Announcements 2/14/11 Prayer Exam 1: last day = tomorrow! We’re likely not going to finish dispersion

today, so extra time on Lab 3: now due Tues Feb 22.

Monday Feb 21 is President’s Day holiday. Tues Feb 22 is a virtual Monday

Complex numbers & traveling waves Traveling wave: A cos(kx – t + )

Write as:

Often:

…or – where “A-tilde” = a complex number, the

phase of which represents the phase of the wave

– often the tilde is even left off

( ) i kx tf t Ae

( ) i kx tif t Ae e

( ) i kx tf t Ae

Reading Quiz Which of the following was not a major

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

Thought Question Which of these are the same?

(1) A cos(kx – t)(2) A cos(kx + t)(3) A cos(–kx – t)

a. (1) and (2)b. (1) and (3)c. (2) and (3)d. (1), (2), and (3)

Which should we use for a left-moving wave: (2) or (3)?

a. Convention: Usually use #3, Aei(-kx-t)

b. Reasons: (1) All terms will have same e-it factor. (2) The sign of the number multiplying x then indicates the direction the wave is traveling.

ˆk k i

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? 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

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” lectures of next week.

Wave packets HW 17-5

Wave packets, cont.

What did we learn?a. To localize a wave in space, you need lots of

frequenciesb. To remove neighboring localized waves, you

need those frequencies to spaced close to each other. (infinitely close, really)