Announcements 10/22/10 Prayer Exam starts next week on Thursday Exam review session, results of voting:

a. Wed Oct 27, 5:30 – 7 pm. Room: C295 (next door) Unknown HW 13 – missing CID/name (turned in ~Oct 11) What are some applications of Fourier transforms?

a. Electronics: circuit response to non-sinusoidal signals (last lecture)

b. Data compression (as mentioned in PpP)c. Acoustics: guitar string vibrations (PpP, today’s

lecture)d. Acoustics: sound wave propagation through dispersive

mediume. Optics: spreading out of pulsed laser in dispersive

mediumf. Optics: frequency components of pulsed laser can

excite electrons into otherwise forbidden energy levelsg. Quantum: “particle in a box” situation, aka “infinite

square well”--wavefunction of an electron

Q&A with Dean Sommerfeldt

Info about the college, upcoming events, changes, concerns

Refreshments and prizes!

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) 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 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!http://www.physics.byu.edu/faculty/colton/courses/phy123-fall10/trianglestring.gif

How about the pulse from HW 23-1?

Any guesses as to what will happen?http://www.physics.byu.edu/faculty/colton/courses/phy123-fall10/squarestring.gif