11/04/2013phy 711 fall 2013 -- lecture 261 phy 711 classical mechanics and mathematical methods...
DESCRIPTION
11/04/2013PHY 711 Fall Lecture 263 Review of wave equation in one-dimension – here (x,t) can describe either a longitudinal or transverse wave. Traveling wave solutions --TRANSCRIPT
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PHY 711 Fall 2013 -- Lecture 26 111/04/2013
PHY 711 Classical Mechanics and Mathematical Methods
10-10:50 AM MWF Olin 103
Plan for Lecture 26:Traveling and standing waves in 1 and 2 dimensions
1. Brief review of 1-d from Chap. 72. Waves in 2-d elastic membranes –
Chap. 8
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PHY 711 Fall 2013 -- Lecture 26 211/04/2013
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PHY 711 Fall 2013 -- Lecture 26 311/04/2013
equation. wave thesatisfies )()(
:or function any for that Note
0 2
22
2
2
ctxgctxfμ(x,t)g(q)f(q)
x c
t
Review of wave equation in one-dimension – here (x,t) can describe either a longitudinal or transverse wave.
Traveling wave solutions --
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PHY 711 Fall 2013 -- Lecture 26 411/04/2013
ctx
ctx
x
x
x
dxxc
ctxctxtx
dxxc
xxg
dxxc
xxf
g(x)f(x)x
dxxc
xgxf
dxdg(x)
dxdf(x)cx)(x,
tμ
xgxfx)μ(x,
x)(x,tμx)μ(x,
')'(21)()(
21),(
')'(1)(21)(
')'(1)(21)(
: and find ,each For
')'(1)()(
)(0
)()()(0 : then
)(0 and )(0 :problem valueInitial
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PHY 711 Fall 2013 -- Lecture 26 511/04/2013
Standing wave solutions of wave equation:
.00with
0 2
22
2
2
μ(L,t),t)μ(x
c t
c kρ(x)k
dxρ(x)d
ρ(x)etx ti
0 where
),( :Assume
22
2
ckL
k
LxAx
sin)(
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PHY 711 Fall 2013 -- Lecture 26 611/04/2013
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PHY 711 Fall 2013 -- Lecture 26 711/04/2013
Extension of ideas to wave motion on a two-dimensional surface – elastic membrane (transverse wave; linear regime).
u(x,y,t)
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PHY 711 Fall 2013 -- Lecture 26 811/04/2013
0///
0
:principle sHamilton'
,,;,,,
,,;,,, :density Lagrangian
2
1
yuyxuxtutu
Ldtδ
dxdytyxtu
yu
xuuL
tyxtu
yu
xuu
t
t
LLLL
L
L
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PHY 711 Fall 2013 -- Lecture 26 911/04/2013
2222
2
22
where 0
0///
21
21,,;,,,
: and constant with membrane elasticfor density Lagrangian
cuctu
yuyxuxtutu
ututyx
tu
yu
xuu
LLLL
L
ckyxk
yxetyxu
cuctu
ti
where 0),(
),(),,(
:solutions waveStanding
where 0
:equation waveldimensiona-Two
22
2222
2
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PHY 711 Fall 2013 -- Lecture 26 1011/04/2013
Consider a rectangular boundary:
a
b
mnmnmn
mn
ckωb
na
mk
byn
axmAyx
bxxyay
sinsin),(
0),()0,(),(),0(:conditionsboundary Clamped
222
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PHY 711 Fall 2013 -- Lecture 26 1111/04/2013
22
211
bak
2 222
21
bak 22 22
222
bak
2 222
12
bak
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PHY 711 Fall 2013 -- Lecture 26 1211/04/2013
More general boundary conditions:
side free"" represents 0
spring with dconstraine side boundard represents
b
bb
u
uu
mnmnmn
mn
ckωb
na
mk
byn
axmAyx
xya
xybxx
sincos),(
0),(),0(),()0,(
:conditionsboundary Mixed
222
222
11
bak
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PHY 711 Fall 2013 -- Lecture 26 1311/04/2013
Consider a circular boundary:R
0),(:),(for conditionsboundary Clamped
Rr
c
krk where 0),(22
integer )2()( :Note
)( :Let
)()(),( :Assume
11
system coordinate lcylindricaIn
2
2
22
22
m
e
rfrrrrr
im
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PHY 711 Fall 2013 -- Lecture 26 1411/04/2013
)(function Neumann lCylindrica )(function Bessel lCylindrica
solutions ential transcendorder withinteger ofequation Bessel
0)(1
:function radialfor equation alDifferenti
22
2
2
2
zNzJ
rfkrm
drd
rdrd
m
m
Consider circular boundary -- continued
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PHY 711 Fall 2013 -- Lecture 26 1511/04/2013
Some properties of Bessel functions
11.620,... 8.417, ,136.5 :210.173,... 7.016, ,832.3 :18.654,... ,520.5 ,406.2 :0
0 functions esselB of Zeros
2
1
0
n
n
n
mnm
zmzmzm
zJ
42cos2)( :form Asymptotic
)(2)()(
)(2)()( :relationsRecursion
2!!1
2)( :series Asending
1
11
11
2
0
mzz
zJ
dzzdJzJzJ
zJzmzJzJ
zmjj
zzJ
zm
mmm
mmm
j
j
jm
m
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PHY 711 Fall 2013 -- Lecture 26 1611/04/2013
Some properties of Bessel functions -- continued
'2
1
2
0
'
2
: functions for theidentity
following theprove topossible isIt :Note
nnmnm
Rmn
mmn
m
mnm
zJRrdrrR
zJrR
zJ
rR
zJ
Rz kr
RzAJ(r)f
rfkrm
drd
rdrd
mnmn
mnmmn
;
0)(1
:function radialfor equation aldifferenti toReturning
22
2
2
2
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PHY 711 Fall 2013 -- Lecture 26 1711/04/2013
r
RzAJ(r)f)(r 01
00101 ,
r
RzAJ(r)f)(r 02
00202 ,
520.5 406.20201 R
kR
k
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PHY 711 Fall 2013 -- Lecture 26 1811/04/2013
cos
cos,
111
1111
rRzAJ
(r)f)(r
cos
cos,
121
1212
rRzAJ
(r)f)(r
016.7 832.31211 R
kR
k
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PHY 711 Fall 2013 -- Lecture 26 1911/04/2013
Maple server evaluation project Request from Rick Matthews, Associate Provost
for Technology and Information Systems The University is exploring the possibility of
making Maple available on a server rather than on individual computersNeed students to exercise the software Need students to assess the performancNeed students to identify issues
Entirely voluntary
Possible pizza party or ??? rewards
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PHY 711 Fall 2013 -- Lecture 26 2011/04/2013
Introduction to algebraic manipulation software
Available from the webpage: http://help.wfu.edu/public/vcl
If you have any trouble with this installation and setup, please contact Ching-Wan Yip at [email protected]
Note: This is a remote server that the university is testing. If you are off campus you will need to use VPN to gain access.
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PHY 711 Fall 2013 -- Lecture 26 2111/04/2013
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PHY 711 Fall 2013 -- Lecture 26 2211/04/2013
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PHY 711 Fall 2013 -- Lecture 26 2311/04/2013