Announcements 2/28/11 Prayer My office hours this week: I’ll likely be in my

lab, room U130, just down the hall from normal office hour location. Find me there.

Exam 2 starts on Saturday Exam review session, results of voting:

a. Friday 3:30 – 5 pm. Room: C261 Next week: I’ll be out of town on Mon. You’ll

have Dr. Gus Hart as a substitute.

Summary of last time

0

0

1( )

L

a f x dxL

0

2 2( )cos

L

nnx

a f x dxL L

0

2 2( )sin

L

nnx

b f x dxL L

01 1

2 2( ) cos sinn n

n n

nx nxf x a a b

L L

The series

How to find the coefficients

Fourier Transform (review)

Do the transform (or have a computer do it)

Answer from computer: “There are several components at different values of k; all are multiples of k=0.01.

k = 0.01: amplitude = 0k = 0.02: amplitude = 0……k = 0.90: amplitude = 1k = 0.91: amplitude = 1k = 0.92: amplitude = 1…”

600 400 200 200 400 600

20

10

10

20

Cos0.9 x Cos0.91 x Cos0.92 x

Cos0.93 x Cos0.94 x Cos0.95 x

Cos0.96 x Cos0.97 x Cos0.98 x

Cos0.99 x Cos1. x Cos1.01 x Cos1.02 x

Cos1.03 x Cos1.04 x Cos1.05 x Cos1.06 x

Cos1.07 x Cos1.08 x Cos1.09 x Cos1.1 xHow does computer know all components will be multiples of k=0.01?

Periodic? “Any function periodic on a distance L can

be written as a sum of sines and cosines like this:”

What about nonperiodic functions? a. “Fourier series” vs. “Fourier transform”b. Special case: functions with finite domain

01 1

2 2( ) cos sinn n

n n

nx nxf x a a b

L L

HW 23-1

“Find y(x) as a sum of the harmonic modes of the string” Why?

Because you know how the string behaves for each harmonic—for fundamental mode, for example:

y = Asin(x/L)cos(1t) --standing wave Asin(x/L) is the initial shape It oscillates sinusoidally in time at frequency 1

If you can predict how each frequency component will behave, you can predict the overall behavior! (You don’t actually have to do that for the HW problem, though.)

HW 23-1, cont.

So, how do we do it? Turn it into part of an infinite repeating

function! Thought question: Which of these two

infinite repeating functions would be the correct choice?(a) (b)

…and what’s the repetition period?

Reading Quiz Section 6.6 was all about the motion of a

guitar string. What was the string’s initial shape?

a. Rectified sine waveb. Sawtooth wavec. Sine waved. Square wavee. Triangle wave

What was section 6.6 all about, anyway?

What will guitar string look like at some later time?

Plan: a. Figure out the frequency components in terms

of “harmonic modes of string”b. Figure out how each component changes in

timec. Add up all components to get how the overall

string changes in time

h

L

initial shape:

Step 1: figure out the frequency components

a0 = ?

an = ?

bn = ?

h

L

h

L

2 2( )sin

" " " "

L

n

L

nxb f x dx

L L

integrate from –L to L:three regions

1

2 3

2 2

region1 region 2 region32 2

2 2 2 2sin sin sin

2 2 2 2

L L L

n

L L L

nx nx nxb mx b dx mx b dx mx b dx

L L L L

2 2

2 2

1 2 2 22 sin 0 sin 2 sin

L L L

n

L L L

h nx h nx h nxb x h dx x dx x h dx

L L L L L L L

Step 1: figure out the frequency components

h

L

h

L

3

2 2

32 cos sin4 4

n

n nh

bn

12 ( 1)

2 2

81 ; odd

nn

hb n

n

Step 2: figure out how each component changes

Fundamental: y = Asin(x/L)cos(1t)

3rd harmonic: y = Asin(3x/L)cos(3t)

5th harmonic: y = Asin(5x/L)cos(5t)

1 = ? (assume velocity and L are known)

= 2f1 = 2(v/1) = 2v/(2L) = v/L

n = ?

h

L

Step 3: put together

Each harmonic has

y(x,t) = Asin(nx/L)cos(n1t)

= Asin(nx/L)cos(nvt/L)

h

L

12 ( 1)

2 21

odd

8( , 0) 1 sin

n

n

h n xf x t

Ln

12 ( 1)

2 21

odd

8( , ) 1 sin cos

n

n

h n x n vtf x t

L Ln

What does this look like? Mathematica!

Step 3: put together

Each harmonic has

y(x,t) = Asin(nx/L)cos(n1t)

= Asin(nx/L)cos(nvt/L)

h

L

12 ( 1)

2 21

odd

8( , 0) 1 sin

n

n

h n xf x t

Ln

12 ( 1)

2 21

odd

8( , ) 1 sin cos

n

n

h n x n vtf x t

L Ln

What does this look like? Mathematica!

How about the pulse from HW 23-1?

Any guesses as to what will happen?

How about the pulse from HW 23-1?

Any guesses as to what will happen?