Announcements 10/21/11 Prayer Chris: no office hours today Due tomorrow: Labs 4-5; term project proposal Due Monday: HW 22 and HW 23

a. See email for hint on HW22-3b. If you can’t get Mathematica to plot things, please come

find me (after giving it a serious attempt). I can usually spot Mathematica errors in 2 minutes or less.

Exam 2 review session: Tuesday 5-6 pm. Room: C460 (probably)

Exam 2 starts on Thursday

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

0 1 0 1 0

2 0 2 0

3 0 3 0

( ) cos sin

cos 2 sin 2

cos 3 sin 3 ...

f x a a k x b k x

a k x b k x

a k x b k x

Written another waywith k0 = 2/L

Building a function by specifying an, bn

Mathematica: build a function…

0N 1N 2N

3N 10N 500N

1 1 2sin

2

nx

n L

Sawtooth Wave, like HW 22-2

(The next few slides from Dr. Durfee)

The Spectrum of a Saw-tooth WaveThe Spectrum of a Saw-tooth Wave

0 10 20 30 40 50 60-0.4

-0.2

0

0.2

0.4

0.6

Am

plitu

de

[m]

k [rad/m]

The Spectrum of a Saw-tooth WaveThe Spectrum of a Saw-tooth Wave

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

Am

plitu

de [

m]

k [rad/m]0 10 20 30 40 50 60

-pi/2

-pi/4

0

Pha

se [

rad]

Electronic “Low-pass filter” “Low pass filter” = circuit which

preferentially lets lower frequencies through.

?Circuit

What comes out?

How to solve: (1) Decompose wave into Fourier series(2) Apply filter to each freq. individually(3) Add up results in infinite series again

Low-Pass Filter – before filterLow-Pass Filter – before filter

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

Am

plitu

de [

m]

k [rad/m]0 10 20 30 40 50 60

0

0.1

0.2

0.3

0.4

0.5

0.6

Am

plitu

de [

m]

k [rad/m]0 10 20 30 40 50 60

0

0.1

0.2

0.3

0.4

0.5

0.6

Am

plitu

de [

m]

k [rad/m]0 10 20 30 40 50 60

-pi

-3 pi/4

-pi/2

-pi/4

0

Pha

se [

rad]

Low-Pass Filter – after filterLow-Pass Filter – after filter

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

Am

plitu

de [

m]

k [rad/m]0 10 20 30 40 50 60

0

0.1

0.2

0.3

0.4

0.5

0.6

Am

plitu

de [

m]

k [rad/m]0 10 20 30 40 50 60

0

0.1

0.2

0.3

0.4

0.5

0.6

Am

plitu

de [

m]

k [rad/m]0 10 20 30 40 50 60

-pi

-3 pi/4

-pi/2

-pi/4

0

Pha

se [

rad]

Low Pass FilterLow Pass Filter

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

y an

d y fil

tere

d

[m]

x [m]

Actual Data from OscilloscopeActual Data from Oscilloscope

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?(assume h, L, and velocity v are known)

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 = b1sin(x/L)cos(1t)

3rd harmonic: y = b3sin(3x/L)cos(3t)

5th harmonic: y = b5sin(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!

Experiment!!

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?

Experiment!!