Announcements 10/19/12 Prayer Learn Smart – info in email Term project proposal due tomorrow Labs 4-5 also due tomorrow Exam 2: starting next Thurs

a. Review session: Tues 5:30-7 pm, place TBA

From warmup Extra time on?

a. The math

Other comments?a. As i almost always do the homework by myself i haven't

had the opportunity to find a partner for the term project. can we spend a minute or two in class and make sure there aren't others in my situation.

b. So im reading the proposal instructions. Can you explain a little more what calculations and predictions you would need?

c. Do I need a log of hours spent on the term project?d. I'd like it to be noted that there are now two discrepancies

in Dr. Durfee's book. The first being that he switched the coefficients for sin and cos, being the only person (as far as we can tell) who does this. Second, he believes the guitar is the ideal musical instrument. This is impossible seeing as the piano is the ideal musical instrument.

Review clicker question:

In the Fourier transform of a periodic function, which frequency components will be present?

a. Just the fundamental frequency, f0 = 1/period

b. f0 and potentially all integer multiples of f0

c. A finite number of discrete frequencies centered on f0

d. An infinite number of frequencies near f0, spaced infinitely close together

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

The graphical “proof” that the functions integrate to zero

Mathematica: Graphical “proof”

(called “orthogonal” functions)

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 frequency 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 Filter (Theory)Low Pass Filter (Theory)

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

What’s y(x,t) for the second harmonic? If you can predict how each frequency component will behave, you can predict the behavior for any shape of wave! (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?

From Warmup What is the main goal of PpP section 6.6? Explain in

your own words what the point is.a. The point in this section was to come up with what

the string of a guitar looks like at any point in time [for a particular starting position]

How does Dr. Durfee achieve that goal? Explain in your own words what he is doing, mathematically.

a. [my answer] The process is to 1) write the initial shape as a sum of Fourier components, 2) attach the appropriate time dependence to each component, then 3) add the components back together to get the time dependence of the string as a whole.

Clicker question: 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!!