l + 1

( r ) +

l

r

R

l

( r ) : ( 4 . 4 . 5 0 )

D u e t o m = 0 P

0

= 1 R

0

d e s c r i b e s a c e n t r a l s y m m e t r i c s p h e r i c a l w a v e , l = 1

d e s c r i b e s a d i p o l e r a d i a t i o n e m i s s i o n a n d l = 2 a q u a d r u p o l e a e r i a l . I n s e r t i n g

Z

r

= r w = e x p ( ; i k r ) ( i k ; 1 = r ) c o s # ( 4 . 4 . 5 1 )

i n t o ( 4 . 4 . 4 0 ) o b t a i n s

E

r

= f E

0

r

1

r

3

+

i k

r

2

c o s #

E

#

= f

E

0

r

2

1

r

3

+

i k

r

2

;

k

2

r

s i n #

B

'

= f B

0

'

1

r

3

+

i k

r

s i n # ( 4 . 4 . 5 2 )

w h e r e f = e x p ( i ! t

; i k r ) a n d E

0

r

B

0

'

a r e p a r t i a l a m p l i t u d e s ( i n t e g r a t i o n c o n -

s t a n t s ) . F r o m ( 4 . 4 . 5 2 ) o n e m a y n d t h a t t h e r a d i a t i o n e n e r g y i n t h e G H z

r a n g e d e p o s i t e d i n t h e h e a d o f a u s e r ( r ! 0 ) i n c r e a s e s w i t h d e c r e a s i n g d i s -

t a n c e r f r o m t h e c e l l u l a r t e l e p h o n e .

F o r a d i p o l e a e r i a l s i t u a t e d a t t h e s u r f a c e o f t h e e a r t h w ( R # ) i s g i v e n b y

4 . 5 ]

w

p r

( r # ) =

X

l

C

l

r

2 k r

H

( 2 )

l + 1 = 2

( k r ) P

l

( c o s # ) r > R : ( 4 . 4 . 5 3 )

F o r e x p ( + i ! t ) H

( 2 )

d e s c r i b e s a n o u t g o i n g w a v e a n d h a s t o s a t i s f y t h e

b o u n d a r y c o n d i t i o n i n i n n i t y

l i m

r ! 1

@ w

@ r

+ i k w

= 0 : ( 4 . 4 . 5 4 )

I n t h e i n t e r i o r o f t h e e a r t h a s t a n d i n g w a v e i s c r e a t e d s o t h a t a t t h e s u r f a c e

r = R t h e b o u n d a r y c o n d i t i o n

X

l

C

l

r

2 k R

H

( 2 )

l + 1 = 2

( k R ) P

l

( c o s # ) =

X

l

D

l

r

2 k R

J

( 2 )

l + 1 = 2

( k R ) P

l

( c o s # )

( 4 . 4 . 5 5 )

m u s t b e s a t i s e d d u e t o c o n t i n u i t y : s o m e o f t h e s e r i e s e x p a n s i o n s o f t y p e

( 4 . 4 . 5 3 ) h a v e v e r y s l o w c o n v e r g e n c e f o r k R > 1 0 0 0 s o t h a t i t i s n e c e s s a r y t o

t r a n s f o r m t h e s e r i e s i n t o a c o m p l e x i n t e g r a l ( W a t s o n t r a n s f o r m a t i o n ) 4 . 5 ] .

T h i s r e p r e s e n t s a n o t h e r e x a m p l e o f a n i n t e g r a l r e p r e s e n t a t i o n o f t h e s o l u t i o n

o f a p a r t i a l d i e r e n t i a l e q u a t i o n , s e e a l s o ( 4 . 4 . 2 4 ) .


7/18/2019 C4029-Ch04


P r o b l e m s

1 . F i n d t h e n o d a l l i n e s a n d b o u n d a r y s h a p e b y p l o t t i n g y ( x ) f r o m t h e

d e g e n e r a t e e i g e n f u n c t i o n s

A

m n

s i n

m x

a

s i n

n y

a

+ A

n m

s i n

n x

a

s i n

m y

a

= 0 ( 4 . 4 . 5 6 )

f o r m = 1 n = 3 A

3 1

=

A

1 3

a = 1 . U s e s i n 3 = s i n c o s 2 +

c o s s i n 2 s i n 2 = 2 s i n c o s 2 c o s

2

= 1 + c o s 2 t o n d y ( x ) =

a r c c o s ( f ( x ) ) . P l a y w i t h v a r i o u s v a l u e s o f A

3 1

.

2 . F i n d t h e t i m e - d e p e n d e n t a s y m m e t r i c s o l u t i o n f o r a c i r c u l a r m e m b r a n e

o f r a d i u s R t h a t i s p l u c k e d a t i t s c e n t e r a t t i m e t = 0 . T h e i n i t i a l

c o n d i t i o n s w i l l t h e n b e

u ( r ' t = 0 ) =

h ( R ; r )

R

@ u ( r ' t = 0 )

@ t

= 0 : ( 4 . 4 . 5 7 )

T h e e i g e n f r e q u e n c i e s !

p

= j

2

p

c = R a n d t h e j

p

a r e d e t e r m i n e d b y t h e

b o u n d a r y c o n d i t i o n ( 2 . 4 . 4 5 ) . S t a r t w i t h

u ( r ' t ) =

X

p

A

p

J

p

!

p

c

r

c o s p ' s i n ( !

p

t +

p

) : ( 4 . 4 . 5 8 )

T h e i n i t i a l c o n d i t i o n s r e s u l t i n

p

= = 2 . M u l t i p l i c a t i o n b y t h e o r t h o -

g o n a l w

l

a n d i n t e g r a t i o n y i e l d

2

Z

0

R

Z

0

h ( R ; r )

R

w

l

r d r d ' =

X

p

A

p

2

Z

0

R

Z

0

w

p

w

l

r d r d ' : ( 4 . 4 . 5 9 )

D u e t o ( 2 . 4 . 4 6 ) a n d

2

R

0

c o s

2

p ' d ' = o n e o b t a i n s t h e e x p a n s i o n c o e -

c i e n t s A

p

f r o m

2

Z

0

R

Z

0

h ( R ; r )

R

J

p

!

p

r

c

r c o s ( p ' ) d r d ' = A

p

2

J

2

p + 1

!

p

c

R

:

( 4 . 4 . 6 0 )

3 . E x t e r i o r c o n t i n u o u s a n d p e r i o d i c e x c i t a t i o n s K c a n b e d e s c r i b e d b y

u =

1

c

2

@

2

u

@ t

2

+ K ( x y ) s i n ! t : ( 4 . 4 . 6 1 )

S h o w t h a t t h e p a r t i c u l a r s o l u t i o n

u ( x y t ) = s i n ! t

X

p

w

p

( x y ) A

p

K ( x y ) =

X

p

K

p

w

p

( x y ) ( 4 . 4 . 6 2 )


7/18/2019 C4029-Ch04



w h e r e w

p

s a t i s f y t h e h o m o g e n e o u s e q u a t i o n , r e s u l t s i n t h e r e s o n a n c e

c o n d i t i o n

A

p

=

c

2

K

p

!

2

; !

2

p

: ( 4 . 4 . 6 3 )

I f t h e w h o l e c i r c u l a r m e m b r a n e i s e x c i t e d b y K ( t ) = K

0

s i n ( ! t + ' ) , t h e

s o l u t i o n i s g i v e n 4 . 8 ] , 4 . 2 ] b y

u ( r t ) = ;

K

0

!

2

1 ;

J

0

( ! r = c )

J

0

( ! R = c )

s i n ( ! t + ) : ( 4 . 4 . 6 4 )

4 . A n e l l i p t i c m e m b r a n e o f e c c e n t r i c i t y e a n d s e m i a x e s a b w i l l b e d e s c r i b e d

b y t h e e l l i p t i c c y l i n d r i c a l c o o r d i n a t e s y s t e m , s e e p r o b l e m 2 o f s e c t i o n

3 . 1 . T h e H e l m h o l t z e q u a t i o n m a y b e s e p a r a t e d i n t o t w o o r d i n a r y

d i e r e n t i a l e q u a t i o n s . D e r i v e t h e s e e q u a t i o n s u s i n g M a t h e m a t i c a . T h e

r e s u l t s h o u l d b e

X

0 0

;

;

p

2

+ e

2

k

2

c o s h

2

u

X = 0

Y

0 0

+

;

p

2

; e

2

k

2

c o s

2

v

Y = 0 ( 4 . 4 . 6 5 )

w h e r e !

2

= c

2

= k

2

a n d p i s t h e s e p a r a t i o n c o n s t a n t . L e t M a t h e m a t i c a

s o l v e t h e s e e q u a t i o n s . T h e r e s u l t s h o u l d b e g i v e n b y M a t h i e u f u n c t i o n s

X ( u ) = c e

1

2

( e

2

k

2

+ 2 p

2

) ;

1

4

e

2

k

2

; i u

+

s e

1

2

( e

2

k

2

+ 2 p

2

)

;

1

4

e

2

k

2

; i u

Y ( v ) = c e

1

2

( e

2

k

2

+ 2 p

2

) ;

1

4

e

2

k

2

v

+

s e

1

2

( e

2

k

2

+ 2 p

2

) ;

1

4

e

2

k

2

; v

: ( 4 . 4 . 6 6 )

T h e s t a b i l i t y o f t h e s e s o l u t i o n s h a d b e e n d i s c u s s e d i n p r o b l e m 3 i n s e c -

t i o n 2 . 6 . F o r m o r e g e n e r a l o r d i n a r y d i e r e n t i a l e q u a t i o n s l i k e ( 2 . 4 . 1 )

w i t h f ( x ) = 0 a n d p e r i o d i c c o e c i e n t s p

1

( x ) p

2

( x ) t h e F l o q u e t t h e o -

r e m g u a r a n t e e s t h e e x i s t e n c e o f a g e n e r a l i z e d p e r i o d i c s o l u t i o n y ( x + ) =

s y ( x ) , w h e r e s i s t h e F l o q u e t e x p o n e n t 1 . 2 ] .

5 . T h e e q u a t i o n ( 4 . 4 . 9 ) h a d b e e n s o l v e d u s i n g M a t h e m a t i c a . T r y t o s o l v e

F

0 0

( r ) +

1

r

F

0

( r ) ; k

2

F ( r ) = 0 ( 4 . 4 . 6 7 )

u s i n g t h e L i e s e r i e s m e t h o d 4 . 4 ] . S i n c e

d F

d r

= Z

d Z

d r

= ;

1

r

; Z + k

2

F


7/18/2019 C4029-Ch04



o n e n d s

F ( r ) = Z

1

( r ) Z = Z

2

( r ) = #

1

( Z

0

Z

1

Z

2

) r = Z

0

#

2

( r ) = ;

1

r

Z + k

2

F

d Z

0

d r

= #

0

= 1

d Z

1

d r

= #

1

( Z

0

Z

1

Z

2

)

d Z

2

d r

= #

2

( Z

0

Z

1

Z

2

) :

W i t h t h e i n i t i a l c o n d i t i o n s r = 0 Z

0

( 0 ) = 0 Z

1

( 0 ) = x

1

Z

2

( 0 ) = x

2

t h e s o l u t i o n i s

Z

i

( r ) =

1

X

= 0

r

!

( D

Z

i

)

r = 0

( 4 . 4 . 6 8 )

w h e r e

D =

1

X

= 0

#

( Z

0

Z

1

: : : Z

n

)

@

@ Z

: ( 4 . 4 . 6 9 )

T h e r e a d e r m i g h t a s k w h y t h i s c o m p l i c a t e d s o r c e r y s i n c e t h e B e s s e l

f u n c t i o n s a s s o l u t i o n s f o r ( 4 . 4 . 9 ) a r e v e r y w e l l k n o w n a n d t a b u l a t e d n u -

m e r i c a l l y . T h e r e a r e , h o w e v e r , n u c l e a r e n g i n e e r i n g p r o b l e m s e x h i b i t i n g

v e r y l a r g e a r g u m e n t s ( 5 0 ) s o t h a t e v e n l a r g e m a i n f r a m e s h a v e p r o b -

l e m s w i t h t h e n u m e r i c a l c a l c u l a t i o n s . N o w t h e L i e s e r i e s m e t h o d o e r s

a w a y t o a v o i d t h e B e s s e l f u n c t i o n b y s p l i t t i n g t h e p r o b l e m i n t o

F

c y l

= F

p l a n e

; F

c o r r e c t i o n

: ( 4 . 4 . 7 0 )

T h e f u n c t i o n s #

m u s t b e h o l o m o r p h i c ( n o s i n g u l a r i t i e s ) . N o w #

2

h a s a

p o l e a t r = 1 , s o t h a t a t r a n s f o r m a t i o n i s n e c e s s a r y . L e t d

= r

; r

; 1

t h e t h i c k n e s s o f t h e - t h r a d i a l d o m a i n , o n e m a y p u t = r ; r

; 1

,

w h e r e a n d r a r e v a r i a b l e s a n d r

; 1

i s t h e r a d i a l d i s t a n c e o f t h e - t h

s u b d o m a i n . T h e n

d Z

0

d

= 1 = #

0

d Z

1

d

= Z

2

= #

1

d Z

2

d

=

Z

2

r

; 1

+

+ k

2

Z

1

= #

2

( 4 . 4 . 7 1 )

a n d t h e n e w o p e r a t o r r e a d s

D =

@

@ Z

0

+ Z

2

@

@ Z

1

+

k

2

Z

1

;

Z

2

r

; 1

+ Z

0

@

@ Z

2

: ( 4 . 4 . 7 2 )

T h e i n i t i a l c o n d i t i o n s a r e n o w F

0

= F ( = 0 ) F

0

( 0 ) = d F ( = 0 ) = d ,

s o t h a t t h e s o l u t i o n r e a d s n o w

Z

1

F ( ) = F

0

c o s h k +

F

0

0

k

s i n h k ;

1

X

= 0

f

!

= F

p l a n e

; F

c o r r

: ( 4 . 4 . 7 3 )


7/18/2019 C4029-Ch04


7/18/2019 C4029-Ch04



c o n d i t i o n d e t e r m i n e s ( ! ) . T h e n =

p

n

2

2

; !

2

= E . I n s e r t i o n i n t o

( 4 . 4 . 7 8 ) y i e l d s

X

0 0

+ X

!

2

b

E

x ; n

2

2

= 0 :

S o l v e t h i s e q u a t i o n u s i n g M a t h e m a t i c a . Y o u s h o u l d g e t A i r y f u n c t i o n s

a s a r e s u l t . D e t e r m i n e t h e t w o i n t e g r a t i o n c o n s t a n t s C 1 ] a n d C 2 ] u s i n g

t h e b o u n d a r y c o n d i t i o n s f o r X . Y o u w i l l g e t t w o h o m o g e n e o u s l i n e a r

e q u a t i o n s f o r C 1 ] a n d C 2 ] a n d t h e v a n i s h i n g o f t h e d e t e r m i n a n t y i e l d s

! . T h e t r a n s c e n d e n t a l e q u a t i o n f o r ! s h o u l d r e a d

A i r y A i

2

6

6

6

4

n

2

2

;

b !

2

E

2 = 3

3

7

7

7

5

A i r y B i

2

6

6

6

4

n

2

2

;

a b !

2

E

;

b !

2

E

2 = 3

3

7

7

7

5

; A i r y B i

2

6

6

6

4

n

2

2

;

b !

2

E

2 = 3

3

7

7

7

5

A i r y B i

2

6

6

6

4

n

2

2

;

a b !

2

E

;

b !

2

E

2 = 3

3

7

7

7

5

= 0 :

F o r a c i r c u l a r m e m b r a n e o f r a d i u s R o n e m i g h t c h o o s e

( r ) =

0

( R

2

; r

2

) + 1

:

S o l v e t h e e q u a t i o n a n d n d t h e e i g e n f r e q u e n c y ! f o r t h e s p e c i a l c a s e

E = 1 R = 1

0

= 1 .

4 . 5 R o d s a n d t h e p l a t e e q u a t i o n

W h e r e a s s t r i n g s a n d m e m b r a n e s a r e t h i n b o d i e s w i t h n e g l i g i b l e r i g i d i t y , r o d s ,

b e a m s a n d p l a t e s a r e r i g i d . T h e i r e l a s t i c c h a r a c t e r i s t i c s a r e d e s c r i b e d b y

t h r e e p a r a m e t e r s : Y o u n g ' s m o d u l u s E o f e l a s t i c i t y , P o i s s o n ' s r a t i o , a n d

t h e s h e a r m o d u l u s G i s d e s c r i b e d b y

G = E = 2 ( 1 + ) : ( 4 . 5 . 1 )

I n t h e S I - s y s t e m o f u n i t s o n e h a s E k g m

; 1

s

; 2

] . U s i n g t h e b o u n d a r y c o n d i -

t i o n s o n t h e s u r f a c e s o f t h e b o d i e s , v a r i a t i o n a l a n d o t h e r m e t h o d s d e r i v e t h e

e q u a t i o n s o f m o t i o n s f o r v a r i o u s t y p e s o f o s c i l l a t i o n s 1 . 2 ] , 4 . 7 ] , 4 . 9 ] , 4 . 1 0 ] .

T h e s e m i g h t b e :


7/18/2019 C4029-Ch04


R o d s a n d t h e p l a t e e q u a t i o n 1 8 9

1 . l o n g i t u d i n a l ( d i l a t i o n a l ) o s c i l l a t i o n s o f a r o d i n t h e x d i r e c t i o n

@

2

u ( x t )

@ t

2

=

E

@

2

u ( x t )

@ x

2

+

1

q

f ( x t ) ( 4 . 5 . 2 )

( q c r o s s s e c t i o n , d e n s i t y , f e x t e r i o r f o r c e )

2 . t r a n s v e r s a l ( b e n d i n g ) v i b r a t i o n s o f a r o d

@

2

u ( x t )

@ t

2

=

E I

q

@

4

u ( x t )

@ x

4

+

1

q

f ( x t ) ( 4 . 5 . 3 )

w h e r e

I = I

y

=

Z

q

Z

y

2

d f o r I

z

=

Z

q

Z

z

2

d f ( 4 . 5 . 4 )

i s t h e b e n d i n g m o m e n t o r t h o g o n a l t o t h e x a x i s . F o r a b e a m o f r e c t a n -

g u l a r c r o s s s e c t i o n a b , o n e h a s

I

y

=

+ a = 2

Z

; a = 2

+ b = 2

Z

; b = 2

y

2

d z d y =

a

3

b

1 2

I

z

=

a b

3

1 2

( 4 . 5 . 5 )

3 . t o r s i o n a l v i b r a t i o n s o f a r o d a r o u n d t h e x a x i s

@

2

u ( x t )

@ t

2

=

G

@

2

u

@ x

2

( 4 . 5 . 6 )

4 . b e n d i n g v i b r a t i o n s o f a p l a t e o f t h i c k n e s s 2 h

@

2

u ( x y t )

@ t

2

= ;

E h

2

3 ( 1

;

2

)

u ( x y t ) ( 4 . 5 . 7 )

( p l a t e e q u a t i o n ) . U s i n g a t i m e d e p e n d e n c e e x p ( i ! t ) , f r o m c a s e t o c a s e

t h e p a r a m e t e r k

4

= 3 ( 1 ;

2

) !

2

= E h

2

i s i n t r o d u c e d , s o t h a t ( 4 . 5 . 7 ) m a y

b e w r i t t e n u ; k

4

u = 0 .

T o b e a b l e t o s o l v e a l l t h e s e e q u a t i o n s w e n e e d b o u n d a r y c o n d i t i o n s . T h e

e n d o f a m a s s i v e b o d y m a y b e f r e e o f f o r c e s , m a y b e s u p p o r t e d b y a n o t h e r

b o d y o r m a y b e c l a m p e d . F o r a r o d w e m a y h a v e t h e l o n g i t u d i n a l o s c i l l a t i o n s :

a . f r e e e n d a t x = l

u

x

( l t ) = 0 ( 4 . 5 . 8 )

b . c l a m p e d e n d a t x = 0

u ( 0 t ) = 0 : ( 4 : 5 : 8 a )

F o r t r a n s v e r s a l o s c i l l a t i o n s o n e a s s u m e s

a . c l a m p e d e n d x = 0

u ( 0 t ) = 0 u

x

( 0 t ) = 0 ( 4 . 5 . 9 )


7/18/2019 C4029-Ch04



b . f r e e e n d x = l

u

x x

( l t ) = 0 u

x x x

( l t ) = 0 ( 4 . 5 . 1 0 )

c . s u p p o r t e d e n d x = l

u ( l t ) = 0 u

x x

( l t ) = 0 ( 4 . 5 . 1 1 )

d . f r e e e n d w i t h a x e d m a s s

m u

t t

( l t ) = E I u

x x x

( l t ) : ( 4 . 5 . 1 2 )

S i n c e t h e e q u a t i o n s c o n t a i n d e r i v a t i v e s w i t h r e s p e c t t o t i m e t , w e n e e d

i n i t i a l c o n d i t i o n s t o o . T h e y m a y b e

u ( x 0 ) = h ( x ) u

t

( x 0 ) = g ( x ) : ( 4 . 5 . 1 3 )

T o r s i o n a l o s c i l l a t i o n s o f a r o d w i t h o n e e n d x = 0 c l a m p e d u ( 0 t ) = 0 a n d

f r e e a t t h e o t h e r e n d x = l a r e d e s c r i b e d b y

u

x

( l t ) = 0 : ( 4 . 5 . 1 4 )

B u t i f a t t h e e n d x = l a b o d y w i t h m o m e n t o f i n e r t i a i s x e d , t h e n

G I u

x

( l t ) =

; u

t t

( l t ) ( 4 . 5 . 1 5 )

i s v a l i d . T h e m o r e c o m p l i c a t e d b o u n d a r y c o n d i t i o n s f o r a p l a t e w i l l b e d i s -

c u s s e d l a t e r .

L e t u s n o w d i s c u s s l o n g i t u d i n a l o s c i l l a t i o n s o f a r o d . U s i n g E = = c

2

,

e q u a t i o n ( 4 . 5 . 2 ) g e t s t h e s a m e f o r m a s t h e e q u a t i o n f o r a s t r i n g ( 4 . 3 . 2 0 ) . I f

b o t h e n d s o f t h e r o d a r e c l a m p e d , w e m a y i m m e d i a t e l y t a k e o v e r a l l s o l u t i o n s

o f ( 4 . 3 . 2 0 ) f o r t h e l o n g i t u d i n a l o s c i l l a t i o n s o f t h e r o d . T h e s a m e i s v a l i d f o r

t o r s i o n a l v i b r a t i o n s f o r c

2

= G = , c o m p a r e ( 4 . 5 . 6 ) a n d ( 4 . 3 . 3 ) . H o w e v e r f o r

t h e f r e e b e n d i n g v i b r a t i o n s o f t h e r o d d e s c r i b e d b y

1

c

2

@

2

u

@ t

2

+

@

4

u

@ x

4

= 0 c

2

=

E I

y

q

( 4 . 5 . 1 6 )

w h e r e c h a s t h e d i m e n s i o n m

2

= s a n d i s n o t a p h a s e v e l o c i t y , b u t i t s s e c o n d

p o w e r . W e n o w h a v e f o r t h e r s t t i m e a p a r t i a l d i e r e n t i a l e q u a t i o n o f f o u r t h

o r d e r . S e p a r a t i n g t i m e d e p e n d e n c e b y u ( x t ) = X ( x ) ( A s i n ! t + B c o s ! t ) ,

s e e s e c t i o n 4 . 3 , w e o b t a i n w i t h ! = c = k

X

0 0 0 0

( x )

; k

2

X ( x ) = 0 : ( 4 . 5 . 1 7 )

S i n c e t h i s e q u a t i o n i s h o m o g e n e o u s a n d h a s c o n s t a n t c o e c i e n t s w e m a y u s e

t h e a n s a t z X ( x ) = e x p (

p

! = c x ) t o g e t t h e g e n e r a l s o l u t i o n

X ( x ) = A

c o s

p

k x + c o s h

p

k x

+ B

c o s

p

k x ; c o s h

p

k x

+ C

s i n

p

k x + s i n h

p

k x

+ D

s i n

p

k x ; s i n h

p

k x

: ( 4 . 5 . 1 8 )


7/18/2019 C4029-Ch04



T a k i n g i n t o a c c o u n t t h e t w o b o u n d a r y c o n d i t i o n s f o r t h e c l a m p e d e n d x = 0

o n e g e t s A = 0 C = 0 a n d f o r t h e c l a m p e d e n d x = l

X ( l ) = B

c o s

p

k l

; c o s h

p

k l

+ D

s i n

p

k l

; s i n h

p

k l

= 0 ( 4 . 5 . 1 9 )

X

0

( l ) = ; k B

s i n

p

k l + s i n h

p

k l

+ k D

c o s

p

k l ; c o s h

p

k l

= 0 : ( 4 . 5 . 2 0 )

T h e s e t w o h o m o g e n e o u s l i n e a r e q u a t i o n s f o r t h e t w o u n k n o w n s B a n d D

p o s s e s s o n l y t h e n a n o n t r i v i a l s o l u t i o n i f t h e d e t e r m i n a n t o f t h e c o e c i e n t s

v a n i s h e s . T h i s c o n d i t i o n d e l i v e r s t h e e i g e n v a l u e e q u a t i o n

c o s

p

k l

c o s h

p

k l

; 1 = 0 ( 4 . 5 . 2 1 )

w h i c h h a s t h e z e r o s a t

p

k l = 4 : 7 3 7 : 8 5 . I f t h e r o d i s c l a m p e d a t o n l y o n e

e n d , b u t t h e o t h e r b e i n g s u p p o r t e d , t h e n

t a n h

p

k l ; t a n

p

k l = 0 ( 4 . 5 . 2 2 )

y i e l d s t h e e i g e n v a l u e . I f b o t h e n d s a r e s u p p o r t e d a n d n o f o r c e s a p p l i e d o n b o t h

e n d s , t h e n ! =

2

c = l

2

=

2

p

E I = q = l

2

. I n o r d e r t o s o l v e t h e i n h o m o g e n e o u s

e q u a t i o n ( 4 . 5 . 3 ) w i t h t h e i n i t i a l c o n d i t i o n s ( 4 . 5 . 1 3 ) , o n e e x p a n d s t h e s o l u t i o n

w i t h r e s p e c t t o t h e e i g e n f u n c t i o n s o f t h e h o m o g e n e o u s p r o b l e m u ( x t ) =

P

n

T

n

( t ) X

n

( x ) . I n s e r t i o n i n t o ( 4 . 5 . 3 ) a n d r e p l a c e m e n t o f X

0 0 0 0

f r o m ( 4 . 5 . 1 7 )

r e s u l t s i n t h e l i n e a r i n h o m o g e n e o u s o r d i n a r y d i e r e n t i a l e q u a t i o n

1

X

n = 1

;

T

0 0

n

( t ) + !

2

n

T

n

( t )

X

n

( x ) = f ( x t ) = q : ( 4 . 5 . 2 3 )

M u l t i p l i c a t i o n b y t h e o r t h o g o n a l X

m

( x ) a n d i n t e g r a t i o n f r o m x = 0 t o x = l

g i v e s

T

0 0

n

( t ) + !

2

n

T

n

( t ) =

R

l

0

f ( x t ) X

n

( x ) d x

q

R

l

0

X

2

n

( x ) d x

= b

n

( t ) : ( 4 . 5 . 2 4 )

T h e i n i t i a l c o n d i t i o n s ( 4 . 5 . 1 3 ) a r e t h e n s a t i s e d b y

u ( x 0 ) = h ( x ) =

1

P

n = 1

T

n

( 0 ) X

n

( x )

u

t

( x 0 ) = g ( x ) =

1

P

n = 1

T

0

n

( 0 ) X

n

( x )

( 4 . 5 . 2 5 )


7/18/2019 C4029-Ch04


7/18/2019 C4029-Ch04



1 . T h e b o u n d a r y o f t h e p l a t e i s c l a m p e d . N o m o t i o n w h a t e v e r i s p o s s i b l e .

T h e n

u ( b o u n d a r y ) = 0 ( 4 . 5 . 3 1 )

a n d

@ u

@ ~n

( b o u n d a r y ) = 0 ( 4 . 5 . 3 2 )

a r e v a l i d . ~n i s t h e n o r m a l u n i t v e c t o r o n t h e b o u n d a r y c u r v e .

2 . I f t h e b o u n d a r y i s s i m p l y i n c u m b e n t o n a s u p p o r t , t h e n o n e h a s

u ( b o u n d a r y ) = 0 ( 4 . 5 . 3 3 )

b u t ( 4 . 5 . 3 2 ) h a s t o b e r e p l a c e d b y

u + ( 1 ; )

@

2

u

@ x

2

c o s

2

# +

@

2

u

@ y

2

s i n

2

# + 2

@

2

u

@ x @ y

s i n # c o s #

= 0

( 4 . 5 . 3 4 )

w h e r e # i s t h e l o c a l a n g l e b e t w e e n t h e x a x i s a n d t h e n o r m a l v e c t o r ~n

o n t h e b o u n d a r y c u r v e . F o r a r e c t a n g u l a r p l a t e o n e h a s # = = 2 f o r a

b o u n d a r y p a r a l l e l t o t h e x a x i s a n d o t h e r w i s e # = 0 . T h e r e f o r e o n e h a s

u +

1

;

@

2

u

@ y

2

= 0 ( 4 . 5 . 3 5 )

f o r t h e p a r t o f t h e b o u n d a r y p a r a l l e l t o t h e x d i r e c t i o n a n d o t h e r w i s e

u +

1 ;

@

2

u

@ x

2

= 0 : ( 4 . 5 . 3 6 )

3 . I f t h e b o u n d a r y i s f r e e , i t m e a n s t h a t n e i t h e r f o r c e s n o r t o r q u e s a r e

e x e r t e d , t h e n ( 4 . 5 . 3 4 ) a n d

@

@ ~n

u + ( 1

; )

@

@ ~s

@

2

u

@ y

2

;

@

2

u

@ x

2

s i n # c o s # +

@

2

u

@ x @ y

( c o s

2

# ; s i n

2

# )

= 0 ( 4 . 5 . 3 7 )

a r e v a l i d . H e r e ~s i s t h e t a n g e n t i a l v e c t o r a l o n g t h e b o u n d a r y c u r v e .

A s e p a r a t i o n s e t u p u ( x y t ) = X ( x )

Y ( y )

T ( t ) a n d h o m o g e n e o u s b o u n d a r y

c o n d i t i o n s u ( x = a y t ) = 0 u ( x y = b t ) = 0 y i e l d t h e s o l u t i o n

u ( x y t ) =

1

X

n = 1

1

X

m = 1

s i n

n x

a

s i n

m y

b

( A

n m

c o s !

n m

t + B

n m

s i n !

n m

t ) ( 4 . 5 . 3 8 )


7/18/2019 C4029-Ch04



a n d t h e e i g e n v a l u e e q u a t i o n

!

2

n m

=

E h

2

4

3 ( 1 ;

2

)

n

2

a

2

+

m

2

b

2

2

: ( 4 . 5 . 3 9 )

T h e s o l u t i o n ( 4 . 5 . 3 8 ) i s c a l l e d N a v i e r s o l u t i o n a n d s a t i s e s t h e b o u n d a r y

c o n d i t i o n s u = 0 u = 0 a t t h e r e c t a n g u l a r b o u n d a r y . I t c a n b e u s e d t o s o l v e

t h e p r o b l e m o f a p l a t e u n d e r l o a d p ( x y ) .

F o r a c i r c u l a r p l a t e t h e s e p a r a t i o n s e t u p u ( r ' t ) = ( ' ) T ( t ) R ( r ) e n d s

u p w i t h t h e e q u a t i o n

d

2

d r

2

+

1

r

d

d r

;

n

2

r

2

+ k

2

d

2

d r

2

+

1

r

d

d r

;

n

2

r

2

; k

2

R ( r ) = 0 ( 4 . 5 . 4 0 )

a n d T ( t ) = C c o s t + D s i n t , w h e r e i s t h e s e p a r a t i o n p a r a m e t e r a n d

k

4

= 3 ( 1 ;

2

) = E h

2

.

O n e s e e s t h a t t h e p r o b l e m f o r t h e p l a t e i s a n a l y t i c a l l y m o r e d i c u l t t h a n

t h a t f o r t h e m e m b r a n e . I t i s n o t p o s s i b l e , f o r e x a m p l e , t o t r e a t t h e g e n e r a l

c a s e o f t h e r e c t a n g u l a r b o u n d a r y i n t e r m s o f f u n c t i o n s k n o w n e x p l i c i t l y . T h e

o n l y p l a t e b o u n d a r y p r o b l e m t h a t h a s b e e n e x p l i c i t l y t r e a t e d i s t h e c i r c u l a r

p l a t e .

F o r a x i a l s y m m e t r y , t h e s o l u t i o n o f t h i s e q u a t i o n i s g i v e n b y B e s s e l a n d

m o d i e d B e s s e l f u n c t i o n s

u ( r ) = A J

0

(

p

k r ) + B I

0

(

p

k r ) ( 4 . 5 . 4 1 )

w h e r e I

0

( k r ) = J

0

( i k r ) . W e n e e d t w o s o l u t i o n s s i n c e t w o b o u n d a r y c o n d i t i o n s

h a v e t o b e s a t i s e d . F o r a c l a m p e d c i r c u l a r p l a t e w i t h r a d i u s R , t h e b o u n d a r y

c o n d i t i o n s d e l i v e r u ( R ) = 0 u

0

( R ) = 0 s o t h a t w e h a v e

A J

0

(

p

k R ) + B I

0

(

p

k R ) = 0

A J

0

0

(

p

k R ) + B I

0

0

(

p

k R ) = 0 : ( 4 . 5 . 4 2 )

T o b e a b l e t o s o l v e t h e s e t w o h o m o g e n e o u s l i n e a r e q u a t i o n s f o r t h e u n k n o w n

p a r t i a l a m p l i t u d e s A a n d B , t h e d e t e r m i n a n t

J

0

(

p

k R ) I

0

0

(

p

k R ) ; J

0

0

(

p

k R ) I

0

(

p

k R ) = 0 ( 4 . 5 . 4 3 )

m u s t v a n i s h . U s i n g

d

d r

J

0

(

p

k r ) = ;

p

k J

1

(

p

k r ) ( 4 . 5 . 4 4 )

a n d a n a l o g o u s l y f o r I

0

t h e d e t e r m i n a n t d e t e r m i n e s t h e e i g e n v a l u e

k = 3 : 1 9 6 2 2 0 6 1 2 . F o r t h e i n i t i a l v a l u e s

u ( r 0 ) = h ( r ) u

t

( r 0 ) = g ( r ) ( 4 . 5 . 4 5 )


7/18/2019 C4029-Ch04



a n d t h e b o u n d a r y v a l u e p r o b l e m

u ( r = R t ) = 0 u

r

( r = R t ) = 0 ( 4 . 5 . 4 6 )

o n e o b t a i n s t h e s o l u t i o n 4 . 8 ]

u ( r t ) =

1

R

2

1

X

n = 1

Z

n

( r )

I

2

0

(

n

) J

2

0

(

n

)

2

4

c o s

2

n

c t

R

2

R

Z

0

h ( ) Z

n

( ) d

+

R

2

c

2

n

s i n

2

n

c t

R

2

R

Z

0

g ( ) Z

n

( ) d

3

5

( 4 . 5 . 4 7 )

w h e r e

Z

( r ) = J

0

r

R

I

0

( ) ; J

0

( ) I

0

r

R

( 4 . 5 . 4 8 )

a n d

n

r e p r e s e n t s t h e n - t h r o o t o f Z

0

( R ) = 0 . T o d e r i v e t h i s e x p r e s s i o n

R

Z

0

Z

2

n

( ) d =

R

6

4

Z

0 0 2

n

; Z

0

n

d

d r

1

r

d

d r

r

Z

n

d r

;

Z

0

n

Z

0 0

n

n

; Z

0

n

Z

0 0 0

n

r = R

= R

2

I

2

0

(

n

) J

2

0

(

n

) ( 4 . 5 . 4 9 )

h a s b e e n u s e d .

U p t o n o w w e t r e a t e d o n l y t i m e - d e p e n d e n t p r o b l e m s . B u t t h e r e i s a p r o b l e m

i n t h e t r a n s i t i o n t o @ = @ t = 0 . D u e t o t h e s u r f a c e b o u n d a r y c o n d i t i o n s o f

m a s s i v e e l a s t i c b o d i e s t h e s i m p l e t r a n s i t i o n @ = @ t = 0 i s n o t p o s s i b l e i n ( 4 . 5 . 3 ) .

T h e e q u a t i o n f o r t h e b e n d i n g o f a g i r d e r h a s t o b e d e r i v e d f r o m t h e b a s i c s .

T h e u s u a l a s s u m p t i o n s f o r t h e b e n d i n g o f a c a n t i l e v e r b e a m o f r e c t a n g u l a r

c r o s s s e c t i o n a b a r e

1 . T h e r e e x i s t s a \ n e u t r a l " c e n t r e l a y e r w i t h i n t h e b e a m t h a t w i l l n o t b e

e x p a n d e d ( e l a s t i c a x i s o r n e u t r a l l a m e n t ) , w h i c h i s d e s c r i b e d b y u ( z ) ,

i f t h e a x i s o f t h e b e a m c o i n c i d e s w i t h t h e z a x i s a n d t h e b e n d i n g i s i n t o

t h e n e g a t i v e x d i r e c t i o n , @ = @ y = 0 .

2 . T h e c r o s s s e c t i o n s i n t h e x y p l a n e a r e a s s u m e d t o c u t t h e e l a s t i c a x i s

a l o n g t h e z d i r e c t i o n p e r p e n d i c u l a r l y .

3 . T h e c r o s s s e c t i o n s w i l l n o t b e d e f o r m e d .

U n d e r t h e s e c o n d i t i o n s t h e d i e r e n t i a l e q u a t i o n f o r t h e e l a s t i c l i n e h a s b e e n

d e r i v e d

d

2

u

x

( z )

d z

2

=

P

E I

( z ; l ) : ( 4 . 5 . 5 0 )


7/18/2019 C4029-Ch04



P i s t h e l o a d a c t i n g a t t h e e n d o f t h e b e a m o f l e n g t h l . I n t e g r a t i o n y i e l d s

u

x

( x z ) =

P

E I

z

3

2 3

;

P l

E I

z

2

2

+ c

1

( x ) z + c

2

( x ) : ( 4 . 5 . 5 1 )

D u e t o t h e b o u n d a r y c o n d i t i o n ( c l a m p e d a t z = 0 )

u ( x 0 ) = 0 @ u ( x 0 ) = @ z = 0 : ( 4 . 5 . 5 2 )

T h i s r e s u l t s i n

u

x

( l ) = ;

P l

3

3 E I

= ;

4 P l

3

a

3

E b

: ( 4 . 5 . 5 3 )

I n t h i s d e r i v a t i o n ( 4 . 5 . 5 ) h a d b e e n u s e d . ( 4 . 5 . 5 3 ) d e n e s t h e p i t c h o f d e e c t i o n

s a g .

T h e b e n d i n g o f a g i r d e r i s a t r a n s v e r s e p r o b l e m . W e s h a l l n o w i n v e s t i g a t e

a l o n g i t u d i n a l p r o b l e m : t h e d e t e r m i n a t i o n o f t h e b u c k l i n g s t r e n g t h a n d t h e

c r i t i c a l c o m p r e s s i v e s t r e n g t h o f a p i l l a r . T h e l o n g i t u d i n a l c o m p r e s s i o n u ( x ) o f

a v e r t i c a l p i l l a r o f c r o s s s e c t i o n q , d e n s i t y , w e i g h t q g c a r r y i n g a l o a d P i s

d e s c r i b e d b y

E I u

0 0 0 0

( x ) + g q

d

d x

( l ; x ) u

0

( x ) ] = 0 ( 4 . 5 . 5 4 )

a n d t h e b o u n d a r y c o n d i t i o n s

u ( 0 ) = u

0

( 0 ) = 0 ( 4 . 5 . 5 5 )

w h i c h d e s c r i b e c l a m p i n g o f t h e l o w e r e n d o n e a r t h ' s s u r f a c e x = 0 a n d

u

0 0

( l ) = 0 u

0 0 0

( l ) = 0 ( 4 . 5 . 5 6 )

a t t h e l o a d f r e e u p p e r e n d . I f t h e r e i s a l o a d P a t x = l , o n e h a s

u

0 0

( l ) = 0 E I u

0 0 0

( l ) = P : ( 4 . 5 . 5 7 )

W e n o w r s t n e g l e c t a l o a d P a n d s o l v e

E I u

0 0 0

( x ) + g q ( l ; x ) u

0

( x ) = 0 ( 4 . 5 . 5 8 )

w h i c h i s o b t a i n e d b y i n t e g r a t i n g ( 4 . 5 . 5 4 ) a n d n e g l e c t i o n o f P a n d o f i n t e g r a t i o n

c o n s t a n t s d e p e n d i n g o n x . A t r a n s f o r m a t i o n u

0

( x ) = v ( x ) A = g q = E I y i e l d s

v

0 0

( x ) + A ( l ; x ) v ( x ) = 0 ( 4 . 5 . 5 9 )


7/18/2019 C4029-Ch04



a n d a s u b s t i t u t i o n = 2

p

A ( l ; x )

3 = 2

= 3 v = w ( )

p

r e s u l t s i n a B e s s e l

e q u a t i o n

w

0 0

( ) +

1

w

0

( ) +

1 ;

1

9

2

w ( ) = 0 ( 4 . 5 . 6 0 )

w i t h t h e s o l u t i o n

w ( ) = C

1

J

1 = 3

( ) + C

2

J

; 1 = 3

( ) : ( 4 . 5 . 6 1 )

T h e b o u n d a r y c o n d i t i o n s u

0 0

( l ) = 0 a n d u ( 0 ) = 0 r e s u l t i n C

1

= 0 C

2

= 0 s o

t h a t u = c o n s t i s a s t a b l e e q u i l i b r i u m i f

J

; 1 = 3

2

3

A

1 = 2

l

2 = 3

= 0 ( 4 . 5 . 6 2 )

i s N O T s a t i s e d . T h e r s t z e r o 1 . 8 7 o f J

; 1 = 3

g i v e s t h e b u c k l i n g l e n g t h l

c

o f

a p i l l a r w i t h n o l o a d

l

k

=

1 : 8 7 3

2

2 = 3

3

s

E I

g q

: ( 4 . 5 . 6 3 )

I f t h e w e i g h t i s n e g l e c t e d ( g = 0 ) , o n e o b t a i n s

l

k

=

2

r

E I

P

( 4 . 5 . 6 4 )

( E u l e r ' s c r i t i c a l l o a d ) . T h e f u l l e q u a t i o n ( 4 . 5 . 5 4 ) m a y b e s o l v e d b y n u m e r i c a l

o r a p p r o x i m a t i o n m e t h o d s , s e e l a t e r i n s e c t i o n 4 . 6 .

A c c o r d i n g t o ( 4 . 5 . 7 ) p l a t e s u n d e r a d i s t r i b u t e d s t a t i c l o a d p ( x y ) a r e d e -

s c r i b e d b y t h e b i h a r m o n i c o p e r a t o r

E h

2

3 ( 1 ;

2

)

u ( x y ) = p ( x y ) : ( 4 . 5 . 6 5 )

S o l u t i o n s o f t h e h o m o g e n e o u s e q u a t i o n ( 4 . 5 . 6 5 ) a r e c a l l e d b i h a r m o n i c . S u c h

f u n c t i o n s a r e x x

2

x

3

y y

2

y

3

x y x

2

y c o s h x c o s y e t c . I f v w a r e h a r -

m o n i c f u n c t i o n s , t h e n x v + w ( x

2

+ y

2

) v + w ( x

2

+ y

2

) l n

p

x

2

+ y

2

, e t c . , a r e

a l s o b i h a r m o n i c . T h e b i h a r m o n i c o p e r a t o r f o r t h e s t r e a m f u n c t i o n a p p e a r s

a l s o i n t h e t h e o r y o f t h e m o t i o n o f s m a l l b o d i e s w i t h i n a v i s c o u s u i d l i k e

b l o o d o r m u c u s . T h e p r o p a g a t i o n o f b a c t e r i a i n b l o o d o r o f s p e r m a c e l l s i n

t h e v i s c i d v a g i n a l m u c u s p r e s e n t s a n i n t e r e s t i n g b o u n d a r y p r o b l e m o f t h e b i -

h a r m o n i c o p e r a t o r 4 . 1 1 ] . T h e r e s e e m s t o e x i s t a s t a t i s t i c a l e v i d e n c e t h a t t h e

p r o b a b i l i t y o f a m a l e b a b y i s t h e h i g h e r , t h e q u i c k e r t h e s p e r m a t o z o o n i s a n d

t h e s h o r t e r t h e t i m e s p a n b e t w e e n o v u l a t i o n a n d f e r t i l i z a t i o n ( s e x d e t e r m i n a -

t i o n b y a b o u n d a r y c o n d i t i o n ) .


7/18/2019 C4029-Ch04



T h e i n h o m o g e n e o u s s t a t i c p l a t e e q u a t i o n ( 4 . 5 . 6 5 ) d e t e r m i n e s t h e d e e c t i o n

o f a p l a t e u n d e r a d i s t r i b u t e d l o a d p ( x y ) . W i t h e x c e p t i o n o f t h e N a v i e r

s o l u t i o n ( 4 . 5 . 3 8 ) , n o e x a c t c l o s e d a n a l y t i c s o l u t i o n s o f ( 4 . 5 . 6 5 ) f o r g e n e r a l

b o u n d a r y c o n d i t i o n s a r e k n o w n f o r r e c t a n g u l a r p l a t e s 4 . 1 0 ] . V a r i a t i o n a l c a l -

c u l u s h a s t o b e u s e d t o n d s o l u t i o n s . A c l o s e d s o l u t i o n o n l y e x i s t s f o r t h e

c i r c u l a r p l a t e . U s i n g a x i a l s y m m e t r y , t h e p l a t e e q u a t i o n r e a d s

u

0 0 0 0

( r ) +

2

r

u

0 0 0

( r ) ;

1

r

2

u

0 0

( r ) +

1

r

3

u

0

( r ) = p ( r )

3

;

1

;

2

E h

2

: ( 4 . 5 . 6 6 )

T o r s t s o l v e t h e h o m o g e n e o u s e q u a t i o n w e m a k e t h e s u b s t i t u t i o n r = e x p

s o t h a t

u

0 0 0 0

( ) ; 4 u

0 0 0

( ) + 4 u

0 0

( ) = 0 ( 4 . 5 . 6 7 )

r e s u l t s . T h e s o l u t i o n o f ( 4 . 5 . 6 1 ) i s t h e n

u ( r ) = c

1

+ c

2

l n r + c

3

r

2

+ c

4

r

2

l n r : ( 4 . 5 . 6 8 )

V a r i a t i o n o f p a r a m e t e r s ( s e e s e c t i o n 2 . 1 ) d e l i v e r s a g e n e r a l s o l u t i o n o f ( 4 . 5 . 6 6 )

i f t h e l o a d s a t i s e s i t s e l f t h e h o m o g e n e o u s e q u a t i o n . F o r c o n s t a n t l o a d p = p

0

a s p e c i a l s o l u t i o n i s g i v e n b y

u ( r ) = c

1

+ c

2

l n r + c

3

r

2

+ c

4

r

2

l n r + r

4

p

0

3 ( 1 ;

2

)

6 4 E h

2

: ( 4 . 5 . 6 9 )

T h i s s o l u t i o n m a y b e s p e c i e d b y t h e f o l l o w i n g b o u n d a r y c o n d i t i o n s

u ( 0 ) = n i t e u

0 0

( 0 ) = n i t e ( 4 . 5 . 7 0 )

a n d b o u n d a r y c l a m p e d a l l a r o u n d

u ( R ) = 0 u

0

( R ) = 0 : ( 4 . 5 . 7 1 )

T h e s e c o n d i t i o n s r e s u l t i n c

2

= 0 c

4

= 0 , s o t h a t t h e d e e c t i o n i s g i v e n b y

u ( r ) =

3 ( 1 ;

2

)

E h

2

p

0

;

R

4

; 2 R

2

r

2

+ r

4

= 6 4 : ( 4 . 5 . 7 2 )

A w a r n i n g s e e m s t o b e n e c e s s a r y . I n t h e s e d a y s o f c o m p u t e r e n t h u s i a s m

m a n y e n g i n e e r s c a l c u l a t e p l a t e s n u m e r i c a l l y . B u t s e v e r a l t i m e s i t h a d b e e n

o v e r s e e n t h a t a p i l l a r s u p p o r t i n g a p l a t e c o n s t i t u t e s a s i n g u l a r i t y a n d t h a t t h e

n o r m a l g r i d u s e d i n n u m e r i c a l c o m p u t a t i o n s s h o u l d h a v e b e e n e x t e n s i v e l y n a r -

r o w e d n e a r t h e p i l l a r . T h e c o n s e q u e n c e o f u s i n g b l a c k b o x c o m p u t e r r o u t i n e s

h a s b e e n a b r e a k d o w n o f a c e i l i n g , c a s e s o f d e a t h a n d l a w s u i t s o v e r m i l l i o n s

o f d o l l a r s .

I f d e e c t i o n s a r e c o m b i n e d w i t h w a r m i n g , t h e n t h e c o e c i e n t o f t h e r m a l

e x p a n s i o n e n t e r s i n t o t h e p l a t e e q u a t i o n 4 . 7 ] .


7/18/2019 C4029-Ch04


P r o b l e m s

1 . S o l v e t h e p r o b l e m d e s c r i b e d b y ( 4 . 5 . 1 7 ) t o ( 4 . 5 . 2 1 ) u s i n g M a t h e m a t i c a :

Clear[X];DSolve[X0 0 0 0 [x]-kˆ2*X[x]==0,X[x],x]

y i e l d s ( 4 . 5 . 1 8 ) i n t h e f o r m

X x ] ! e

;

p

k x

C 2 ] + e

p

k x

C 4 ] + C 1 ] C o s

p

k x ] + C 3 ] S i n

p

k x ]

T o b e a b l e t o h a n d l e t h e s o l u t i o n , w e r e d e n e b y c o p y a n d p a s t e

X[x_]= e

;

p

k x C[2]+e

p

k x C[4]

+C[1]Cosp

k x +C[3]Sin

p

k x ];

a n d d e r i v e i t

Y[x_]=D[X[x],x]

r e s u l t i n g i n

; e

;

p

k x

p

k C 2 ] + e

p

k x

p

k C 4 ] +

p

k C 3 ] C o s

p

k x ]

;

p

k C 1 ] S i n

p

k x ]

T h i s i s e q u i v a l e n t t o ( 4 . 5 . 1 8 ) . T h e n e x t s t e p i s t h e s a t i s f a c t i o n o f t h e

b o u n d a r y c o n d i t i o n s a t x = 0 . W e d o t h i s b y t y p i n g

X[0]==0

a n d o n e o b t a i n s a c o n d i t i o n f o r t h e i n t e g r a t i o n c o n s t a n t s

C 1 ] + C 2 ] + C 4 ] = = 0 o r

C[4]==-C[1]-C[2]

A n d t h e v a n i s h i n g o f t h e d e r i v a t i v e

Ya t x = 0 y i e l d s

Y[0]==0 o r C[3]==2*C[2]+C[1]

T h e s e e x p r e s s i o n s f o r

C[3]a n d

C[4]w i l l n o w b e i n s e r t e d i n t o

X[x]

( a n d l a t e r a l s o i n t o

Y[x]) . T h e r e s u l t i s a r e d e n i t i o n

Clear[X, a11, a12];

X[x_]=e

;

p

k x C[2]+e

p

k x *(-C[1]-C[2])+C[1]*

Cos[p

k x ]+(2*C[2]+C[1])*Sin[p

k x ];

T o b e a b l e t o l l t h e d e t e r m i n a n t l a t e r , w e h a v e t o f a c t o r o u t C[1] a n d

C[2]. T h i s i s d o n e b y

Collect[X[l],C[1]]( 4 . 5 . 7 3 )

y i e l d i n g

e

;

p

k l

C 2 ] ; e

p

k l

C 2 ] + 2 C 2 ] S i n

p

k l ]

+ C 1 ]

; e

p

k l

+ C o s

h

p

k l

i

+ S i n

p

k l ]

N o w o n e c a n r e a d o a n d d e n e

a11=(-ep

k l +Cos[p

k l

+Sin[p

k l

]);


7/18/2019 C4029-Ch04



a n d a n a l o g o u s l y

a12=(e;

p

k l -\Ep

k l +2 Sin[p

k l ]);

R e d e n i t i o n o f t h e d e r i v a t i v e

Yb y i n s e r t i n g f o r

C[3]a n d

C[4]r e s u l t s

i n

e

p

k x

p

k ( ; C 1 ] ; C 2 ] ) ; e

;

p

k x

p

k C 2 ] +

p

k ( C 1 ] + 2 C 2 ] ) C o s

p

k x ] ;

p

k C 1 ] S i n

p

k x ]

N o w o n e i s a b l e t o s a t i s f y t h e b o u n d a r y c o n d i t i o n ( 4 . 5 . 2 0 )

Collect[Y[l],C[1]]

y i e l d s t h e i n p u t

a21=

; e

p

k l

+ C o s

p

k l ] ; S i n

p

k l ]

a n d a n a l o g o u s l y

a22=

; e

;

p

k l

; e

p

k l

+ 2 C o s

p

k l ]

N o w w e h a v e a l l e l e m e n t s o f t h e m a t r i x M a n d d e n e

M={{a11,a12},{a21,a22}}/. Sqrt[k]*l->y ( 4 . 5 . 7 4 )

a n d a t t h e s a m e t i m e w e a r e r e p l a c i n g t h e a r g u m e n t

p

k l b y y .

W e c a n s i m p l i f y t h e e x p r e s s i o n a n d d e n e a n e w f u n c t i o n F ( y ) c o n t a i n -

i n g t h e d e t e r m i n a n t o f t h e m a t r i x M

Simplify[%];

Clear[F];F[y_]=Det[M];l=1.;( 4 . 5 . 7 5 )

N o w w e a r e a b l e t o v e r i f y ( 4 . 5 . 2 1 ) . T h e c o m m a n d

FindRoot[F[y]==0,{y,4.0}]

s t a r t s a t y = 4 : 0 a n d n d s 4 . 7 3 0 0 4 .

2 . A r o d i s c l a m p e d a t x = 0 a n d f r e e a t x = l . S o l v e t h e h o m o g e -

n e o u s e q u a t i o n ( 4 . 5 . 2 ) f o r l o n g i t u d i n a l o s c i l l a t i o n s u s i n g t h e b o u n d a r y

c o n d i t i o n s ( 4 . 5 . 8 ) u

x

( l t ) = 0 a n d ( 4 . 5 . 8 a ) u ( 0 t ) = 0 . D e r i v e t h e e i g e n -

f r e q u e n c i e s !

n

= ( 2 n ; 1 ) c = 2 l a n d t h e s o l u t i o n

u ( x t ) =

1

X

n = 1

s i n k

n

x ( A

n

c o s c k

n

t + B

n

s i n c k

n

t )

k

n

= ( 2 n

; 1 ) x = 2 l : ( 4 . 5 . 7 6 )

3 . I f a m a s s m i s x e d a t t h e e n d x = l o f a r o d , t h e n t h e b o u n d a r y

c o n d i t i o n i s m o d i e d a n d r e a d s m u

t t

( l t ) = ; E q u

x

( l t ) . I f t h e i n i -

t i a l c o n d i t i o n s a r e u ( x 0 ) = h ( x ) = h

0

x = l a n d u

t

( x 0 ) = 0 d e r i v e t h e

s o l u t i o n

u ( x t ) =

4 u

0

l

1

X

n = 1

c o s ( c k

n

t )

s i n k

n

l s i n k

n

x

k

n

( 2 k

n

l + s i n 2 k

n

l )

( 4 . 5 . 7 7 )


7/18/2019 C4029-Ch04



a n d t h e e i g e n v a l u e e q u a t i o n i s

c o t k l = m k = q : ( 4 . 5 . 7 8 )

4 . I n v e s t i g a t e t h e l o n g i t u d i n a l o s c i l l a t i o n s o f a r o d o f l e n g t h l f o r t h e i n i t i a l

c o n d i t i o n s u ( x 0 ) = h ( x ) u

t

( x 0 ) = g ( x ) , i f t h e e n d x = 0 i s c l a m p e d

a n d t h e e n d x = l i s f r e e 4 . 8 ] . T h e s o l u t i o n i s

u ( x t ) =

2

l

1

X

n = 0

2

4

c o s

2 n + 1

2 l

c t

l

Z

0

h ( ) s i n

2 n + 1

2 l

d +

2 l

( 2 n + 1 ) c

s i n

2 n + 1

2 l

c t

l

R

0

g ( ) s i n

2 n + 1

2 l

d

#

s i n

2 n + 1

2 l

x

: ( 4 . 5 . 7 9 )

5 . A b r i d g e s u p p o r t e d a t b o t h e n d s x = 0 x = l i s l o a d e d b y a f o r c e P

t h a t c r o s s e s t h e b r i d g e w i t h c o n s t a n t s p e e d v

0

. T h e l o a d i n g f u n c t i o n f

i s a s s u m e d t o a c t w i t h i n t h e s m a l l i n t e r v a l "

f ( x t ) =

P = " v

0

t x v

0

t + "

0 0 x < v

0

t v

0

t + " l

a n d

l

Z

0

f ( x t ) d t =

v

0

t + "

Z

v

0

t

P = " d x = P :

T h e t r a n s v e r s e d e e c t i o n ( v e r t i c a l t o t h e b r i d g e i n x d i r e c t i o n ) i s t h e n

g i v e n b y

u ( x t ) =

2 P

q l

1

X

n = 1

s i n ( n x = l )

!

2

n

; !

2

n

( s i n !

n

t ; !

n

s i n ( !

n

t ) = !

n

) ) ( 4 . 5 . 8 0 )

w h e r e

!

n

=

n

2

2

l

2

s

E I

q

: ( 4 . 5 . 8 1 )

I f o n e o f t h e e x c i t i n g f r e q u e n c i e s ! = n v

0

= l i s n e a r o r e q u a l t o t h e

e i g e n f r e q u e n c i e s !

n

o f t h e b r i d g e , r e s o n a n c e a n d b r e a k a g e w i l l o c c u r .

6 . D e r i v e ( 4 . 5 . 6 4 ) b y s o l v i n g ( 4 . 5 . 5 4 ) f o r g = 0 a n d c a l c u l a t e t h e b u c k l i n g

l e n g t h l

c

o f a p i l l a r a c c o r d i n g t o ( 4 . 5 . 6 3 ) , ( 4 . 5 . 6 4 ) . A s s u m e t h e d i m e n -

s i o n s a = 0 : 1 m , b = 0 : 1 2 m , u s e ( 4 . 5 . 5 ) , q = a b = 2 : 5 k N / m

3

E = 3 0

G N / m

2

( r e i n f o r c e d c o n c r e t e ) a n d v a r y P i n ( 4 . 5 . 6 4 ) .

7 . S o l v e t h e p l a t e e q u a t i o n ( 4 . 5 . 7 ) f o r a t i m e d e p e n d e n c e

e x p ( i ! t ) . U s e

k

4

= !

2

3 ( 1 ;

2

) = E h

2

. N o b o u n d a r y o r i n i t i a l c o n d i t i o n s a r e g i v e n .

S p l i t t h e o p e r a t o r ; k

4

i n t o ( + k

2

) ( ; k

2

) a n d s o l v e u + k

2

u =

0 u ; k

2

u = 0 . V e r i f y t h e s o l u t i o n , w h i c h h a s b e e n p r e p a r e d t o b e


7/18/2019 C4029-Ch04



u s e d b y a c o l l o c a t i o n p r o c e d u r e ( s e c t i o n 4 . 8 ) . R e c a l l e q u a t i o n ( 1 . 1 . 5 5 )

a n d n d a n a n a l o g o u s s o l u t i o n o f ( 4 . 5 . 7 )

u ( x y ) =

X

u

h

A

n

c o s

p

k

2

; b

2

n

x

c o s ( b

n

y )

+ B

n

c o s h

p

k

2

; b

2

n

x

c o s h ( b

n

y )

i

: ( 4 . 5 . 8 2 )

V e r i f y t h i s s o l u t i o n b y i n s e r t i n g i t i n t o t h e p l a t e e q u a t i o n .

8 . A c c o r d i n g t o ( 1 . 4 . 1 6 ) t h e f r e e v i b r a t i o n s o f a p l a t e w i t h v a r y i n g t h i c k n e s s

h ( x ) a r e d e s c r i b e d b y

E h

2

1 2 ( 1 ;

2

)

u ( x y t ) +

o

u

t t

+

E

1 2 ( 1 ;

2

)

6 h h

x

( u

x x x

+ u

x y y

) ]

+ 3 u

x x

( 2 h

2

x

+ h h

x x

) + 3 u

y y

( 2 h

2

x

+ h h

x x

)

= 0 : ( 4 . 5 . 8 3 )

A s s u m e h ( x ) = a x + b a n d a t i m e d e p e n d e n c e c o s ( ! t ) a n d t r y t o s o l v e

( 4 . 5 . 8 3 ) .

4 . 6 A p p r o x i m a t i o n m e t h o d s

I n p r a c t i c a l a p p l i c a t i o n s i n p h y s i c s a n d e n g i n e e r i n g , m a n y p r o b l e m s a r i s e t h a t

c a n n o t b e s o l v e d a n a l y t i c a l l y . A n i m m e d i a t e u s e o f a b l a c k b o x n u m e r i c a l

r o u t i n e m a y g i v e n o s a t i s f y i n g r e s u l t s o r m a y e v e n b r i n g a b o u t t h e b r e a k d o w n

o f a c o n s t r u c t i o n . I t i s a l w a y s a d v a n t a g e o u s t o d i s p o s e o f a n a p p r o x i m a t e

a n a l y t i c e x p r e s s i o n . I t c a n b e o b t a i n e d b y :

1 . c o l l o c a t i o n i n s e c t i o n 1 . 3 , e q u a t i o n ( 1 . 3 . 1 0 ) ,

2 . s u c c e s s i v e a p p r o x i m a t i o n i n s e c t i o n 2 . 7 , e q u a t i o n ( 2 . 7 . 2 8 ) ,

3 . a v e r a g i n g m e t h o d i n s e c t i o n 2 . 7 , e q u a t i o n ( 2 . 7 . 3 1 ) ,

4 . m u l t i p l e t i m e s c a l e s i n s e c t i o n 2 . 7 , e q u a t i o n ( 2 . 7 . 5 4 ) ,

5 . W K B m e t h o d i n s e c t i o n 2 . 7 , e q u a t i o n ( 2 . 7 . 6 4 ) ,

6 . t h e m o m e n t m e t h o d i n s e c t i o n 3 . 2 , e q u a t i o n ( 3 . 2 . 4 4 ) ,

7 . e x p a n s i o n o f t h e s o l u t i o n i n t o a p o w e r s e r i e s , e q u a t i o n ( 4 . 6 . 6 1 ) .

I n m a n y p r a c t i c a l p r o b l e m s d e s c r i b e d b y p a r t i a l d i e r e n t i a l e q u a t i o n s , a

s m a l l p a r a m e t e r " < < 1 , t h e s o - c a l l e d p e r t u r b a t i o n t e r m , a p p e a r s . L e t u s


7/18/2019 C4029-Ch04


A p p r o x i m a t i o n m e t h o d s 2 0 3

c o n s i d e r a n e x a m p l e . W e a s s u m e p ( x ) = 1 a n d ( x ) = 1 + " ( x ) i n ( 4 . 3 . 3 4 )

a n d o b t a i n f o r t h e i n h o m o g e n e o u s s t r i n g

@

2

u ( x t )

@ x

2

=

1 + " ( x )

c

2

@

2

u ( x t )

@ t

2

: ( 4 . 6 . 1 )

I f t h e s t r i n g i s c l a m p e d a t b o t h e n d s x = 0 a n d x = l o n e m a y e x p e c t a d i s c r e t e

e i g e n v a l u e s p e c t r u m !

n

l i k e i n s e c t i o n 4 . 3 , e q u a t i o n ( 4 . 3 . 1 6 ) . T o s o l v e ( 4 . 6 . 1 )

w e a s s u m e

u ( x t ) = f ( x ) c o s ( ! t + ' ) ( 4 . 6 . 2 )

a n d o b t a i n

d

2

f ( x )

d x

2

= ;

!

2

c

2

( 1 + " ( x ) ) f ( x ) : ( 4 . 6 . 3 )

S i n c e w e k n o w t h e s o l u t i o n ( 4 . 3 . 1 5 ) f o r " = 0 , w e s e t u p

f ( x ) = s i n

n x

l

+ g ( x ) ! = !

n

( 1 + ) : ( 4 . 6 . 4 )

S i n c e t h e p e r t u r b a t i o n s " g a r e c o n s i d e r e d t o b e s m a l l , o n e m a y n e g l e c t

" g

2

g " a n d h i g h e r o r d e r s . I n s e r t i o n o f ( 4 . 6 . 4 ) i n t o ( 4 . 6 . 3 ) r e s u l t s i n

g

0 0

( x ) +

!

2

n

c

2

g = ;

n

2

2

l

2

" ( x ) + 2 ] s i n

n x

l

: ( 4 . 6 . 5 )

E x p a n d i n g g ( x ) i n t o a s e r i e s o f t h e e i g e n f u n c t i o n s o f t h e u n p e r t u r b e d p r o b l e m ,

( s a t i s f y i n g t h e b o u n d a r y c o n d i t i o n s )

g ( x ) =

1

X

= 1

a

s i n

x

l

( 4 . 6 . 6 )

a n d i n s e r t i n g i n t o ( 4 . 6 . 5 ) y i e l d s

1

X

= 1

!

2

n

;

2

2

c

2

l

2

a

s i n

x

l

= ; !

2

n

" ( x ) + 2 ] s i n

n x

l

: ( 4 . 6 . 7 )

I f o n e m u l t i p l i e s b y s i n ( n x = l ) , i n t e g r a t e s o v e r 0 t o l a n d t a k e s o r t h o g o n a l i t y

f o r 6= n i n t o a c c o u n t , o n e o b t a i n s

= ;

1

l

l

Z

0

" ( x ) s i n

2

n x

l

d x : ( 4 . 6 . 8 )

I n t h i s d e r i v a t i o n

1

l

l

Z

0

s i n

2

n x

l

d x =

1

2

( 4 . 6 . 9 )


7/18/2019 C4029-Ch04



h a s b e e n u s e d . M u l t i p l i c a t i o n o f ( 4 . 6 . 7 ) b y s i n ( m x = l ) a n d i n t e g r a t i o n y i e l d s

f o r m 6= n t h e e x p a n s i o n c o e c i e n t s a

m

o f ( 4 . 6 . 7 )

a

m

=

!

2

n

!

2

m

; !

2

n

2

l

l

Z

0

" ( x ) s i n

n x

l

s i n

m x

l

d x : ( 4 . 6 . 1 0 )

W e t h u s h a v e t h e s o l u t i o n o f ( 4 . 6 . 1 ) f o r a d i s c r e t e s p e c t r u m . F o r t h e i n v e s t i -

g a t i o n o f t h e c o n t i n u o u s s p e c t r u m w e r e p l a c e t h e s t a n d i n g w a v e s ( 4 . 6 . 4 ) b y

t r a v e l l i n g w a v e s

u ( x t ) = A e

i k ( x ; c t )

+ u

1

( x t ) = u

0

+ u

1

( 4 . 6 . 1 1 )

a n d r e c e i v e b y i n s e r t i o n a n d n e g l e c t i o n o f h i g h e r o r d e r t e r m s " u

1

0

@

2

u

1

@ x

2

;

1

c

2

@

2

u

1

@ t

2

= ; " A k

2

e

i k ( x ; c t )

"

c

2

@

2

u

0

@ t

2

: ( 4 . 6 . 1 2 )

T h i s e q u a t i o n c a n b e s o l v e d b y

u

1

( x t ) = v ( x ) e

; i k c t

( 4 . 6 . 1 3 )

l e a d i n g t o

v

0 0

+ k

2

v = ; " A k

2

e

i k x

: ( 4 . 6 . 1 4 )

F o r h i g h e r a c c u r a c y o n e c a n u s e m o r e t e r m s : u = u

0

+ u

1

+ u

2

: : : . T h i s w o r k s

a l s o f o r m o r e d i m e n s i o n s i n ( x y t )

@

2

u

@ x

2

+

@

2

u

@ y

2

;

1

c

2

@

2

u

@ t

2

=

" ( x y )

c

2

@

2

u

; 1

@ t

2

: ( 4 . 6 . 1 5 )

F o r t i m e - d e p e n d e n t p e r t u r b a t i o n s o n e c a n u s e

u ( x t ) =

1

X

= 1

a

( t ) s i n

x

l

: ( 4 . 6 . 1 6 )

W h a t h a p p e n s i f w e h a v e a d e g e n e r a t e e i g e n v a l u e p r o b l e m l i k e i n s e c t i o n 3 . 4 ?

W e s h a l l s e e t h a t t h e p e r t u r b a t i o n a b o l i s h e s t h e d e g e n e r a t i o n . W e i n v e s t i g a t e

a s q u a r e m e m b r a n e o f l e n g t h a w i t h t h e u n p e r t u r b e d s o l u t i o n

u

m n

( x y t ) = A

m n

s i n

m x

a

s i n

n y

a

c o s ( !

m n

t ; '

m n

) : ( 4 . 6 . 1 7 )

T o s o l v e t h e p e r t u r b e d m e m b r a n e e q u a t i o n ( 4 . 6 . 1 5 ) w e s e t u p

u ( x y t ) =

h

s i n

m x

a

s i n

n y

a

+ s i n

n x

a

s i n

m y

a

+ g ( x y )

i

c o s ( ! t ; ' ) : ( 4 . 6 . 1 8 )

H e r e

! = !

m n

( 1 + ) !

m n

=

c

a

p

m

2

+ n

2

( 4 . 6 . 1 9 )


7/18/2019 C4029-Ch04



i s t h e p e r t u r b e d e i g e n v a l u e . I n s e r t i o n a n d n e g l e c t i o n o f h i g h e r o r d e r s y i e l d

g +

!

2

m n

c

2

g = ;

!

2

m n

c

2

( " + 2 )

s i n

m x

a

s i n

n y

a

+ s i n

m x

a

s i n

n y

a

: ( 4 . 6 . 2 0 )

T o s o l v e t h i s e q u a t i o n w e s e t u p

g ( x y ) =

1

X

= 1

1

X

= 1

a

s i n

x

a

s i n

y

a

: ( 4 . 6 . 2 1 )

M u l t i p l y i n g ( 4 . 6 . 2 0 ) b y

s i n

m x

a

s i n

n y

a

o r s i n

n x

a

s i n

m y

a

a n d i n t e g r a t i o n r e s u l t i n

( "

m n

+ 2 ) +

m n

= 0

m n

+ ( "

m n

+ 2 ) = 0 : ( 4 . 6 . 2 2 )

H e r e t h e a b b r e v i a t i o n s

"

m n

=

4

a

2

a

Z

0

a

Z

0

" ( x y ) s i n

2

m x

a

s i n

2

n y

a

d x d y

m n

=

4

a

2

a

Z

0

a

Z

0

" ( x y ) s i n

m x

a

s i n

n x

a

s i n

m y

a

s i n

n y

a

d x d y

h a d b e e n u s e d . ( 4 . 6 . 2 2 ) i s a h o m o g e n e o u s s y s t e m o f t w o l i n e a r e q u a t i o n s f o r

t h e u n k n o w n p a r t i a l a m p l i t u d e s a n d . T o s o l v e t h i s s y s t e m , t h e d e t e r -

m i n a n t o f t h e c o e c i e n t s m u s t v a n i s h . W e t h u s o b t a i n f o r t h e e i g e n v a l u e

p e r t u r b a t i o n

1 2

=

1

4

; ( "

m n

+ "

n m

)

q

( "

m n

; "

n m

)

2

+ 4

2

m n

: ( 4 . 6 . 2 3 )

T h i s r e s u l t i s i n t e r e s t i n g w e h a v e t w o s o l u t i o n s ! = !

m n

+

1

! = !

m n

+

2

,

t h e d e g e n e r a t i o n ( o n e e i g e n v a l u e b e l o n g i n g t o t w o e i g e n f u n c t i o n s ) i s s u s -

p e n d e d b y t h e p e r t u r b a t i o n .

T h e p r o c e d u r e t h a t w e j u s t d i s c u s s e d m a y b e f o r m a l i z e d . L e t

L f u g + " S f u g = N f u g ( 4 . 6 . 2 4 )

b e a l i n e a r s e l f - a d j o i n t d i e r e n t i a l o p e r a t o r e q u a t i o n s u b j e c t e d t o a n h o m o g e -

n e o u s b o u n d a r y c o n d i t i o n R f u g = 0 . T h e n

k

=

0 k

+ "

1 k

+ "

2

2 k

+ : : : '

k

= '

0 k

+ " '

1 k

+ "

2

'

2 k

+ : : : ( 4 . 6 . 2 5 )


7/18/2019 C4029-Ch04



i s u s e f u l . F o l l o w i n g t h e p r o c e d u r e d e s c r i b e d a b o v e a n d u s i n g t h e o r t h o g o n a l i t y

o f t h e e i g e n f u n c t i o n s

b

Z

a

'

0 k

1 k

N f '

0 k

g ; S f '

0 k

g ] d x = 0 ( 4 . 6 . 2 6 )

t h e r s t p e r t u r b a t i o n o f t h e e i g e n v a l u e i s g i v e n b y

1 k

=

b

R

a

'

0 k

S f '

0 k

g d x

b

R

a

'

0 k

N f '

0 k

g d x

( 4 . 6 . 2 7 )

s o t h a t t h e p e r t u r b e d e i g e n v a l u e i s g i v e n b y

k

0 k

+ "

1 k

( 4 . 6 . 2 8 )

a n d t h e e i g e n f u n c t i o n s o f r s t o r d e r a r e d e t e r m i n e d b y

L f '

1 k

g ;

0 k

N f '

1 k

g =

1 k

N f '

0 k

g ; S f '

0 k

g

R f '

1 k

g = 0 : ( 4 . 6 . 2 9 )

I n s e c t i o n 4 . 5 w e t r e a t e d a p i l l a r u n d e r l o a d P a n d d e f o r m e d b y i t s o w n

w e i g h t q g . T h e p r o b l e m d e s c r i b e d b y ( 4 . 5 . 5 4 ) h a d b e e n s o l v e d f o r t w o c a s e s :

1 . n o l o a d P . E q u a t i o n ( 4 . 5 . 5 8 ) h a d b e e n s o l v e d a n d t h e c r i t i c a l l e n g t h

( 4 . 5 . 6 3 ) h a d b e e n d e r i v e d ,

2 . n o w e i g h t , b u t w i t h a l o a d P , s e e ( 4 . 5 . 6 4 ) a n d p r o b l e m 6 o f s e c t i o n 4 . 5 .

N o w w e s h a l l h a n d l e t h e f u l l p r o b l e m ( 4 . 5 . 5 4 ) w i t h t h e b o u n d a r y c o n d i t i o n s

( 4 . 5 . 5 5 ) a n d ( 4 . 5 . 5 7 ) . W e r e w r i t e ( 4 . 5 . 5 4 ) i n t h e f o r m

u

0 0 0 0

+ " ( l ; x ) u

0 0

; x u

0

] + u

0 0

= 0 ( 4 . 6 . 3 0 )

w h e r e

" = g q = E I = P = E I ( 4 . 6 . 3 1 )

d e n o t e t h e p e r t u r b a t i o n p a r a m e t e r a n d t h e e i g e n v a l u e r e s p e c t i v e l y . F i r s t w e

s o l v e t h e u n p e r t u r b e d e q u a t i o n

u

0 0 0 0

=

; u

0 0

( 4 . 6 . 3 2 )

w i t h t h e f o u r h o m o g e n e o u s b o u n d a r y c o n d i t i o n s ( 4 . 5 . 5 5 ) a n d ( 4 . 5 . 5 6 ) . W e

o b t a i n

u

0

( x ) = 1 ; c o s

r

4 n

2

l

2

x ( 4 . 6 . 3 3 )


7/18/2019 C4029-Ch04



a n d

0 n

= 4 n

2

= l

2

n = 1 2 : : : : ( 4 . 6 . 3 4 )

F i n a l l y , ( 4 . 6 . 2 7 ) y i e l d s

n

= l = 2

n

4 n

2

= l

2

+ " l = 2 ( 4 . 6 . 3 5 )

a n d

P =

4

2

l

2

E I ;

g q l

2

( 4 . 6 . 3 6 )

f o r E u l e r ' s c r i t i c a l l o a d . T h e b u c k l i n g l e n g t h i s n o w g i v e n b y

l =

3

s

8

2

E I

g q

4 : 3

3

s

E I

g q

( 4 . 6 . 3 7 )

c o m p a r e ( 4 . 5 . 6 3 ) .

P e r t u r b a t i o n t h e o r y h a s a l s o b e e n a p p l i e d t o i n v e s t i g a t e s m a l l a l t e r a t i o n s

o f t h e b o u n d a r y ( F e e n b e r g m e t h o d ) o r o f t h e b o u n d a r y c o n d i t i o n s 1 . 2 ] .

P r o b l e m s o f t h i s k i n d a r e h a n d l e d t o d a y b y n u m e r i c a l m e t h o d s . I n t h e

e x a m p l e s t r e a t e d h e r e , w e a s s u m e d t h a t t h e l i m i t " ! 0 d o e s n o t c o n s i d e r a b l y

m o d i f y t h e d i e r e n t i a l e q u a t i o n . T h i s i s n o t t h e c a s e f o r s i n g u l a r p e r t u r b a t i o n

p r o b l e m s . C o n s i d e r

; "

2

u

0 0

+ c ( x u ) = 0 u ( 0 ) = 0 u ( 1 ) = 0 ( 4 . 6 . 3 8 )

" ( x y ) ;

x

( x y ) = s i n y : ( 4 . 6 . 3 9 )

T h i s e q u a t i o n d e s c r i b e s a v i s c o u s p l a n e o w 3 . 9 ] . P r o b l e m s o f t h i s t y p e b e -

c o m e e v e n m o r e c o m p l i c a t e d , i f t h e b o u n d a r y c u r v e d o e s n o t c o i n c i d e w i t h

c u r v e s o f a c o o r d i n a t e s y s t e m i n w h i c h t h e p e r t i n e n t p a r t i a l d i e r e n t i a l e q u a -

t i o n i s s e p a r a b l e . T h u s , t h e b o u n d a r y c o n d i t i o n s m a y r e a d

= 0 @ = @ x = 0 ( 4 . 6 . 4 0 )

f o r a b o u n d a r y d e s c r i b e d b y y = a x 0

x

x

1

a n d a l s o f o r a b o u n d a r y

y = b x + c x

0

x

x

1

. F u r t h e r m o r e

= 0 @

2

= @ y

2

= 0 f o r y = 0 : ( 4 . 6 . 4 1 )

T h e u n p e r t u r b e d s o l u t i o n o f ( 4 . 6 . 3 8 ) i s n o w g i v e n b y

0

( x y ) =

; x s i n y + f ( y ) s i n y : ( 4 . 6 . 4 2 )

T h e b a s i c i d e a i s n o w t h e a s s u m p t i o n o f a n i n n e r a n d o u t e r s o l u t i o n . T h e

o u t e r s o l u t i o n i s a s s u m e d t o b e r e s p o n s i b l e f o r l o c a t i o n s f a r o u t s i d e o f t h e

c l o s e d t r i a n g u l a r i n n e r d o m a i n d e t e r m i n e d b y ( 4 . 6 . 4 0 ) a n d ( 4 . 6 . 4 1 ) . T h e i n n e r

s o l u t i o n h a s t o d e s c r i b e t h e b e h a v i o r w i t h i n t h e c l o s e d d o m a i n . T o s a t i s f y

t h e a s s u m p t i o n t h a t t h e o u t e r s o l u t i o n d e c r e a s e s w i t h i n c r e a s i n g d i s t a n c e


7/18/2019 C4029-Ch04



n o r m a l t o t h e t h r e e b o u n d a r i e s , i t i s o f a d v a n t a g e t o i n t r o d u c e c o o r d i n a t e s

o r t h o g o n a l t o t h e b o u n d a r i e s . A s a n e x a m p l e , w e c o n s i d e r t h e l i n e y = a x .

T h e n t h i s l i n e s h o u l d b e v e r t i c a l t o t h e a x i s a n d i t s n o r m a l i s d e n e d b y

= "

n

( y ; a x ) = y '

1

= '

1

( ) : ( 4 . 6 . 4 3 )

m u s t v a n i s h a t t h e b o u n d a r y y = a x . T h e n t h e p e r t i n e n t b o u n d a r y c o n d i t i o n

r e a d s

0

( a y y ) + '

1

( 0 ) = 0

@

0

( a y y )

@ x

+ "

n

@ '

1

( 0 )

@

= 0 : ( 4 . 6 . 4 4 )

T h e n ( 4 . 6 . 3 9 ) r e a d s

;

1 + a

2

2

"

4 n + 1

'

1

; "

n

'

1

= 0 : ( 4 . 6 . 4 5 )

F o r t h e c o n t r i b u t i o n s o f t h e t w o l e a d i n g t e r m s t o b e o f t h e s a m e o r d e r o f

m a g n i t u d e i n " , o n e m u s t h a v e 4 n + 1 = n n = ; 1 = 3 a n d ( 4 . 6 . 4 5 ) b e c o m e s

;

1 + a

2

2

'

1

; '

1

= 0 : ( 4 . 6 . 4 6 )

T h i s m a y b e s o l v e d b y a n e x p a n s i o n w i t h r e s p e c t t o p o w e r s "

1 = 3

'

1

= G

1

( ) e x p

h

e

2 i = 3

i

+ G

2

( ) e x p

h

e

4 i = 3

i

( 4 . 6 . 4 7 )

w i t h = ( 1 + a

2

)

; 3 = 2

. I f t h e o t h e r t w o b o u n d a r i e s a r e t r e a t e d a n a l o g o u s l y

t h e s o l u t i o n t o ( 4 . 6 . 3 9 ) r e a d s 3 . 9 ]

( x y ) =

; x + f ( y ) ] s i n y + G

1

( y ) e x p

h

e

2 i = 3

i

+ G

2

( y ) e x p

h

e

4 i = 3

i

+ G

3

( y ) e x p

h

~

i

( 4 . 6 . 4 8 )

~

= "

n

( x + x

0

; a y ) : ( 4 . 6 . 4 9 )

N o w G

1

G

2

G

3

h a v e t o b e c h o s e n t o s a t i s f y a l l b o u n d a r y c o n d i t i o n s f o r

( x y ) .

O t h e r a u t h o r s p r e f e r t o s o l v e s i n g u l a r p e r t u r b a t i o n p r o b l e m s u s i n g s p l i n e

m e t h o d s 4 . 1 2 ] . T h e L i e s e r i e s m e t h o d m e n t i o n e d i n ( 2 . 5 . 4 4 ) a n d ( 2 . 7 . 6 8 )

r e p r e s e n t s a n o t h e r p e r t u r b a t i o n m e t h o d . I n t h e N A S A r e p o r t 2 . 3 ] a l l d e t a i l s

a r e g i v e n o n t h e s o l u t i o n o f t h e t h r e e b o d y p r o b l e m S u n , J u p i t e r a n d 8 t h

M o o n o f J u p i t e r . I t c o u l d b e d e m o n s t r a t e d t h a t p e r t u r b a t i o n m e t h o d s b a s e d

o n ( 2 . 7 . 6 8 ) c o n v e r g e v e r y r a p i d l y 4 . 1 3 ] , 2 . 1 1 ] . T h e p e r t u r b a t i o n m e t h o d

a p p l i e d o n t h e J u p i t e r M o o n h a s b e e n t e s t e d b y t h e p r o g r a m L I E S E 2 . 4 ]

a n d t h e r e s u l t s o b t a i n e d h a d b e e n c o m p a r e d b y K o v a l e v s k y , P a r i s , t o t h e

r e s u l t s o b t a i n e d b y t h e C o w e l l m e t h o d . C o m p u t a t i o n s w i t h a n I B M 6 5 0 g a v e

d e v i a t i o n s i n t h e c o o r d i n a t e s a n d v e l o c i t i e s , o b t a i n e d w i t h a c a l c u l a t i n g s t e p

t = 5 d f o r 1 0 0 d a y s f o r w a r d a n d b a c k w a r d , l e s s t h a n 5 0 1 0

; 1 0

L ( L =


7/18/2019 C4029-Ch04



a s t r o n o m i c a l u n i t 1 4 9 5 . 0 4 2 0 1 1 0

1 0

c m ) . W i t h t h e L i e m e t h o d c a l c u l a t i o n s

f o r a g a i n 1 0 0 d a y s f o r w a r d a n d b a c k w a r d w i t h a s t e p t = 1 d , t h e l o c a l

d e v i a t i o n h a s b e e n 1 5 1 0

; 1 0

L i n A a c h e n , G e r m a n y , o n a S I E 2 0 0 2 c o m p u t e r

a n d h a s l a t e r b e e n i m p r o v e d a t t h e U n i v e r s i t y o f W i s c o n s i n .

P r o b l e m s

1 . W e k n o w t h a t t h e s o l u t i o n s o f t h e m e m b r a n e e q u a t i o n a r e d e s c r i b e d i n

a n u n a m b i g u o u s w a y b y t w o h o m o g e n e o u s b o u n d a r y c o n d i t i o n s . N o w

t h e q u e s t i o n m a y a r i s e i f p e r t u r b a t i o n t h e o r y a d m i t s a t h i r d b o u n d a r y

c o n d i t i o n . L e t u s i n v e s t i g a t e i f t h i s i s p o s s i b l e . W e a s s u m e a s q u a r e

m e m b r a n e o f l a t e r a l l e n g t h a a n d s i t u a t e d i n s u c h a m a n n e r t h a t t h e

p o i n t o f i n t e r s e c t i o n o f t h e t w o d i a g o n a l s c o i n c i d e w i t h t h e o r i g i n o f

t h e c a r t e s i a n c o o r d i n a t e s y s t e m . L e t u s n o w a s s u m e t h e f o l l o w i n g t h r e e

b o u n d a r y c o n d i t i o n s :

u

a

2

y

= 0 u

x

a

2

= 0 u ( 0 0 ) = 0 : ( 4 . 6 . 5 0 )

T h e l a s t c o n d i t i o n h a s t h e m e a n i n g t h a t t h e c e n t e r o f t h e m e m b r a n e i s

x e d . W e e x p r e s s t h i s b y a m o d i c a t i o n o f t h e m e m b r a n e e q u a t i o n

u + k

2

u ( 1 ; ( x ) ( y ) ) = 0 : ( 4 . 6 . 5 1 )

u ( x y ) = u

0

( x y ) + u

1

( x y ) +

2

u

2

( x y ) + : : :

k

2

= E

0

+ E

1

+

2

E

2

+ : : :

u

0

=

2

a

c o s

x

a

c o s

y

a

u

n m

( x y ) =

2

a

c o s

2 n + 1

a

x c o s

2 m + 1

a

y : ( 4 . 6 . 5 2 )

S o l u t i o n :

E

0

=

2

2

a

2

E

m n

= E

n m

=

2 n + 1

a

2

2

+

2 m + 1

a

2

2

u

1

( 0 0 ) = ;

X

n m

2

a

2

1

n

2

+ m

2

+ n + m

= ;

2

a

2

X

n

1

n

2

+ n

X

m

1

1 +

m

2

+ m

n

2

+ n

< ;

2

a

2

1

2

X

n

n

n

2

+ n

= ;

1

a

2

X

n

1

n + 1

! ; 1 : ( 4 . 6 . 5 3 )


7/18/2019 C4029-Ch04



D i s c u s s t h e f a c t t h a t u

1

d i v e r g e s s i n c e

X

m

1

1 +

m

2

+ m

n

2

+ n

>

n

2

:

2 . T h e S c h r o e d i n g e r e q u a t i o n f o r a n e l e c t r o n m

e

i n a p o t e n t i a l U

+

8

2

m

e

h

2

( E ; U ) = 0 ( 4 . 6 . 5 4 )

d e s c r i b e s t h e c o m p l e x s c a l a r e l d ( x y z ) , w h i c h g i v e s t h e p r o b a b i l i t y

d t o l o c a l i z e a n e l e c t r o n a t t h e p o i n t x y z w i t h i n d w i t h t h e

t o t a l e n e r g y E . h i s P l a n c k ' s c o n s t a n t 6 . 6 2 6

1 0

; 3 4

J s . F o r a n e l e c t r o n

i n a n h y d r o g e n a t o m , t h e a t t r a c t i v e p o t e n t i a l b e t w e e n t h e e l e c t r o n a n d

t h e h y d r o g e n n u c l e u s i s g i v e n b y U = + e

2

= r . U s i n g t h e F r o b e n i u s

m e t h o d f o r t h e r - d e p e n d e n c e a n d t h e k n o w l e d g e o f s p h e r i c a l f u n c t i o n s

o n e m a y d e r i v e t h e s p h e r i c a l s o l u t i o n

( r ' # ) =

X

m l

L

2 l + 1

B ; 1 ; l

( ) P

m

l

( c o s # ) e x p ( i m ' ) ( 4 . 6 . 5 5 )

w h e r e

8

2

m

e

E

h

2

= ;

1

r

2

0

r

0

2

8

2

m

e

e

2

h

2

= B

2 r

r

0

= ( 4 . 6 . 5 6 )

a n d t h e L a r e L a g u e r r e p o l y n o m i a l s ( 2 . 3 . 2 4 ) . T h e e i g e n v a l u e s a r e

g i v e n b y ( B a l m e r f o r m u l a )

E

n

= ;

2

2

m

e

e

4

n

2

h

2

: ( 4 . 6 . 5 7 )

D u e t o

n ; 1

X

l = 0

( 2 l + 1 ) = n

2

( 4 . 6 . 5 8 )

t h e h y d r o g e n p r o b l e m i s ( n

2

; 1 ) f o l d d e g e n e r a t e d . S h o w t h a t a n h y d r o -

g e n a t o m p l a c e d i n t o a n h o m o g e n e o u s c o n s t a n t m a g n e t i c e l d H g a i n s

e n e r g y E ! E ; e h H m = 4 m

e

a n d t h a t t h e e i g e n v a l u e w i l l b e p e r t u r b e d

E =

e h H

4 m

e

m = ;

B

H m ; l m l

a n d t h e d e g e n e r a t i o n i s p a r t i a l l y s u s p e n d e d .

B

i s B o h r ' s m a g n e t o n ,

m n o w t h e m a g n e t i c q u a n t u m n u m b e r .

3 . I n a c r y s t a l l a t t i c e t h e p o t e n t i a l a c t i n g o n a n e l e c t r o n m a y b e p e r i o d i c :

U = U

0

+ U

1

, U

1

= ; A c o s 2 x = d , w h e r e d i s t h e u n i t c e l l d i m e n s i o n


7/18/2019 C4029-Ch04



( l a t t i c e p a r a m e t e r ) . S h o w t h a t t h e s e t u p ( x ) = e x p 2 i ( b y + c z ) y i e l d s

E

1

= E ; U

0

; h

2

( b

2

+ c

2

) = 2 m

e

= x = d = 8 d

2

m

e

E

1

= h

2

=

8 d

2

m

e

A = h

2

a n d n a l l y a M a t h i e u e q u a t i o n

d

2

u

d

2

+ ( + c o s 2 ) u = 0 ( 4 . 6 . 5 9 )

i s o b t a i n e d . D u e t o i t s s t a b i l i t y r e g i o n s i n t e r e s t i n g p h y s i c a l c o n s e q u e n c e s

a p p e a r .

( a ) < ; o r E

1

< ; A : n o e l e c t r o n e n e r g y v a l u e s e x i s t t h a t a r e

s m a l l e r t h a n t h e p o t e n t i a l e n e r g y m i n i m u m ,

( b ) i f ; < < + o r ; A < E

1

< + A : e l e c t r o n s a r e b o u n d i n

p o t e n t i a l w e l l s a n d o s c i l l a t e ,

( c ) i f > o r E

1

> A : e l e c t r o n s m o v e f r e e l y

( d ) i f E

1

< A , a b a n d s p e c t r u m ( B r i l l o u i n z o n e s ) e x i s t s t h a t b r o a d -

e n s w i t h i n c r e a s i n g 4 . 1 4 ] .

4 . S o l v e

y

0 0

+ ( 1 + x

2

) y + 1 = 0 y (

1 ) = 0 ( 4 . 6 . 6 0 )

u s i n g a p o w e r s e r i e s 1 . 7 ]

y ( x ) =

1

X

= 0

a

x

2

: ( 4 . 6 . 6 1 )

E q u a t i o n ( 4 . 6 . 6 0 ) d e s c r i b e s t h e b e n d i n g o f a c l a m p e d s t r u t w i t h a d i s -

t r i b u t e d t r a n s v e r s e l o a d P . I n ( 4 . 6 . 6 1 ) o n l y e v e n p o w e r s a r e i n c l u d e d

d u e t o t h e s y m m e t r y o f t h e d i e r e n t i a l e q u a t i o n . I n s e r t i o n o f ( 4 . 6 . 6 1 )

i n t o ( 4 . 6 . 6 0 ) r e s u l t s i n

1

X

= 1

( 2 + 2 ) ( 2 + 1 ) a

+ 1

+ a

+ a

; 1

] x

2

+ a

0

+ 2 a

1

+ 1 = 0 ( 4 . 6 . 6 2 )

s o t h a t

1 + a

0

+ 2 a

1

= 0 ( 4 . 6 . 6 3 )

a

; 1

+ a

+ ( 2 + 2 ) ( 2 + 1 ) a

+ 1

= 0 = 1 2 3 : : : ( 4 . 6 . 6 4 )

w h i l e t h e b o u n d a r y c o n d i t i o n d e m a n d s

y ( 1 ) =

p

X

= 0

a

= 0 ( 4 . 6 . 6 5 )

a n d y i e l d s a n e q u a t i o n f o r a

0

. H e r e p m a y b e 1 2 : : : 7 ( o r m o r e ) . E x -

p r e s s i n g a l l c o e c i e n t s b y a

0

, t h e n ( 4 . 6 . 6 3 ) a n d ( 4 . 6 . 6 4 ) y i e l d


7/18/2019 C4029-Ch04


7/18/2019 C4029-Ch04



I n t e g r a t i o n y i e l d s

u ( x ) = c s i n h x +

1

x

Z

0

s i n h ( x ; ) f ( ) d ( 4 . 6 . 6 9 )

w h e r e c f o l l o w s f r o m u ( 1 ) = 0 . F u r t h e r m o r e ,

v = c

1

e x p x ] p

1

( x ) + c

2

e x p ; x ] p

2

( x )

+

1

2

2

x

Z

0

n

e x p ( x ; ) ] p

1

( x ) p

2

( )

; e x p ; ( x ; ) ] p

1

( ) p

2

( x )

o

g ( ) d : ( 4 . 6 . 7 0 )

c

1

c

2

a r e d e t e r m i n e d b y ( 4 . 6 . 6 8 ) a n d f o r p

1

p

2

o n e o b t a i n s

p

1

( x ) =

3

;

3

2

x

+

3

x

2

p

2

( x ) =

3

+

3

2

x

+

3

x

2

:

T h e i n t e g r a l s c o n t a i n e d i n ( 4 . 6 . 6 9 ) a n d ( 4 . 6 . 7 0 ) a r e q u i t e l a b o r i o u s . I t

i s a d v a n t a g e o u s t o e x p a n d f ( x ) a n d g ( x ) a c c o r d i n g t o a s y s t e m o f o r -

t h o n o r m a l f u n c t i o n s

f ( x ) =

1

X

n = 0

C

n

1 n

( x ) g ( x ) =

1

X

n = 0

D

n

1 n

( x ) :

1

R

0

1 n

( x )

1 m

( x ) d x =

n m

1

R

0

3 n

( x )

3 m

( x ) d x =

n m

n m

=

0 f o r n 6= m

1 f o r n = m

a n d

c

n

=

1

Z

0

f ( x )

1 n

( x ) d x d

n

=

1

Z

0

g ( x )

1 n

( x ) d x n = 0 1 2 : : : :

U s i n g L a g u e r r e p o l y n o m i a l s w i t h ! 2 k = 0 a n d u s i n g t h e R o d -

g r i g u e z f o r m u l a ( 2 . 3 . 2 4 ) o n e o b t a i n s

1 n

( x ) =

p

2

n !

e x p x ]

d

n

d x

n

( e x p ; 2 x ] x

n

) =

p

2 L

k

n

( x 2 )

3 n

( x ) =

2

p

2

n !

p

( n + 1 ) ( n + 2 )

e x p x ]

x

d

n

d x

n

;

e x p ; 2 x ] x

n + 2

n = 0 1 2 : : :

u

0

( x ) = v

0

( x ) =

1

p

2

1 0

( x ) :

U s i n g t h e f u n c t i o n s o n e c a n s o l v e t h e d i e r e n t i a l e q u a t i o n s i t e r a t i v e l y

o r u s i n g c o l l o c a t i o n m e t h o d s .


7/18/2019 C4029-Ch04


4 . 7 V a r i a t i o n a l c a l c u l u s

V a r i a t i o n a l m e t h o d s a l l o w t h e d e v e l o p m e n t o f n e w p r o c e d u r e s t o s o l v e v a r i o u s

m a t h e m a t i c a l p r o b l e m s . I n s e c t i o n 1 . 4 , e q u a t i o n ( 1 . 4 . 6 ) , w e b r i e y d i s c u s s e d

t h e f u n d a m e n t a l p r o b l e m o f t h e c a l c u l u s o f v a r i a t i o n a n d w e u s e d t h e E u l e r

e q u a t i o n ( 1 . 4 . 7 ) . N o w i t i s t i m e t o d e r i v e t h e s e e q u a t i o n s . A c c o r d i n g t o ( 1 . 4 . 6 )

f o r a g i v e n f u n c t i o n a l F ( x y y

0

( x ) y

0 0

( x ) : : : y

n

( x ) ) , w h e r e y ( x ) i s u n k n o w n ,

t h e i n t e g r a l

J =

x

2

Z

x

1

F ( x y y

0

y

0 0

: : : ) d x ( 4 . 7 . 1 )

s h o u l d b e m a d e a n e x t r e m u m , s o t h a t y ( x ) m a y b e d e t e r m i n e d . T h e e x i s t e n c e

o f a n e x t r e m u m o f J i s g u a r a n t e e d b y t h e W e i e r s t r a s s t h e o r e m . T h i s s t a t e s

t h a t e a c h c o n t i n u o u s f u n c t i o n p o s s e s s e s a n e x t r e m u m w i t h i n a c l o s e d i n t e r v a l .

L e t u s c o n s i d e r a s i m p l e e x a m p l e 4 . 1 5 ] . T h e F e r m a t p r i n c i p l e o f g e o -

m e t r i c a l o p t i c s s t a t e s t h a t l i g h t p r o p a g a t i n g w i t h i n a n i n h o m o g e n e o u s ( o r

h o m o g e n e o u s ) m e d i u m o f r e f r a c t i v e i n d e x n t a k e s t h a t p a t h t h a t i s t h e s h o r t -

e s t i n t i m e . F o r m u l a t e d a s a v a r i a t i o n a l p r o b l e m t h e p r i n c i p l e r e a d s J !

e x t r e m u m

J =

Z

d t =

Z

d s

c ( x y z )

= c

0

Z

n ( x y z ) d s : ( 4 . 7 . 2 )

T o s o l v e p r o b l e m s o f t h i s t y p e w e \ v a r i a t e " J :

J =

x

2

Z

x

1

F ( x y y

0

: : : y

( n )

) d x = 0 : ( 4 . 7 . 3 )

A s s u m i n g t h a t s u c h a f u n c t i o n y ( x ) e x i s t s , w e w r i t e

y ( x ) + " y ( x ) y ( x

1

) = y ( x

2

) = 0 : ( 4 . 7 . 4 )

I t i s a p p a r e n t t h a t t h e l i m i t " ! 0 i n d u c e s t h e f u n c t i o n a l . T h e r e f o r e

J =

x

2

Z

x

1

F

x y + " y y

0

+ " y

0

: : : y

( n )

+ " y

( n )

d x ( 4 . 7 . 5 )

a n d

d J

d "

" = 0

= 0 : ( 4 . 7 . 6 )


7/18/2019 C4029-Ch04


V a r i a t i o n a l c a l c u l u s 2 1 5

D u e t o t h e s m a l l n e s s o f " , w e e x p a n d t h e f u n c t i o n a l F i n t o a T a y l o r s e r i e s

J =

x

2

Z

x

1

F ( x y y

0

: : : ) + "

@ F

@ y

y + "

@ F

@ y

0

y

0

+ : : : + "

@ F

@ y

( n )

y

( n )

+ : : : "

2

+ : : :

d x : ( 4 . 7 . 7 )

P e r f o r m i n g ( d J = d " )

" = 0

a n d i n t e g r a t i n g w e o b t a i n

x

2

Z

x

1

@ F

@ y

0

y

0

d x =

@ F

@ y

0

y

x

2

x

1

;

x

2

Z

x

1

y

d

d x

@ F

@ y

0

d x : ( 4 . 7 . 8 )

D u e t o y ( x

1

) = 0 y ( x

2

) = 0 , t h e ] t e r m v a n i s h e s

x

2

Z

x

1

y

@ F

@ y

;

d

d x

@ F

@ y

0

+

d

2

d x

2

@ F

@ y

0 0

+ : : : + ( ; 1 )

n

d

n

d x

n

@ F

@ y

( n )

d x = 0 : ( 4 . 7 . 9 )

T h i s i n t e g r a l v a n i s h e s t h e n a n d o n l y i f t h e e x p r e s s i o n i n b r a c e s v a n i s h e s . T h i s

g i v e s t h e E u l e r e q u a t i o n ( 1 . 4 . 7 ) . F o r o n e f u n c t i o n y ( x ) i t r e a d s

@ F

@ y

;

d

d x

@ F

@ y

0

+ : : : + ( ; 1 )

n

d

n

d x

n

@ F

@ y

( n )

= 0 : ( 4 . 7 . 1 0 )

T h e s e f u n c t i o n a l s a r e c l o s e l y c o n n e c t e d t o i m p o r t a n t d i e r e n t i a l e q u a t i o n s o f

p h y s i c s a n d e n g i n e e r i n g . T h e f u n c t i o n a l

F ( u u

x

u

y

) =

1

2

;

u

2

x

+ u

2

y

; k

2

u

2

( 4 . 7 . 1 1 )

y i e l d s t h e H e l m h o l t z e q u a t i o n u

x x

+ u

y y

+ k

2

u = 0 : I t c a n b e s h o w n 3 . 1 3 ] ,

4 . 1 6 ] t h a t e v e r y o r d i n a r y d i e r e n t i a l e q u a t i o n o f s e c o n d o r d e r y

0 0

= F ( x y y

0

)

h a s a n a s s i g n e d f u n c t i o n a l r e p r o d u c i n g t h e d i e r e n t i a l e q u a t i o n .

T h e s o - c a l l e d d i r e c t m e t h o d s o e r a d i r e c t s o l u t i o n o f t h e v a r i a t i o n a l p r o b l e m

( 4 . 7 . 1 ) w i t h o u t a n y r e f e r e n c e t o t h e p e r t i n e n t d i e r e n t i a l e q u a t i o n . T h u s t h e

d i r e c t m e t h o d s o e r m e t h o d s t o s o l v e b o u n d a r y v a l u e p r o b l e m s e v e n i n s u c h

c a s e s , t h a t a n a n a l y t i c s o l u t i o n o f t h e d i e r e n t i a l e q u a t i o n i s n o t p o s s i b l e .

T h e r i t z m e t h o d i s o n e o f t h e s i m p l e s t d i r e c t m e t h o d s . I t u s e s t h e s u c c e s s i v e

s e t u p

u

n

( x y ) = u

0

( x y ) +

n

X

= 1

c

n

u

( x y ) : ( 4 . 7 . 1 2 )


7/18/2019 C4029-Ch04



N o w o n e m a y i n t e r p r e t J a s a f u n c t i o n a l o f t h e n e w v a r i a b l e s c

n

a n d u s i n g

@ J = @ c

n

= 0 y i e l d s t h e R i t z e q u a t i o n s f o r t h e d e t e r m i n a t i o n o f t h e c

n

. I n

t h e R i t z m e t h o d t h e f u n c t i o n a l F m u s t b e k n o w n . A c c o r d i n g t o G a l e r k i n

t h i s i s n o t n e c e s s a r y . O n e m a y u s e

@ J

@ c

n

=

b

Z

a

d

Z

c

@ F

@ u

@ u

@ c

n

+

@ F

@ u

x

@ u

x

@ c

n

+

@ F

@ u

y

@ u

y

@ c

n

d y d x

=

b

Z

a

d

Z

c

@ F

@ u

u

n

+

@ F

@ u

x

u

n x

+

@ F

@ u

y

u

n y

d y d x : ( 4 . 7 . 1 3 )

P a r t i a l i n t e g r a t i o n y i e l d s

@ J

@ c

n

=

@ F

@ u

x

u

b

a

+

@ F

@ u

y

u

d

c

+

b

Z

a

d

Z

c

@ F

@ u

;

d

d x

@ F

@ u

x

;

d

d y

@ F

@ u

y

u d y d x ( 4 . 7 . 1 4 )

s i n c e t h e u

n

( x y ) h a v e t o s a t i s f y t h e r e c t a n g u l a r b o u n d a r y c o n d i t i o n s u

n

= 0

f o r x = a b y = c d . T h e b r a c k e t s w i t h i n t h e i n t e g r a l ( 4 . 7 . 1 4 ) v a n i s h a n d

a r e i d e n t i c a l w i t h t h e p a r t i a l d i e r e n t i a l e q u a t i o n L f u g = 0 . T h u s w e h a v e

t h e G a l e r k i n e q u a t i o n s

b

Z

a

d

Z

c

L f u g u

n

( x y ) d y d x = 0 : ( 4 . 7 . 1 5 )

T h e c o e c i e n t s c

n

h a v e t o b e d e t e r m i n e d i n s u c h a m a n n e r , t h a t t h e d i e r -

e n t i a l e q u a t i o n L f u g i s o r t h o g o n a l t o t h e t r i a l f u n c t i o n s u

n

. T o s o l v e u = 0

f o r a n e l l i p s e w i t h t h e s e m i - a x e s a a n d b w i t h t h e i n h o m o g e n e o u s b o u n d a r y

c o n d i t i o n u

0

= x

2

+ y

2

o n e m a y u s e t h e R i t z t r i a l f u n c t i o n s

u

n

( x y ) = x

2

+ y

2

+

x

2

a

2

+

y

2

b

2

; 1

;

c

n 0

+ c

n 1

x + c

n 2

y + c

n 3

x

2

+ : : :

: ( 4 . 7 . 1 6 )

T h e r s t a p p r o x i m a t i o n y i e l d s

u

1

= x

2

+ y

2

+ c

1 0

x

2

a

2

+

y

2

b

2

; 1

@ u

1

@ x

= 2 x

1 +

c

1 0

a

2

@ u

1

@ y

= 2 y

1 +

c

1 0

b

2

: ( 4 . 7 . 1 7 )


7/18/2019 C4029-Ch04



T h e n

J =

Z Z

4 x

2

1 +

c

1 0

a

2

2

+ 4 y

2

1 +

c

1 0

b

2

2

d x d y : ( 4 . 7 . 1 8 )

E x e c u t i n g t h e i n t e g r a t i o n s

Z Z

d x d y = a b

Z Z

x

2

d x d y =

a

3

b

4

Z Z

y

2

d x d y =

a b

3

4

( 4 . 7 . 1 9 )

a n d a p p l y i n g ( 4 . 7 . 1 4 ) o n e o b t a i n s

c

1 0

= ;

2 a

2

b

2

a

2

+ b

2

u

1

( x y ) = x

2

+ y

2

;

2 a

2

b

2

a

2

+ b

2

x

2

a

2

+

y

2

b

2

; 1

: ( 4 . 7 . 2 0 )

I n s e c t i o n 1 . 2 , T a b l e 1 . 1 w e s a w t h a t t h e D i r i c h l e t p r o b l e m f o r a n e l l i p t i c

e q u a t i o n l i k e u = 0 i s s o l v a b l e f o r o n e c l o s e d b o u n d a r y . N o a n a l y t i c s o l u t i o n

s e e m s t o e x i s t f o r t w o c l o s e d b o u n d a r i e s . B u t d i r e c t m e t h o d s o f v a r i a t i o n a l

c a l c u l u s a r e a b l e t o s o l v e a b o u n d a r y p r o b l e m o f a t w o - f o l d c o n n e c t e d d o m a i n

4 . 1 7 ] . T h e d o m a i n i s g i v e n b y a s q u a r e o f l e n g t h a = 4 w i t h a c u t - o c i r c l e

o f r a d i u s R = 1 , s e e F i g u r e 4 . 1 4 . A s s u m i n g t h e b o u n d a r y v a l u e s

u ( x y ) = x

2

+ y

2

( 4 . 7 . 2 1 )

o n t h e b o u n d a r i e s a s w e l l a s o f t h e s q u a r e a n d t h e c i r c l e , t h e a p p r o p r i a t e

t r i a l f u n c t i o n s w i l l b e

u

n

( x y ) = x

2

+ y

2

+

;

x

2

+ y

2

; 1

;

x

2

; 4

;

y

2

; 4

n

X

= 0

c

u

( x y ) : ( 4 . 7 . 2 2 )

T h e b o u n d a r y c o n d i t i o n s ( 4 . 7 . 2 1 ) a r e s a t i s e d b y t h e s e t r i a l f u n c t i o n s f o r t h e

s q u a r e b o u n d a r i e s x = 2 y = 2 a n d t h e c i r c l e b o u n d a r y x

2

+ y

2

= 1 .

T a k i n g i n t o a c c o u n t t h e s y m m e t r i e s o f t h e p r o b l e m , o n e m a y w r i t e

5

X

= 0

c

u

( x y ) = c

0

+ c

1

( x + y ) + c

2

;

x

2

+ y

2

+ c

3

x y + c

4

;

x

3

+ y

3

+ c

5

;

x

2

y + x y

2

: ( 4 . 7 . 2 3 )

U s i n g t h e s y m m e t r y o f t h e o u t e r b o u n d a r y a n d t h e i n t e g r a l s

I ( n ) =

Z Z

x

n

d x d y =

8

>

>

>

>

<

>

>

>

>

:

2

n + 2

n + 1

;

( n

; 1 ) ! !

( n + 2 ) ! !

2

f o r n e v e n

2

n + 2

n + 1

;

( n ; 1 ) ! !

( n + 2 ) ! !

f o r n o d d

( 4 . 7 . 2 4 )

o n e o b t a i n s t h e n u m e r i c a l v a l u e s p r e s e n t e d i n T a b l e 4 . 1 .


http://c4029-ch01.pdf/

7/18/2019 C4029-Ch04



T a b l e 4 . 1 . R e s u l t s f r o m ( 4 . 7 . 2 3 )

a p p r o x i m a t i o n N r 0 N r 1 N r 2 N r 3

c

0

0 . 0 6 2 5 0 . 1 0 2 7 0 . 1 8 1 2 0 . 0 9 6 9

c

1

- ; 0 : 0 1 6 6 ; 0 : 0 7 1 0 0 . 0 5 2 1

c

2

- - ; 0 : 0 0 0 5 ; 0 : 0 6 7 3

c

3

- - 0 . 0 4 4 2 ; 0 : 0 4 7 4

c

4

- - - 0 . 0 1 3 1

c

5

- - - 0 . 0 1 8 6

J 1 4 1 . 5 5 2 0 1 4 0 . 7 5 4 0 1 3 9 . 6 4 4 0 1 3 9 . 6 1 7 0

U s i n g t h e v a l u e s o f t h e l a s t c o l u m n t h e s o l u t i o n m a y b e w r i t t e n

f[x_,y_]=xˆ2+yˆ2+(xˆ2+yˆ2-1)*(xˆ2-4)*(yˆ2-4)*

(0.0969+0.0521*(x+y)-0.0673*(xˆ2+yˆ2)-0.0474*

x*y+0.0131*(xˆ3+yˆ3)+0.0186*(xˆ2*y+x*yˆ2))( 4 . 7 . 2 5 )

P1=Plot3D[f[x,y],{x,-2,2},{y,-2,2},

Shading->False,ClipFill->None,

PlotRange-> {0.,8.},PlotPoints->100]

F u n c t i o n ( 4 . 7 . 2 5 ) h a s b e e n p l o t t e d i n F i g u r e 4 . 1 4 t o g e t h e r w i t h t h e t w o

b o u n d a r i e s . I n p r o b l e m 1 o f t h i s s e c t i o n w e w i l l d i s c u s s t h e m e t h o d s t o c r e a t e

F i g u r e 4 . 1 4 . D u e t o t h e q u i t e r o u g h a p p r o x i m a t i o n a n a d d i t i o n a l s m a l l d o m a i n

h a s b e e n c u t o u t .

F i g u r e 4 . 1 4

T w o b o u n d a r i e s f o r t h e L a p l a c e e q u a t i o n

T h e R i t z m e t h o d i s a p p a r e n t l y c a p a b l e t o n d a g o o d a p p r o x i m a t i o n t o

t h e e i g e n f u n c t i o n s . T h e R a y l e i g h m e t h o d i s b e t t e r s u i t e d t o n d e i g e n v a l u e

a p p r o x i m a t i o n s 1 . 2 ] , 3 . 1 4 ] , 1 . 7 ] . W e c o n s i d e r a n e q u a t i o n a n d i t s f u n c t i o n a l

d

d x

p ( x )

d u

d x

; q ( x ) u ( x ) + ( x ) u ( x ) = 0

J =

R

p u

0 2

+ q u

2

; u

2

d x : ( 4 . 7 . 2 6 )


7/18/2019 C4029-Ch04



T h e n t h e s m a l l e s t e i g e n v a l u e i s g i v e n b y t h e R a y l e i g h c o e c i e n t

=

b

Z

a

;

p u

0 2

+ q u

2

d x /

b

Z

a

u

2

d x : ( 4 . 7 . 2 7 )

L e t u s t r y a n e x a m p l e . A c c o r d i n g t o ( 2 . 4 . 4 5 ) a c i r c u l a r m e m b r a n e o f r a d i u s

R h a s t h e l o w e s t e i g e n v a l u e

p

= 2 : 4 0 4 8 2 5 = R . N o w f o r t h e H e l m h o l t z

e q u a t i o n i n p o l a r c o o r d i n a t e s o n e m a y c h o o s e t h e t r i a l f u n c t i o n

u = 1 ; r = R : ( 4 . 7 . 2 8 )

I n s e r t i n g u ( r ) i n t o

2

Z

0

R

Z

0

u

0 2

( r ) r d r d '

2

Z

0

R

Z

0

u

2

( r ) r d r d ' ( 4 . 7 . 2 9 )

o n e g e t s

p

=

r

R

2

= 6

=

p

6 = R = 2 : 4 4 9 = R : ( 4 . 7 . 3 0 )

A t h i r d p r o c e d u r e i s t h e T r e f f t z m e t h o d . I n s t e a d o f d o m a i n i n t e g r a l s

l i k e ( 4 . 7 . 2 9 ) , o n l y b o u n d a r y i n t e g r a l s h a v e t o b e c a l c u l a t e d . T h i s m e t h o d i s

s i m i l a r t o t h e b o u n d a r y e l e m e n t m e t h o d 4 . 1 8 ] . A n o t h e r p r o c e d u r e i s t h e l e a s t

s q u a r e s m e t h o d l o o k i n g f o r a m i n i m u m o f (

R

( u

0

; u )

2

d x )

1 = 2

, w h e r e u

0

; u

i s t h e d i e r e n c e b e t w e e n t h e s o l u t i o n o f t h e d i e r e n t i a l e q u a t i o n a n d t h e t r i a l

f u n c t i o n 1 . 7 ] . H e r e t h e b o u n d a r y m a x i m u m p r i n c i p l e 1 . 7 ] h e l p s . F o r e l l i p t i c

p a r t i a l d i e r e n t i a l e q u a t i o n s i t s t a t e s t h a t t h e d i e r e n c e u

0

; u i s e q u a l o r

s m a l l e r t h a n t h e d i e r e n c e b e t w e e n t h e a p p r o x i m a t e s o l u t i o n a n d t h e g i v e n

b o u n d a r y v a l u e s . L e t u s c o n s i d e r a n e x a m p l e f o r t h e T r e f f t z m e t h o d :

u ( x y ) = b

u ( 1 y ) = u ( x 1 ) = 0 : ( 4 . 7 . 3 1 )

F i r s t t h e P o i s s o n e q u a t i o n w i l l b e h o m o g e n i z e d :

u ( x y ) = b ( x

2

+ y

2

) = 4 + v ( x y ) v = 0

v = ; b ( x

2

+ y

2

) = 4 f o r x = 1 y = 1 : ( 4 . 7 . 3 2 )

N o w w e d e n e t h e t r i a l f u n c t i o n f o r a s q u a r e

v = ( x

4

; 6 x

2

y

2

+ y

4

) : ( 4 . 7 . 3 3 )

U s e o f t h e s e c o n d G r e e n t h e o r e m

Z

( v

r u

; u

r v ) d f =

Z

( v u

; u v ) d ( 4 . 7 . 3 4 )


7/18/2019 C4029-Ch04



a n d o f u = 0 v = 0 l e a d s t o t h e b o u n d a r y i n t e g r a l o v e r t h e s q u a r e

Z

v

@

@ x

v ;

@ v

@ x

v

d x = 0 ( 4 . 7 . 3 5 )

o r

1

Z

0

1

4

b

;

1 + y

2

;

4 ; 1 2 y

2

+

;

4 ; 1 2 y

2

;

1 ; 6 y

2

+ y

4

d y = 0

s o t h a t = 7 b = 1 4 4 r e s u l t s .

V a r i a t i o n a l m e t h o d s m a y s o l v e i n t e g r a l e q u a t i o n s t o o . I n w a v e m e c h a n i c s

t h e c o n n e c t i o n b e t w e e n a p o t e n t i a l V b e t w e e n t h e p a r t i c l e s a n d t h e s c a t t e r i n g

c r o s s s e c t i o n i s i m p o r t a n t . F r o m s c a t t e r i n g e x p e r i m e n t s o n e m a y c a l c u l a t e

t h e p o t e n t i a l 4 . 1 9 ] . L e t u s c o n s i d e r t h e i n t e g r a l e q u a t i o n

y ( r ) = y

0

( r ) + "

1

Z

0

K ( r r

0

) y ( r

0

) d r

0

: ( 4 . 7 . 3 6 )

H e r e " i s a p h y s i c a l c o n s t a n t , t h e k e r n e l K ( r r

0

) m a y b e s y m m e t r i c a n d e x -

p r e s s i b l e b y a s c a t t e r i n g p o t e n t i a l V a n d a G r e e n f u n c t i o n . N o w t w o i n t e g r a l s

w i l l b e d e n e d :

I

1

=

1

Z

0

y

2

( r ) d r ; "

1

Z

0

1

Z

0

y ( r ) K ( r r

0

) y ( r

0

) d r d r

0

( 4 . 7 . 3 7 )

I

2

=

1

Z

0

y ( r ) y

0

( r ) d r : ( 4 . 7 . 3 8 )

I f y ( r ) s o l v e s ( 4 . 7 . 3 6 ) , t h e n I

1

= I

2

. T h i s c a n b e s h o w n b y m u l t i p l y i n g ( 4 . 7 . 3 6 )

b y y ( r ) a n d i n t e g r a t i o n . N o w l e t u s v a r i a t e y ( r ) ! y ( r ) + y ( r ) , t h e n

I

1

=

1

Z

0

2 y ( r ) y ( r ) d r ; "

1

Z

0

1

Z

0

y ( r ) K ( r r

0

) y ( r

0

) d r d r

0

( 4 . 7 . 3 9 )

I

2

=

1

Z

0

y ( r ) y

0

( r ) d r ( 4 . 7 . 4 0 )

s i n c e y

2

= 2 y y . ( T h e v a r i a t i o n s y m b o l i s s u b j e c t e d t o t h e s a m e r u l e s a s

t h e d i e r e n t i a l ) . I f y ( r ) i s a s o l u t i o n o f t h e i n t e g r a l e q u a t i o n , t h e n I

1

= 2 I

2

o r ( I

1

; 2 I

2

) = 0 o r a c c o r d i n g t o t h e S c h w i n g e r v a r i a t i o n a l p r i n c i p l e :

I

1

I

2

2

= 0 : ( 4 . 7 . 4 1 )

S u c h v a r i a t i o n a l m e t h o d s a r e a b l e t o s o l v e n o n s e p a r a b l e p a r t i a l d i e r e n t i a l

e q u a t i o n s .


7/18/2019 C4029-Ch04


P r o b l e m s

1 . T h e r e a d e r i s i n v i t e d t o p r o d u c e t h e r i g h t - h a n d s i d e o f F i g u r e 4 . 1 4 b y

t y p i n g

P1=Plot3D[f[x,y],{x,-2,2},

{y,-2,2},Shading->False,ClipFill->None,

PlotRange-> {0.,8.},PlotPoints->100]( 4 . 7 . 4 2 )

ClipFill->None

a v o i d s t h e c l o s i n g o f h o l e s .

T h e c i r c l e i n t h i s g u r e m a y b e c r e a t e d b y

Clear[CG]; CG = Graphics[Circle[{0., 0.}, 1.],

AspectRatio -> Automatic];

Show[CG]

w h e r e AspectRatio -> Automatic a v o i d s p e r s p e c t i v e d e f o r m a t i o n .

T h e s q u a r e w i l l b e c r e a t e d b y

Clear[Q];

Q=Line[{{-2.,-2.},{2.,-2},{2.,2.},{-2.,2.},

{-2.,-2.}}]T h e n t h e c o m b i n a t i o n o f c i r c l e a n d s q u a r e c a n b e d o n e b y t h e c o m m a n d

QG=Graphics[Q];

QQG=Show[QG,CG,AspectRatio->Automatic]

F i n a l l y t h e c o m m a n d

Show[GraphicsArray[{QQG, P1}]]

p r o d u c e s F i g u r e 4 . 1 4 .

2 . C o n s i d e r t h e t r a n s v e r s a l o s c i l l a t i o n s o f a h e a v y r o d w i t h n o l o a d P

d e s c r i b e d b y ( 4 . 5 . 5 4 ) i n t h e f o r m

c

2

u

x x x x

; !

2

u + g ( l ; x ) u

x x

; g u

x

= 0 : ( 4 . 7 . 4 3 )

C a l c u l a t e

J

1

=

Z Z

u

c

2

u

x x x x

+ g ( l ; x ) u

x x

; g u

x

d x d y

J

2

=

; !

2

Z Z

u d x d y :

C o m b i n i n g R a y l e i g h w i t h G a l e r k i n w r i t e

= !

2

=

l

Z

0

u

c

2

u

x x x x

+ g ( l ; x ) u

x x

; g u

x

d x /

0

@

;

l

Z

0

u

2

d x

1

A

:

( 4 . 7 . 4 4 )


7/18/2019 C4029-Ch04



A s s u m e t h e t r i a l f u n c t i o n ,

u ( x ) = a ( l

3

x

; 2 l x

3

+ x

4

) ( 4 . 7 . 4 5 )

w h i c h s a t i s e s u ( 0 ) = 0 u ( l ) = 0 u

0 0

( 0 ) = 0 u

0 0

( l ) = 0 . C a l c u l a t e

=

9 : 8 7 7

l

2

s

E I

q

1

; 0 : 0 5 1

g q l

3

E I

o r

=

9 : 8 7 7

l

2

s

E I

q

f o r g = 0 :

3 . S o l v e t h e m e m b r a n e v i b r a t i o n s c

2

( u

x x

+ u

y y

) + !

2

u ( x y ) = 0 f o r

u ( ; l y ) = 0 u ( l y ) = 0 u ( x ; l ) = 0 u ( x l ) = 0 u s i n g t h e t r i a l f u n c -

t i o n a ( l

2

; x

2

) ( l

2

; y

2

) ( R i t z : ! 2 : 2 3 6 c = l ) .

4 . S o l v e u + k

2

u = 0 u ( ; l y ) = 0 u ( l y ) = 0 u ( x ; l ) = 0 u ( x l ) =

0 u (

; l y ) = 0 u ( l y ) = 0 u ( x

; l ) = 0 u ( x l ) = 0 ( N a v i e r

s o l u t i o n s ) u s i n g ( l

2

; x

2

) ( l

2

; y

2

) ( R i t z : ! 5 : 2 4 4 c = l

2

) .

5 . S o l v e

u ( x y ) = 0 f o r ; 1 x + 1 ; 1 y + 1 ( 4 . 7 . 4 6 )

w i t h

u = 1 + y

2

f o r x = 1 u = x

2

+ 1 f o r y = 1 : ( 4 . 7 . 4 7 )

T h e s o l u t i o n s c o s x c o s h y o f t h e L a p l a c e e q u a t i o n c a n n o t b e u s e d

a s t r i a l f u n c t i o n s , s i n c e c o s h ( 2 n + 1 ) y = 2 i n c r e a s e s w i t h i n c r e a s i n g n . U s e

s y m m e t r i c h a r m o n i c p o l y n o m i a l s .

v

1

= R e ( x + i y )

4

= x

4

; 6 x

2

y

2

+ y

4

( 4 . 7 . 4 8 )

v

2

= R e ( x + i y )

8

= x

8

; 2 8 x

6

y

2

+ 7 0 x

4

y

4

; 2 8 x

2

y

6

+ y

8

( 4 . 7 . 4 9 )

a n d t h e t r i a l f u n c t i o n

v = a

0

+ a

1

v

1

+ a

2

v

2

: ( 4 . 7 . 5 0 )

U s i n g t h e b o u n d a r y m a x i m u m p r i n c i p l e o n e m a y d e m a n d

j

;

x

2

+ 1

;

;

a

0

+ a

1

;

x

4

; 6 x

2

+ 1

j ! M i n ( 4 . 7 . 5 1 )

a s r s t a p p p r o x i m a t i o n . I f t h e e r r o r m a y b e 0 . 0 2 5 a n d a s s u m i n g t h e t w o

c o l l o c a t i o n p o i n t s ( 0 , 1 ) a n d ( 1 , 1 ) o n e h a s f r o m ( 4 . 7 . 5 1 )

x = 0 : 1 ; ( a

0

+ a

1

) = 0 : 0 2 5

x = 1 : 2 ; ( a

0

+ a

1

; 6 a

1

+ a

1

) = 0 : 0 2 5


7/18/2019 C4029-Ch04


C o l l o c a t i o n m e t h o d s 2 2 3

s o t h a t a

0

= 1 : 1 7 5 a

1

= ; 0 : 2 . S h o w t h a t t h e n e x t a p p r o x i m a t i o n d e -

m a n d i n g a n e r r o r 0 : 0 0 5 r e s u l t s i n a

0

= 1 : 1 7 4 4 6 5 a

1

= ; 0 : 1 8 4 7 3 8 a

2

= 0 . 0 0 5 0 8 7 4 4 . P l o t u = x

2

+ y

2

a n d t h e t r i a l f u n c t i o n ( 4 . 7 . 5 0 ) .

6 . I n p r o b l e m 5 t h e d i e r e n t i a l e q u a t i o n ( 5 . 4 . 1 ) h a d b e e n s o l v e d e x a c t l y

a n d t h e b o u n d a r y c o n d i t i o n s h a d b e e n s a t i s e d b y t h e t r i a l f u n c t i o n

( o u t e r c o l l o c a t i o n ) . N o w t r y i n t e r i o r c o l l o c a t i o n . S a t i s f y t h e b o u n d a r y

c o n d i t i o n s e x a c t l y a n d t h e t r i a l f u n c t i o n s h o u l d a p p r o x i m a t e l y s a t i s f y

t h e d i e r e n t i a l e q u a t i o n . U s e

u = x

2

+ y

2

+ ( 1

; x

2

) ( 1

; y

2

) = 1 + x

2

y

2

v = u + ( 1 ; x

2

) ( 1 ; y

2

)

X

n m

a

n m

x

2 n

y

2 m

:

C h o o s e t h e c o l l o c a t i o n p o i n t s ( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 0 ) , ( 1 , 1 ) a n d v = 0 .

R e s u l t : a

0 0

= a

1 0

= a

0 1

= 0 : 1 4 2 8 5 7 a

1 1

= 0 :

4 . 8 C o l l o c a t i o n m e t h o d s

I f a p a r t i a l d i e r e n t i a l e q u a t i o n i s s e p a r a b l e o r r e d u c i b l e i n t o o r d i n a r y d i e r -

e n t i a l e q u a t i o n s a s d i s c u s s e d i n c h a p t e r 3 a n d i f t h e b o u n d a r i e s c o i n c i d e w i t h

c o o r d i n a t e l i n e s o f a c o o r d i n a t e s y s t e m i n w h i c h t h e p a r t i a l d i e r e n t i a l e q u a -

t i o n i s s e p a r a b l e , t h e n t h e s o l u t i o n o f a b o u n d a r y p r o b l e m i s e a s y . T h e r e o c c u r

h o w e v e r m a n y p r o b l e m s t h a t a r e n o t w e l l p o s e d o r u n s e p a r a b l e . I n p r i n c i p l e ,

h o w e v e r , t h e g e n e r a l s o l u t i o n o f a p a r t i a l d i e r e n t i a l e q u a t i o n m u s t c o n t a i n

t h e s o l u t i o n o f a n y b o u n d a r y p r o b l e m . N o t o n l y t h e o r e t i c a l c o n s i d e r a t i o n s b u t

a l s o t h e s u c c e s s o f n u m e r i c a l m e t h o d s l i k e n i t e d i e r e n c e s , n i t e e l e m e n t s o r

b o u n d a r y e l e m e n t s l e a d t o t h i s c o n c l u s i o n . H o w e v e r , v e r y o f t e n t h e p r o b -

l e m i s h o w t o n d t h e s o l u t i o n . I n s o m e c a s e s , i f f o r i n s t a n c e t h e b o u n d a r y

h a s s h a r p c o r n e r s o r l a r g e c u r v a t u r e , t h e g r i d o f d i s c r e t e p o i n t s , t h e m e s h e s ,

m u s t b e v e r y n e a n d v e r y l a r g e m a t r i c e s a p p e a r i n t h e r e s p e c t i v e c o d e g i v i n g

n u m e r i c a l p r o b l e m s a n d c a u s i n g l o n g a n d t i m e - w a s t i n g p r o g r a m m i n g e o r t s .

T h e r e a r e p r o b l e m s f o r i n s t a n c e i n p l a s m a p h y s i c s , w h e r e t h e n u m e r i c a l t h r e e -

d i m e n s i o n a l s o l u t i o n o f a p l a s m a c o n n e m e n t e x i s t s , b u t m a t h e m a t i c i a n s d e n y

t h e p o s s i b i l i t y o f s o l v i n g t h e p e r t i n e n t e q u a t i o n s .

H a v i n g i n m i n d t h e t h e o r e m t h a t a n a n a l y t i c s o l u t i o n o f a n e l l i p t i c e q u a t i o n

e x i s t s o n l y f o r o n e c l o s e d b o u n d a r y , l e t u s n d a c l a s s i c a t i o n s c h e m e o f s u c h

b o u n d a r y v a l u e p r o b l e m s :

1 . T h e p a r t i a l d i e r e n t i a l e q u a t i o n i s s e p a r a b l e :


7/18/2019 C4029-Ch04



1 . 1 o n e s m o o t h b o u n d a r y : t h e p r o b l e m i s w e l l p o s e d a n d t h e s o l u t i o n i s

a n a l y t i c d e g e n e r a t i o n o f e i g e n f u n c t i o n s i s a d m i t t e d a n d d e n i t i o n

o f n o d a l l i n e s a s b o u n d a r y i s p o s s i b l e , s e e ( 4 . 4 . 6 ) ,

1 . 2 t w o s m o o t h b o u n d a r i e s l i k e i n a n a n n u l a r c i r c u l a r m e m b r a n e : t h e

s o l u t i o n i s n o l o n g e r a n a l y t i c a n d c o n t a i n s s i n g u l a r i t i e s ,

1 . 3 t w o s m o o t h c l o s e d b o u n d a r i e s b e l o n g i n g t o t w o d i e r e n t s e p a r a -

b l e c o o r d i n a t e s y s t e m s ( \ n o n u n i f o r m p r o b l e m s " ) , h o l e s w i t h i n t h e

d o m a i n l i k e i n ( 4 . 7 . 2 1 ) , s e e F i g u r e 4 . 1 4 ,

1 . 4 t h r e e b o u n d a r y c o n d i t i o n s f o r a n e l l i p t i c e q u a t i o n o f s e c o n d o r -

d e r d e l i v e r i n g a d i v e r g e n t a p p r o x i m a t e s o l u t i o n a s i n p r o b l e m 1 o f

s e c t i o n 4 . 6 ,

1 . 5 n o s m o o t h J o r d a n c u r v e s a s b o u n d a r i e s , b u t c o r n e r s e t c . , s e e F i g -

u r e s 4 . 2 a n d 4 . 3 ,

1 . 6 a s p e c i a l s i t u a t i o n i s g i v e n , i f t h e p a r t i a l d i e r e n t i a l e q u a t i o n i s

s e p a r a b l e , b u t t h e b o u n d a r y c o n d i t i o n s c a n n o t b e s a t i s e d i n t h e

a c t u a l c o o r d i n a t e s y s t e m , s e e s e c t i o n 5 . 1

2 . T h e p a r t i a l d i e r e n t i a l e q u a t i o n i s N O T s e p a r a b l e :

2 . 1 w e l l p o s e d , n o h o l e s , n o c o r n e r s , o n l y o n e b o u n d a r y

2 . 2 t w o o r m o r e b o u n d a r i e s , c o r n e r s .

I n c a s e s 1 . 2 t o 1 . 5 a n d f o r 2 . , a p p r o x i m a t e o r v a r i a t i o n a l m e t h o d s m i g h t h e l p ,

b u t v e r y o f t e n o n l y n u m e r i c a l m e t h o d s c a n s o l v e t h e p r o b l e m :

a . n i t e d i e r e n c e s 1 1 ] , 1 . 7 ] , 3 . 3 6 ] , 4 . 2 0 ] t o 4 . 2 4 ]

b . n i t e e l e m e n t m e t h o d s ( F E M ) 4 . 2 5 ] t o 4 . 2 7 ] , 4 . 2 9 ] , 4 . 3 0 ] ( a p p r o p r i a t e

f o r t h e w h o l e d o m a i n )

c . b o u n d a r y e l e m e n t m e t h o d s ( B E M ) 4 . 1 8 ] , 4 . 2 7 ] , 4 . 2 8 ] ( a p p l i c a b l e o n l y

i f t h e d o m a i n i s h o m o g e n e o u s , n o d e n s i t y d i s t r i b u t i o n o r l o c a l l y v a r y i n g

c o e c i e n t s i n t h e d i e r e n t i a l e q u a t i o n

d . c o l l o c a t i o n m e t h o d s 1 . 7 ] , 4 . 3 1 ] , 4 . 3 2 ] , 4 . 4 2 ] a s u s e d i n s e c t i o n 1 . 3 a n d

i n p r o b l e m 5 o f s e c t i o n 4 . 7

e . L i e s e r i e s m e t h o d s 4 . 1 3 ] , 4 . 2 0 ]

f . v a r i a t i o n a l m e t h o d s , s e e ( 4 . 7 . 2 2 ) .

U p t o n o w w e m a i n l y c o n s i d e r e d a n a l y t i c a l s o l u t i o n s . L e t u s n o w i n v e s t i -

g a t e t h e i n u e n c e o f a s i n g u l a r i t y i n t h e s o l u t i o n . W e r s t d i s c u s s a s i m p l e

h o m o g e n e o u s p r o b l e m o f t h e L a p l a c e e q u a t i o n f o r a r e c t a n g l e a

b

U ( x y ) = 0 ( 4 . 8 . 1 )

U ( a y ) = 0 ; b y + b ( 4 . 8 . 2 )


7/18/2019 C4029-Ch04



U ( x b ) = f ( x ) = c o s ( x = a ) ; a x + a ( 4 . 8 . 3 )

w h e r e w e s e t l a t e r a = 2 b = 1 , s e e F i g u r e 4 . 1 5 . T h e s o l u t i o n t o ( 4 . 8 . 1 ) t o

( 4 . 8 . 3 ) i s g i v e n b y

U ( x y ) =

1

X

n = 0

A

n

c o s

( 2 n + 1 ) x

2 a

c o s h

( 2 n + 1 ) y

2 a

( 4 . 8 . 4 )

w h e r e

A

n

=

1

a c o s h

( 2 n + 1 ) b

2 a

+ a

Z

; a

f ( ) c o s

( 2 n + 1 )

2 a

d : ( 4 . 8 . 5 )

T h e s o l u t i o n ( 4 . 8 . 4 ) i s s h o w n i n F i g u r e 4 . 1 5 .

F i g u r e 4 . 1 5

I n h o m o g e n e o u s b o u n d a r y p r o b l e m ( 4 . 8 . 4 )

T h e f o l l o w i n g M a t h e m a t i c a c o m m a n d r e a l i z e d t h e c a l c u l a t i o n s ( 4 . 8 . 5 ) a n d

g e n e r a t e d F i g u r e 4 . 1 5 .

A[n_]=Integrate[Cos[Pi*x/2.]*Cos[(2*n+1)*Pi*x/4.],

{x,-2.,2.}]/(2*Cosh[(2*n+1)*Pi/4.]);

Clear[U];U[x_,y_]=Sum[A[n]*Cos[(2*n+1)*Pi*x/4.]

*Cosh[(2*n+1)*Pi*y/4.],{n,50}];

A[17]

3 . 8 3 6 2 4 1 0

; 1 4

Plot3D[U[x,y], {x,-2.,2.},{y,-1.,1.},

PlotRange->{-1.,1.0},PlotPoints->40]


7/18/2019 C4029-Ch04



T h e t r a n s i t i o n t o a h o m o g e n e o u s b o u n d a r y p r o b l e m f ( x ) ! 0 r e s u l t s i m -

m e d i a t e l y i n t h e t r i v i a l s o l u t i o n U ( x y ) = 0 , s i n c e a l l A

n

! 0 . T h i s c o n r m s

t h e t h e o r e m t h a t t h e L a p l a c e e q u a t i o n h a s o n l y t h e t r i v i a l ( a n a l y t i c ) s o l u -

t i o n f o r h o m o g e n e o u s b o u n d a r y c o n d i t i o n s . C o u l d a s i n g u l a r i t y m o d i f y t h e

s i t u a t i o n ? I t i s k n o w n t h a t t h e l o g a r i t h m o f a s q u a r e r o o t i s a s i n g u l a r p a r -

t i c u l a r s o l u t i o n o f ( 4 . 8 . 1 ) , s e e p r o b l e m 1 i n t h i s s e c t i o n . W e a d d s u c h a t e r m

t o ( 4 . 8 . 4 ) s o t h a t

U ( x y ) =

1

X

n = 0

A

n

c o s

( 2 n + 1 ) x

2 a

c o s h

( 2 n + 1 ) y

2 a

+

1

X

n = 0

B

n

c o s

( 2 n + 1 ) y

2 b

c o s h

( 2 n + 1 ) x

2 b

; l n

r

x

2

+ y

2

a

2

+ b

2

: ( 4 . 8 . 6 )

T h e n t h e h o m o g e n e o u s b o u n d a r y c o n d i t i o n s y i e l d 4 . 3 3 ]

A

n

=

1

a c o s h

( 2 n + 1 ) b

2 a

+ a

Z

; a

l n

r

x

2

+ b

2

a

2

+ b

2

c o s

( 2 n + 1 ) x

2 a

d x

B

n

=

1

b c o s h

( 2 n + 1 ) a

2 b

+ b

Z

; b

l n

r

a

2

+ y

2

a

2

+ b

2

c o s

( 2 n + 1 ) y

2 b

d y : ( 4 . 8 . 7 )

T h e l o g a r i t h m i c t e r m s a t i s e s t h e h o m o g e n e o u s b o u n d a r y c o n d i t i o n s i n t h e

c o r n e r s . F o r n = 0 : : : 3 t h e b o u n d a r y c o n d i t i o n s a r e s a t i s e d w i t h a n a c c u -

r a c y 1 0

; 4

. T h e s a m e r e s u l t m a y b e o b t a i n e d b y a c o l l o c a t i o n m e t h o d . E i g h t

c o l l o c a t i o n p o i n t s ( x

i

b ) 0 x

i

a ( a y

i

) 0 y

i

b i = 1 : : : 4 p r o d u c e

e i g h t l i n e a r e q u a t i o n s f o r t h e A

n

( a n d B

n

)

3

X

n = 0

A

n

c o s

( 2 n + 1 ) x

i

4

c o s h

( 2 n + 1 )

4

= l n

r

x

2

i

+ 1

5

( 4 . 8 . 8 )

w h i c h a v o i d s t h e n u m e r i c a l i n t e g r a t i o n s ( 4 . 1 . 7 ) . F i g u r e 4 . 1 6 s h o w s t h e s o l u t i o n

( 4 . 8 . 6 ) .


7/18/2019 C4029-Ch04


F i g u r e 4 . 1 6

H o m o g e n e o u s b o u n d a r y p r o b l e m o f t h e L a p l a c e e q u a t i o n

H a v i n g s e e n t h e g o o d r e s u l t s o b t a i n e d b y t h e c o l l o c a t i o n m e t h o d i t m i g h t

b e a p p r o p r i a t e t o d i s c u s s t h i s m e t h o d i n d e t a i l . W h e r e s h o u l d t h e c o l l o c a t i o n

p o i n t s b e c h o s e n ? F

e j e r a n s w e r s : f o r a c i r c l e t h e y s h o u l d b e d e t e r m i n e d b y

t h e u n i t r o o t s i n t h e c o m p l e x p l a n e a n d f o r o t h e r b o u n d a r i e s t h e c o n f o r m a l

m a p p i n g m a y h e l p 4 . 3 4 ] t o 4 . 3 6 ] . T h e q u e s t i o n o f t h e c o n v e r g e n c e o f c o l l o -

c a t i o n m e t h o d s h a s b e e n d i s c u s s e d b y G r i s v a r d 4 . 3 7 ] ( g u a r a n t e e s t h e c o n -

v e r g e n c e o f t h e s o l u t i o n b e t w e e n c o l l o c a t i o n p o i n t s t o w a r d t h e r e a l s o l u t i o n )

a n d E i s e n s t a t 4 . 3 8 ] . C o l l o c a t i o n m e t h o d s 4 . 3 9 ] , 4 . 4 0 ] u s e m a i n l y e x a c t

s o l u t i o n s ( l i k e h a r m o n i c p o l y n o m i a l s e t c . ) o f t h e p e r t i n e n t p a r t i a l d i e r e n t i a l

e q u a t i o n s l i k e t h e L a p l a c e o r H e l m h o l t z e q u a t i o n s .

C o l l o c a t i o n m e t h o d s m a y a l s o b e u s e d t o d e n e n o d a l l i n e s a s n e w b o u n d -

a r i e s . I t i s e a s y t o v e r i f y t h a t

U ( x y ) = A c o s k x + B c o s

p

k

2

;

2

x

c o s y ( 4 . 8 . 9 )

i s a s o l u t i o n o f t h e t w o - d i m e n s i o n a l H e l m h o l t z e q u a t i o n U

x x

+ U

y y

+ k

2

U = 0 ,

c o m p a r e ( 1 . 1 . 5 5 ) . N o w t h e g e o m e t r i c a l l o c u s o f a l l p o i n t s ( x

i

y

i

) a t w h i c h

U ( x

i

y

i

) = U ( x

i

y

i

( x

i

) ) = 0 r e p r e s e n t s a n o d a l l i n e . F r o m ( 4 . 8 . 9 ) o n e o b t a i n s

a n o d a l l i n e

y ( x ) =

1

a r c c o s

"

; A c o s k x

B c o s (

p

k

2

;

2

x )

#

: ( 4 . 8 . 1 0 )

P l a y i n g w i t h t h e t h r e e p a r a m e t e r s A = B k m a y g e n e r a t e v a r i o u s b o u n d a r i e s .

O n t h e o t h e r h a n d w e c a n a s s u m e a c i r c u l a r b o u n d a r y y =

p

R

2

; x

2

i n


7/18/2019 C4029-Ch04



c a r t e s i a n c o o r d i n a t e s a n d w e c o u l d t r y t o d e r i v e t h e e i g e n v a l u e k o f a c i r c u l a r

m e m b r a n e 4 . 3 1 ] . L e t u s c o n s i d e r a c i r c u l a r m e m b r a n e o f r a d i u s R ( = 1 c m ) .

T h e e i g e n v a l u e c a n b e c a l c u l a t e d e x a c t l y i n c y l i n d r i c a l c o o r d i n a t e s r ' . F o r

a x i a l s y m m e t r y , o r @ u = @ ' = 0 o n e o b t a i n s k = 2 : 4 0 5 . C a n w e r e p r o d u c e t h i s

v a l u e w i t h t h e n e w m e t h o d ? W e c h o o s e t h e c a r t e s i a n s o l u t i o n o f u

x x

+ u

y y

+

k

2

u = 0 w h i c h i s s y m m e t r i c i n x a n d y . W r i t i n g a F O R T R A N p r o g r a m f o r

( 4 . 8 . 1 0 ) w i t h t w o D O - l o o p s ( f o r v a r i a b l e s k a n d ) w e n d f o r R = 1 t h a t

= 2 : 0 7 8 0 a n d k = 2 : 4 0 1 9 a n d t h a t ( 4 . 8 . 1 0 ) d e s c r i b e s a c i r c l e . N o w w e h a v e

t o c o n t r o l i f ( 4 . 8 . 1 0 ) d e s c r i b e s a c o n t i n u o u s f u n c t i o n w i t h n o s i n g u l a r i t y a n d

i f t h e a c c u r a c y w i t h w h i c h i t d e s c r i b e s t h e c i r c l e x

2

+ y

2

= R

2

i s s a t i s f y i n g ,

w h i c h w a s t h e c a s e . W h e n w e s o l v e d w i t h t w o \ s e p a r a t i o n c o n s t a n t s " a n d

w i t h t h e c o l l o c a t i o n p o i n t s ( 0 , 1 ) ( 1 , 0 ) ( 0 . 7 0 7 1 , 0 . 7 0 7 1 ) w e o b t a i n e d t h e r e s u l t s

1

= 2 : 2 2 2 0 9 6 4 7 3

2

= 0 : 9 1 9 4 9 6 1 8 1 a n d k = 2 : 4 0 4 8 2 5 5 5 7 4 1 5 . T h e e x a c t

v a l u e s k g i v e n f r o m t h e z e r o o f t h e B e s s e l f u n c t i o n J

0

i s 2 : 4 0 4 8 2 5 5 5 7 0 6 9 6 .

W e t h u s a l s o h a v e a m e t h o d t o c a l c u l a t e h i g h e r z e r o e s o f B e s s e l f u n c t i o n s :

N u m b e r o f c o l l o c a t i o n p o i n t s e i g e n v a l u e e x a c t v a l u e s

2 2 . 4 0 4 1 9 2 . 4 0 4 8 2 5 5 5 7 6

3 2 . 4 0 4 8 2 5 5 5 7 4 2 . 4 0 4 8 2 5 5 5 7 6

3 5 . 5 2 0 0 7 8 1 1 0 3 5 . 5 2 0 0 7 8 1 1 0 3

4 8 . 6 5 3 7 2 7 9 1 2 9 1 8 . 6 5 3 7 2 7 9 1 2 9

5 1 1 . 7 9 1 5 3 4 4 3 9 0 1 1 1 . 7 9 1 5 3 4 4 3 9 1

6 1 4 . 9 3 0 9 1 7 7 0 8 4 9 1 4 . 9 3 0 9 1 7 7 0 8 6

7 1 8 . 0 7 1 0 6 3 9 6 7 9 1 1 8 . 0 7 1 0 6 3 9 6 7 9

8 2 1 . 2 1 1 6 3 6 6 2 9 8 8 2 1 . 2 1 1 6 3 6 6 2 9 9

F o r a g r e a t e r n u m b e r o f c o l l o c a t i o n p o i n t s , t h e m e t h o d s t o s o l v e a s e t o f

t r a n s c e n d e n t a l e q u a t i o n s f a i l o r g i v e o n l y t h e t r i v i a l s o l u t i o n . W e a r e t h e r e f o r e

u s i n g a n o t h e r m e t h o d t h a t a v o i d s t h e s o l u t i o n o f n o n l i n e a r o r t r a n s c e n d e n t a l

e q u a t i o n s .

W e n o w c o n s i d e r a g e n e r a l i z a t i o n o f ( 4 . 8 . 9 ) i n t h e f o r m

N

X

n = 1

A

n

c o s

p

k

2

;

2

n

x

i

c o s

n

q

1 ; x

2

i

= 0 i = 1 : : : P : ( 4 . 8 . 1 1 )

W i t h t h e b o u n d a r y c o n d i t i o n o n a c i r c l e a n d a s o l u t i o n i n c a r t e s i a n c o o r d i -

n a t e s w e h a v e

u ( r = R = 1 ) = 0 y =

p

1

; x

2

u ( x y ) =

p

1 ; x

2

=

N

X

n = 1

A

n

c o s

p

k

2

;

2

n

x

i

c o s

n

q

1 ; x

2

i

= 0 :

( 4 . 8 . 1 2 )

k : e i g e n v a l u e , N : n u m b e r o f p a r t i a l s o l u t i o n s a n d t h e A

n

a r e t h e a m p l i t u d e s

o f t h e p a r t i a l m o d e s . T h e n u m b e r P o f c o l l o c a t i o n p o i n t s x

i

y

i

i = 1 : : : P


7/18/2019 C4029-Ch04



o n t h e c i r c l e x

2

i

+ y

2

i

= 1 . F o r A

1

= 1 a n d t h e r e m a i n i n g 2 N u n k n o w n s

A

2

: : : A

n

1

;

N

a n d k b y u s e o f a c o m p u t e r c o d e f o r N = 2 P = 2 N = 4 ,

w e o b t a i n

1

= 2 : 2 2 2 0 9

2

= 0 : 9 1 9 4 9 A

2

= 1 a n d k = 2 : 4 0 4 8 2 5 5 ( e x a c t

v a l u e u p t o 6 d i g i t s ) . F o r x e d P t h e a c c u r a c y r e m a i n s x e d . C h o o s i n g

a r b i t r a r i l y t h e

n

n = 1 2 : : : N a n d A

1

= 1 , o n l y t h e P = N u n k n o w n s

A

n

n = 2 : : : N a n d t h e e i g e n v a l u e k r e m a i n t o b e d e t e r m i n e d . T h e n ( 4 . 8 . 1 1 )

c o n s t i t u t e s a s y s t e m o f h o m o g e n e o u s l i n e a r e q u a t i o n s f o r t h e A

n

. T o n d a

n o n t r i v i a l s o l u t i o n , t h e d e t e r m i n a n t D ( k )

D = j D

i n

j = j c o s

p

k

2

;

2

n

x

i

c o s

n

p

1 ; x

2

i

j = 0

i = 1 : : : P n = 1 : : : P ( 4 . 8 . 1 3 )

h a s t o v a n i s h . T h i s d e t e r m i n e s k . S o l v i n g n o w f o r t h e A

n

n = 2 : : : N A

1

= 1

o n e o b t a i n s t h e s o l u t i o n . T h e

n

h a d b e e n c h o s e n a r b i t r a r i l y ,

2

n

< k

2

.

A s i m p l e F O R T R A N p r o g r a m d o e s t h e j o b t o s o l v e u

x x

+ u

y y

+ k

2

u = 0

f o r a c l a m p e d c i r c u l a r m e m b r a n e . T h e p r o g r a m C O L L O C . f r e a d s f o r P = 8

c o l l o c a t i o n p o i n t s a n d 8 a r b i t r a r y s e p a r a t i o n c o n s t a n t s B E T A . T h e n u m b e r P

s h o u l d n o t b e v e r y l a r g e ( < 2 0 ) . I f t h e n u m b e r o f c o l l o c a t i o n p o i n t s i s l a r g e ,

i n a c c u r a c i e s , o s c i l l a t i o n s m a y a p p e a r a n d t h e a c c u r a c y g o e s d o w n .

C23456PROGRAM COLLOC

IMPLICIT NONE

INTEGER P,I,J,III,IT,L

PARAMETER (IT=80,P=8)

INTEGER INDX(P)

DOUBLE PRECISION DETN(P,P),DET(P,P),EIGENVALUE,

*BETA(P),DELTA

DOUBLE PRECISION DEIGENVALUE,D,DETV(IT),X0(P),

*Y0(P)

OPEN (UNIT=2, FILE="COLLOCRes")

C GUESS OF EIGENVALUE

C L=1

C IF(L.GT.1) GOTO 22

WRITE(6,*)’GIVE GUESS OF EIGENVALUE’

READ (5,*) EIGENVALUE

22 DEIGENVALUE=EIGENVALUE/10

WRITE(2,*)’FIRST EIGENVALUE= ’,EIGENVALUE

DELTA=EIGENVALUE/P

BETA(1)=EIGENVALUE-0.0001

DO 44 I=1,P-1

BETA(I+1)=BETA(I)-DELTA

44 CONTINUE

DO 444 I=1,P

WRITE(2,*)’ BETA(’,I,’)=’,BETA(I)


7/18/2019 C4029-Ch04



444 CONTINUE

CALL DATA (P,X0,Y0,L)

C WRITE(2,*) ’ EIGENVALUE = , DETERMINANT = ’

DO 999 III=1,IT

CALL FILLUP (DETN,EIGENVALUE,BETA,X0,Y0,P)

CALL LUD(DETN,P,INDX,D)

DO 20 J=1,P

D=D*DETN(J,J)

20 CONTINUE

DETV(III)=D

IF(III.EQ.1)GOTO 999

C IS THERE A CHANGE OF SIGN ?

IF(DETV(III)*DETV(III-1).LT.0.)THEN

DEIGENVALUE=-DEIGENVALUE/2

ENDIF

C WRITE(2,77) EIGENVALUE,DETV(III)

C 77 FORMAT(1X, D19.13,3X,D19.13)

EIGENVALUE=EIGENVALUE+DEIGENVALUE

999 CONTINUE

WRITE(2,*)’LAST EIGENVALUE: ’

WRITE(6,*)’LAST EIGENVALUE: ’

55 FORMAT(F16.10)

CALL FILLUP (DETN,EIGENVALUE,BETA,X0,Y0,P)

C WRITE(2,55) EIGENVALUE-DEIGENVALUE

WRITE(6,55) EIGENVALUE-DEIGENVALUE

C FILL DET WHICH WILL BE MODIFIED IN LUD AND

C DETN WILL BE NEEDED

DO 1 I=1,P

DO 2 J=1,P

DET(I,J)=DETN(I,J)

2 CONTINUE

1 CONTINUE

CALL LINEQ (DET,DETN)

C LOOK FOR NEXT EIGENVALUE

C L=L+1

C IF(L.GT.1)THEN

C EIGENVALUE=EIGENVALUE+0.1

C IF(L.GT.6)GOTO 100

C GOTO 22

C ENDIF

100 WRITE(6,*)’I HAVE FINISHED’

WRITE(6,*) CHAR(7)

STOP

END


7/18/2019 C4029-Ch04


7/18/2019 C4029-Ch04



IF (ABS(A(I,J)).GT.AAMAX) AAMAX=ABS(A(I,J))

11 CONTINUE

IF (AAMAX.EQ.0.) PAUSE ’Singular matrix.’

VV(I)=1./AAMAX

12 CONTINUE

DO 19 J=1,P

DO 14 I=1,J-1

SUM=A(I,J)

DO 13 K=1,I-1

SUM=SUM-A(I,K)*A(K,J)

13 CONTINUE

A(I,J)=SUM

14 CONTINUE

AAMAX=0.

DO 16 I=J,P

SUM=A(I,J)

DO 15 K=1,J-1

SUM=SUM-A(I,K)*A(K,J)

15 CONTINUE

A(I,J)=SUM

DUM=VV(I)*ABS(SUM)

IF (DUM.GE.AAMAX) THEN

IMAX=I

AAMAX=DUM

ENDIF

16 CONTINUE

IF (J.NE.IMAX)THEN

DO 17 K=1,P

DUM=A(IMAX,K)

A(IMAX,K)=A(J,K)

A(J,K)=DUM

17 CONTINUE

D=-D

VV(IMAX)=VV(J)

ENDIF

INDX(J)=IMAX

IF(A(J,J).EQ.0.)A(J,J)=TINY

IF(J.NE.P)THEN

DUM=1./A(J,J)

DO 18 I=J+1,P

A(I,J)=A(I,J)*DUM

18 CONTINUE

ENDIF

19 CONTINUE

RETURN


7/18/2019 C4029-Ch04



END

SUBROUTINE LINEQ (DET,DETN)

C23456SOLVES SYSTEM HOMOG. LINEAR EQUATIONS

IMPLICIT NONE

INTEGER P,I,J,JF,L,K,NF,IF

PARAMETER(P=8)

DOUBLE PRECISION AF(P-1,P-1),BF(P-1),A(P),

*DETN(P,P),BOUNDARY(P)

DOUBLE PRECISION C(P-1,P-1),X(P-1),PQ,DET(P,P)

INTEGER INDX(P-1)

A(P)=1.000000

C REDUCTION OF SIZE OF HOMOG. LINEAR SYSTEM

NF=P-1

DO 501 IF=1,NF

BF(IF)=-DET(IF,P)

DO 501 JF=1,NF

AF(IF,JF)=DET(IF,JF)

501 CONTINUE

DO 12 L=1,NF

DO 11 K=1,NF

C(K,L)=AF(K,L)

11 CONTINUE

12 CONTINUE

CALL LUD(C,NF,INDX,PQ)

DO 13 L=1,NF

X(L)=BF(L)

13 CONTINUE

CALL LUB(C,NF,INDX,X)

DO 33 I=1,NF

A(I)=X(I)

33 CONTINUE

WRITE(2,*) ’PARTIAL AMPLITUDES ARE: ’

WRITE(6,*) ’PARTIAL AMPLITUDES ARE: ’

WRITE(2,’(1X,6F12.4)’) (A(L), L=1,P)

WRITE(6,’(1X,6F12.4)’) (A(L), L=1,P)

DO 10 I=1,P

BOUNDARY(I)=0.

DO 10 J=1,P

BOUNDARY(I)=BOUNDARY(I)+DETN(I,J)*A(J)

10 CONTINUE

DO 4 I=1,P

WRITE(2,*)’BOUNDARY(’,I,’)= ’, BOUNDARY(I)

WRITE(6,*)’BOUNDARY(’,I,’)= ’, BOUNDARY(I)

4 CONTINUE


7/18/2019 C4029-Ch04



RETURN

END

SUBROUTINE LUB(A,NF,INDX,B)

C23456SOLVES REDUCED LINEAR EQUATIONS

IMPLICIT NONE

INTEGER NF,I,LL,J,II,INDX(NF)

DOUBLE PRECISION A(NF,NF),B(NF),SUM

II=0

DO 12 I=1,NF

LL=INDX(I)

SUM=B(LL)

B(LL)=B(I)

IF (II.NE.0)THEN

DO 11 J=II,I-1

SUM=SUM-A(I,J)*B(J)

11 CONTINUE

ELSE IF (SUM.NE.0.) THEN

II=I

ENDIF

B(I)=SUM

12 CONTINUE

DO 14 I=NF,1,-1

SUM=B(I)

IF(I.LT.NF)THEN

DO 13 J=I+1,NF

SUM=SUM-A(I,J)*B(J)

13 CONTINUE

ENDIF

B(I)=SUM/A(I,I)

14 CONTINUE

RETURN

END

T h e e x e c u t a b l e l e m a y b e c a l l e d c i r c m e m a n d t h e r e s u l t s a r e w r i t t e n t o

t h e l e C O L L O C R e s t h a t r e a d s

Output on the screen from program circmem

GIVE GUESS OF EIGENVALUE

2.4

LAST EIGENVALUE:

2.4048255577

PARTIAL AMPLITUDES ARE:

.1773 .4925 -.6585 1.8660 -2.7769 3.4474


7/18/2019 C4029-Ch04



-2.4165 1.0000

BOUNDARY( 1)= -4.440892098500626E-16

BOUNDARY( 2)= .0

BOUNDARY( 3)= .0

BOUNDARY( 4)= 1.387778780781446E-17

BOUNDARY( 5)= -8.326672684688674E-17

BOUNDARY( 6)= .0

BOUNDARY( 7)= 1.110223024625157E-16

BOUNDARY( 8)= 1.271205363195804E-13

I HAVE FINISHED

T h e m e t h o d p r e s e n t e d m a y b e s u b j e c t e d t o t h e f o l l o w i n g c r i t i c i s m :

1 . I s i t t r u e t h a t o n e o b t a i n s t h e c o r r e c t e i g e n f r e q u e n c y i n t h e c a s e o f a

m e m b r a n e w i t h a b o u n d a r y t h a t i s n o t a c i r c l e a n d o f w h i c h t h e e i g e n v a l u e i s

u n k n o w n ? H e r e o n e h a s s o m e c o n t r o l f r o m t h e F a b e r t h e o r e m , w h i c h s t a t e s

4 . 1 ] t h a t f r o m a l l m e m b r a n e s ( p l a t e s ) o f a r b i t r a r y f o r m b u t w i t h t h e s a m e

s u r f a c e o r s a m e c i r c u m f e r e n c e t h e c i r c u l a r m e m b r a n e ( p l a t e ) h a s t h e l o w e s t

e i g e n v a l u e . S l i g h t l y m o d i f y i n g a c i r c u l a r s h a p e i n t o a n e l l i p t i c o r C a s s i n i

c u r v e s h a p e g i v e s a c o n t r o l . T h e i m p l i c i t p l o t t i n g o f t h e h o m o g e n e o u s b o u n d -

a r y c o n d i t i o n s l i k e ( 4 . 8 . 1 2 ) a l s o g i v e s c o n d e n c e i n t h e c o r r e c t n e s s o f t h e

s o l u t i o n , s i n c e ( 4 . 8 . 1 2 ) m u s t d e l i v e r t h e o r i g i n a l b o u n d a r y s h a p e , i n o u r c a s e

a c i r c l e . O n t h e o t h e r h a n d , G r i s v a r d ' s t h e o r e m g u a r a n t e e s t h e c o n v e r g e n c e

o f t h e s o l u t i o n b e t w e e n c o l l o c a t i o n p o i n t s t o w a r d t h e r e a l s o l u t i o n .

2 . F i n a l l y , f o r n o n c i r c u l a r b o u n d a r i e s t h e s h a p e s m a y b e a c c e p t e d a s n o d a l

l i n e s o f a l a r g e r c i r c u l a r b o u n d a r y . T h e n C o u r a n t ' s t h e o r e m g u a r a n t e e s

a n a c c e p t a b l e s o l u t i o n f o r t h e l o w e s t e i g e n v a l u e . I t s t a t e s 4 . 4 1 ] : F o r a s e l f -

a d j o i n t , s e c o n d - o r d e r d i e r e n t i a l e q u a t i o n a n d a h o m o g e n e o u s b o u n d a r y c o n -

d i t i o n a l o n g a c l o s e d d o m a i n , t h e n o d a l l i n e s o f t h e n - t h e i g e n v a l u e d i v i d e

t h e d o m a i n i n t o n o m o r e t h a n n s u b d o m a i n s . T h e l o w e s t e i g e n v a l u e d o e s n o t

d i v i d e t h e d o m a i n .

3 . O n e m a y a l s o c r i t i c i z e t h a t t h e a r b i t r a r y c h o i c e o f t h e \ s e p a r a t i o n c o n -

s t a n t s " c o u l d l e a d t o w r o n g o r o n l y v e r y a p p r o x i m a t e r e s u l t s .

I f n o i n c r e a s i n g f u n c t i o n s l i k e c o s h ( a p p e a r s i n t h e s o l u t i o n o f t h e p l a t e

e q u a t i o n , s e e ( 4 . 5 . 8 4 ) ) a r e p r e s e n t , c o l l o c a t i o n m e t h o d s y i e l d a c c e p t a b l e r e -

s u l t s . T o d i s c u s s t h e d e p e n d e n c e o f t h e r e s u l t s o n t h e a r b i t r a r y c h o i c e o f t h e

s e p a r a t i o n c o n s t a n t s

n

(

2

n

< k

2

) i n ( 4 . 8 . 1 2 ) i t i s n e c e s s a r y t o m a k e s e v e r a l

c a l c u l a t i o n s w i t h v a r y i n g s e t s o f

n

. F o r t h i s p u r p o s e w e p r e s e n t a M a t h e m a t -

i c a p r o g r a m . I t i s e x e c u t e d i n c a r t e s i a n c o o r d i n a t e s a n d s o l v e s t h e m e m b r a n e

e q u a t i o n u

x x

+ u

y y

+ k

2

u = 0 f o r t h e h o m o g e n e o u s b o u n d a r y c o n d i t i o n s o f a

c l a m p e d c i r c u l a r m e m b r a n e .

T h e M a t h e m a t i c a p r o g r a m C o l m e i g v . r e a d s a s f o l l o w s . F i r s t n c o l l o c a t i o n

p o i n t s a r e c h o s e n o n t h e c i r c u l a r b o u n d a r y o f r a d i u s 1 . I n o r d e r t h a t

Piw i l l

b e u s e d w i t h 6 d i g i t s a c c u r a c y o n e m a y w r i t e N[Pi,6]


7/18/2019 C4029-Ch04



(* Colmeigv.: Eigenvalue problem for

circular membrane.Calculation in cartesian coordinates.

1.step: give n collocation points and separation

constants b *)

n=4; dth=N[Pi/(2*n),6];

Table[x[l]=Cos[l*dth],{l,1,n}];

Table[y[l]=Sin[l*dth],{l,1,n}];

Table[{x[l],y[l]},{l,1,n}];

(* Are the collocation points on the circle? *)

Table[x[l]ˆ2+y[l]ˆ2,{l,1,n}]

A f t e r c o n t r o l i f t h e c h o s e n n c o l l o c a t i o n p o i n t s x

l

y

l

a r e r e a l l y s i t u a t e d o n t h e

b o u n d a r y x

2

l

+ y

2

l

= 1 , a n a r b i t r a r y r s t g u e s s o f t h e e i g e n v a l u e , t h e s t e p i n

t h e e i g e n v a l u e i s g i v e n a n d t h e s e p a r a t i o n c o n s t a n t s b

l

a r e c h o s e n .

firsteigenv=2.000000000000; delta=firsteigenv/n;

b[1]=firsteigenv-0.0000001;

Table[b[l+1]=b[l]-delta,{l,1,n-1}];( 4 . 8 . 1 4 )

Table[b[l],{l,1,n}];

I n t h e s e c o n d s t e p t h e e i g e n v a l u e i s c a l c u l a t e d f r o m t h e h o m o g e n e o u s b o u n d -

a r y c o n d i t i o n

(* 2.step: calculate the eigenvalue from

homogeneous boundary conditions *)

Clear[eigenvalue];

f[eigenvalue_]=Det[Table[Cos[b[mi]*y[ipi]]*

Cos[Sqrt[eigenvalueˆ2-b[mi]ˆ2]*x[ipi]],

{ipi,1,n},{mi,1,n}]];

(* 3.step: make a plot to find another guess

for the first eigenvalue.Try 2 methods for

finding roots of the determinant *)

Plot[f[eigenvalue],{eigenvalue,0.,3.}]

T h e p l o t ( n o t s h o w n h e r e ) p r e s e n t s t h e c u r v e f ( e i g e n v a l u e ) , w h i c h c r o s s e s

f = 0 a t e i g e n v a l u e = 2 . 4 .

T h e n

FindRoot[f[eigenvalue]==0,

{eigenvalue,2.0}] //Timing ( 4 . 8 . 1 5 )

f 0 . 3 6 S e c o n d , f e i g e n v a l u e ! 2 . 4 0 5 9 3 g g

r e n e s t h e v a l u e a n d i n f o r m s t h e u s e r t h a t t h i s c a l c u l a t i o n s n e e d e d 0 . 3 6 s e c .

I n o r d e r t o c a l c u l a t e t h e u n k n o w n p a r t i a l a m p l i t u d e s

Ampt h e m a t r i x m i s

d e n e d a n d c a l c u l a t e d

(* 4.step: define the matrix M *) eigenvalue=2.40593;

Clear[m];m=Table[j*k,{j,1,n},{k,1,n}];

Table[m[[ipi,mi]]=Cos[b[mi]*y[ipi]]*

Cos[Sqrt[eigenvalueˆ2-b[mi]ˆ2]*x[ipi]],{ipi,1,n},


7/18/2019 C4029-Ch04



{mi,1,n}];nf=n-1;

Det[m]

3 . 0 3 6 1 1 0

; 9

.

S i n c e t h e v a l u e o f t h e d e t e r m i n a n t i s n e a r l y z e r o ( 1 0

; 9

) , t h e s y s t e m o f l i n e a r

e q u a t i o n s f o r t h e a m p l i t u d e s i s h o m o g e n e o u s a n d o n e h a s t o r e w r i t e t h e l i n e a r

s y s t e m b y r e d u c i n g t h e o r d e r n o f t h e d e t e r m i n a n t t o n

f

= n ; 1 . T h e l a s t r h s

c o l u m n o f t h e m a t r i x m b e c o m e s t h e r h s o f t h e l i n e a r s y s t e m f o r n

f

u n k n o w n s

a n d a r b i t r a r i l y t h e l a s t u n k n o w n A

n

i s s e t t o 1 .

(* 5.step: solve the homogeneous linear equations *)

bbf=Table[-m[[ifit,n]],{ifit,1,nf}];

rdutn=Table[m[[ifit,klfit]],{ifit,1,nf},{klfit,1,nf}];

B=LinearSolve[rdutn,bbf]; ( 4 . 8 . 1 6 )

An={1.}; A=Table[B[[lk]],{lk,1,nf}];

(*6.step: check satisfaction of the homogeneous

boundary condition.*)

Amp=Join[A,An];

boundary = m . Amp ( 4 . 8 . 1 7 )

f 0 . , - 6 . 9 3 8 8 9 1 0

; 1 7

, 0 . , - 0 . 0 0 0 1 1 4 7 7 5 g .

T h e o u t p u t o b t a i n e d b y i n s e r t i n g t h e u n k n o w n s i n t o t h e s y s t e m o f l i n e a r

e q u a t i o n s d e m o n s t r a t e s t h a t t h e r s t t h r e e u n k n o w n s s a t i s f y e x a c t l y t h e l i n e a r

s y s t e m , b u t a s i s v e r y w e l l k n o w n , t h e c h o s e n u n k n o w n A

n

s a t i s e s t h e s y s t e m

o n l y r o u g h l y . I n p r o b l e m 1 o f t h i s s e c t i o n w e w i l l d i s c u s s t h e n e w c o m m a n d s

i n ( 4 . 8 . 1 5 ) t o ( 4 . 8 . 1 7 ) . T o c h e c k i f t h e n u m e r i c a l s o l u t i o n

f ( x y ) =

n

X

m = 1

A m p

m i

c o s ( b

m i

y ) c o s

q

e i g e n v

2

; b

2

m i

x

( 4 . 8 . 1 8 )

s a t i s e s t h e b o u n d a r y c o n d i t i o n s , a n u m e r i c a l a n d a g r a p h i c a l c h e c k a r e m a d e :

fxy[x_,y_]:=Sum[Amp[[mi]]*Cos[b[mi]*y]*

Cos[Sqrt[eigenvalueˆ2-b[mi]ˆ2]*x],{mi,1,n}]

Do[Print[fxy[x[l],y[l]]],{l,n}]

0 .

- 9 . 7 1 4 4 5 1 4 6 5 4 7 0 1 2

; 1 7

- 2 . 2 2 0 4 4 6 0 4 9 2 5 0 3 1 3

; 1 6

- 0 . 0 0 0 1 1 4 7 7 5

plot1=ContourPlot[fxy[x,y],{x,-1.05,1.05},{y,-1.05,1.05},

ContourShading->False,ContourSmoothing->2,

PlotPoints->60,Contours->{0},DisplayFunction->Identity];

plot2=ListPlot[Table[{x[i],y[i]},{i,1,n}],

Frame->True, AspectRatio->1.,

PlotRange->{{-1.,1.},{-1.,1.}},

PlotStyle->PointSize[1/60],DisplayFunction->Identity];

Show[plot1,plot2,DisplayFunction->$DisplayFunction]


7/18/2019 C4029-Ch04



T h e s e c o m m a n d s g e n e r a t e f ( x y ) = 0 d e s c r i b i n g t h e c i r c u l a r b o u n d a r y a n d

s h o w t h e l o c a t i o n o f t h e c o l l o c a t i o n p o i n t s , s e e F i g u r e 4 . 1 7

O t h e r e x a m p l e s o f t h e u s e o f c o l l o c a t i o n m e t h o d s w i l l b e g i v e n i n c h a p t e r 5 .

A w a r n i n g i s n e c e s s a r y : w h e n u s i n g c o l l o c a t i o n m e t h o d s v e r y o f t e n a m a t r i x m

( a n d i t s d e t e r m i n a n t ) h a v e t o b e c a l c u l a t e d . C a r e i s n e c e s s a r y t h a t t h e m a t r i x

b e w e l l - c o n d i t i o n e d . A r r a n g e m e n t s i n t h e m a t r i x s h o u l d b e m a d e s u c h t h a t

t h e H a d a m a r d c o n d i t i o n n u m b e r K s h o u l d b e 0 : 1 < K < 1 4 . 4 3 ] - 4 . 4 5 ]

a n d 2 ] . S e e a l s o p r o b l e m 1 o f t h i s s e c t i o n a n d ( 5 . 2 . 4 0 ) .

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

F i g u r e 4 . 1 7

C i r c u l a r m e m b r a n e i n c a r t e s i a n c o o r d i n a t e s

K =

d e t M

1

2

: : :

n

( 4 . 8 . 1 9 )

w h e r e

i

=

q

m

2

i 1

+ m

2

i 2

+ : : : + m

2

i n

: ( 4 . 8 . 2 0 )

M o r e d e t a i l s c a n b e f o u n d i n p r o b l e m 1 o f t h i s s e c t i o n . C o l l o c a t i o n m e t h o d s

s h o u l d b e u s e d w i t h c a r e t o a v o i d d i s a p p o i n t m e n t s .


7/18/2019 C4029-Ch04


P r o b l e m s

1 . T h e n e w c o m m a n d s ( 4 . 8 . 1 5 ) t o ( 4 . 8 . 1 7 ) s e e m t o b e s e l f e x p l a n a t o r y . T h e

r s t c o m m a n d

m=Table

g e n e r a t e s t h e m a t r i x m a n d t h e s e c o n d c o m m a n d

Table[m[ipi,mi]]

l l s t h e e l e m e n t s m

i p i m i

o f t h e m a t r i x . ( 4 . 8 . 1 6 ) m a y f a i l o r g i v e e r r o -

n o e o u s r e s u l t s f o r n o t w e l l - c o n d i t i o n e d m a t r i c e s . T h e n e c e s s a r y f a c t o r -

i n g o f a m a t r i x i s d i s c u s s e d i n 2 ] . T h u s

LUFactor[m]( 4 . 8 . 2 1 )

g i v e s t h e L U ( l o w e r a n d u p p e r t r i a n g u l a r i z a t i o n o f a m a t r i x ) d e c o m p o -

s i t i o n a n d

LUSolve[lu,b]( 4 . 8 . 2 2 )

s o l v e s a l i n e a r s y s t e m r e p r e s e n t e d b y lu a n d t h e r i g h t - h a n d s i d e b.

T h e p a c k a g e

LinearAlgebra

GaussianElimination c o n t a i n s t h e

t w o l a s t c o m m a n d s . T r y 2 ]

a={{5,3,0},{7,9,2},{-2,-8,-1}} ( 4 . 8 . 2 3 )

T o d e n e t h e m a t r i x a . Y o u m a y a l s o u s e t h e T a b l e c o m m a n d ( 4 . 8 . 1 6 )

o r

MatrixForm[a={{5,3,0},{7,9,2},{-2,-8,-1}}] ( 4 . 8 . 2 4 )

T h e n t h e c o m m a n d lu=LUFactor[a] ( 4 . 8 . 2 5 )

s h o u l d y i e l d

( (

5

7

1 2

1 9

;

2 2

1 9

)

n

7 9 2

o

(

;

2

7

;

3 8

7

;

3

7

) )

n

2 3 1

o

.

A s s u m e n o w t h e e x i s t e n c e o f a n i n h o m o g e n e o u s l i n e a r s y s t e m

lu*x=b, w h e r e

xa r e t h e t h r e e u n k n o w n s a n d a s s u m e f o r t h e r h s v e c t o r

b={6,-3,7}t h e n

LUSolve[lu,b]( 4 . 8 . 2 6 )

y i e l d s t h e t h r e e u n k n o w n s

n

7 5

4 4

;

3 7

4 4

;

8 1

2 2

o

T o v e r i f y t h e s o l u t i o n w r i t e a.% ( 4 . 8 . 2 7 )

w h i c h s h o u l d g i v e f 6 ; 3 7 g . T h i s m e t h o d h a d b e e n u s e d i n t h e f o r m

boundary = m . Amp ( 4 . 8 . 2 8 )

2 . U s i n g M a t h e m a t i c a v e r i f y t h e s e t w o s o l u t i o n s :

( 4 . 8 . 6 ) s o l v e s t h e L a p l a c e e q u a t i o n ,


7/18/2019 C4029-Ch04



( 4 . 8 . 9 ) s o l v e s t h e H e l m h o l t z e q u a t i o n . U s e

VV[x,y]=Cos[Pi*x/a]*Cosh[Pi*y/a]+Cosh[Pi*x/b]*

Cos[Pi*y/b]+Log[Sqrt[(xˆ2+yˆ2)/(aˆ2+bˆ2)]];

Chop[Simplify[D[VV[x,y],{x,2}]+D[VV[x,y],{y,2}]]]

0

3 . P l a y ( a n d p l o t ) w i t h v a r i o u s v a l u e s o f A = B k i n ( 4 . 8 . 1 0 ) .

4 . U s i n g t h e M a t h e m a t i c a p l o t c o m m a n d s , c r e a t e F i g u r e s 4 . 1 5 a n d 4 . 1 6 .

5 . U s e t h e F O R T R A N p r o g r a m C O L L O C a n d / o r t h e M a t h e m a t i c a p r o g r a m

C o l m e i g v w i t h v a r i o u s s e t s o f t h e s e p a r a t i o n c o n s t a n t s

n

( B E T A o r b ) .

E s t a b l i s h a t a b l e d e s c r i b i n g h o w t h e e i g e n v a l u e k d e p e n d s o n t h e n u m b e r

P o r n o f c o l l o c a t i o n p o i n t s , o n t h e v a l u e s o f

n

(

2

n

< k

2

) a n d o n t h e

r s t g u e s s o f t h e e i g e n v a l u e E I G E N V A L U E ( r e s p . r s t e i g e n v ) .

c4029-ch04

Documents