Announcements 10/7/11 Prayer We’re likely not going to finish dispersion

today, so you might want to wait until after Monday before doing Lab 3 (Dispersion).

I just got the exams from the Testing Center, TA & I will work on grading them today & this weekend.

Non Sequitur

Reading Quiz Which of the following was not a major

topic of the reading assignment?a. Dispersionb. Fourier transformsc. Reflectiond. Transmission

Reflection/transmission at boundaries: The setup

Why are k and the same for I and R? (both labeled k1 and 1) “The Rules” (aka “boundary conditions”)

a. At boundary: f1 = f2

b. At boundary: df1/dx = df2/dx

Region 1: light string Region 2: heavier string

in-going wave transmitted wave

reflected wave

1 1( )i k x tIA e

1 1( )i k x tRA e

2 2( )i k x tTA e

1 1 1 1( ) ( )1

i k x t i k x tI Rf A e A e 2 2( )

2i k x t

Tf A e

Goal: How much of wave is transmitted and reflected? (assume k’s and ’s are known)

x = 0

1 1 1 1 1cos( ) cos( )I I R Rf A k x t A k x t 2 2 2cos( )T Tf A k x t

Boundaries: The math

1 1 1 1 2 2( 0 ) ( 0 ) ( 0 )i k t i k t i k tI R TA e A e A e

2 2( )2

i k x tTf A e

x = 0

1 20 0B.C.1:

x xf f

1 1 2i t i t i tI R TA e A e A e

I R TA A A and 1 2

1 1 1 1( ) ( )1

i k x t i k x tI Rf A e A e

Goal: How much of wave is transmitted and reflected?

Boundaries: The math

1 1 2( ) ( ) ( )1 1 2

0 0

i k x t i k x t i k x tI R T

x xik A e ik A e ik A e

2( )2

i k x tTf A e

x = 0

1 2

0 0

B.C.2:x x

df df

dx dx

1 1 2i t i t i t

I R Tik A e ik A e ik A e

1 1 2I R Tk A k A k A

1 1( ) ( )1

i k x t i k x tI Rf A e A e

Goal: How much of wave is transmitted and reflected?

Boundaries: The math

Like: and

How do you solve?

x = 0

1 1 2I R Tk A k A k A I R TA A A

Goal: How much of wave is transmitted and reflected?

x y z 3 3 5x y z

2 equations, 3 unknowns??

Can’t get x, y, or z, but can get ratios!y = -0.25 x z = 0.75 x

Boundaries: The results

Recall v = /k, and is the same for region 1 and region 2. So k ~ 1/v

Can write results like this:

x = 0

1 2

1 2

R

I

A k kr

k kA

Goal: How much of wave is transmitted and reflected?

1

1 2

2T

I

A kt

k kA

2 1

1 2

R

I

A v vr

v vA

2

1 2

2T

I

A vt

v vA

“reflection coefficient” “transmission coefficient”

The results….

Special Cases

Do we ever have a phase shift in reflected or transmitted waves?

a. If so, when? And what is it? What if v2 = 0?

a. When would that occur? What if v2 = v1?

a. When would that occur?

x = 0

2 1

1 2

R

I

A v vr

v vA

2

1 2

2T

I

A vt

v vA

The results….

Power

Recall: (A = amplitude)

Region 1: and v are same… so P ~ A2

Region 2: and v are different… more complicated…but energy is conserved, so easy way is:

x = 0

2 21

2P A v

2R

I

PR r

P

21T

I

PT r

P

r,t = ratio of amplitudesR,T = ratio of power/energy

Quick Writing We saw that

A1cos(kx + 1) + A2cos(kx + 2) gives you a cosine wave with the same k, and hence wavelength.

If you add a third, fourth, fifth, etc., such cosine wave, you still get a simple cosine wave. See

How can you then add together cosine waves to get more complicated shapes? Or can you?

Wave packets HW 17-5

Wave packets, cont.

Results:a. To localize a wave in space, you need lots of spatial

frequencies (k values)b. To remove neighboring localized waves (i.e. to

make it non-periodic), you need those frequencies to spaced close to each other. (infinitely close, really)

Dispersion A dispersive medium: velocity is different for

different frequenciesa. Any real-world examples?

Why do we care? a. Real waves are often not shaped like sine

waves.– Non sine-wave shapes are made up of

combinations of sine waves at different frequencies.

b. Real waves are not infinite in space or in time.– Finite waves are also made up of combinations

of sine waves at different frequencies.Focus on (b) for now… (a) is the main topic of the “Fourier transform” lectures