Announcements 10/5/12 Prayer Handout – Adding together two cosine waves Colloquium: Did you notice “Fourier

transforms”? I just got the exams from the Testing Center,

TA & I will work on grading them today & this weekend.

Non Sequitur

From warmup

Extra time on?a. how exactly can an amplitude absorb a

complex number when it itself is not complex? Is it related to the way you lump a constant into +C after taking an integral?

Other comments?a. (none in particular)

Adding together two cosine waves

In short: “The amplitude and phase of the answer were completely determined in the step where we added the amplitudes & phases of the original two cosine waves, as vectors.”

Don’t worry about writing each step completely. a.Don’t write “Real( )”b.Don’t write “e i (3x)”

HW 16.5: Solving Newton’s 2nd Law

Simple Harmonic Oscillator (ex.: Newton 2nd Law for mass on spring)

Guess a solution like

what it means, really:

there’s an understood “Real{ … }”

2

2

d x kx

mdt

( ) i tx t Ae

( ) cos( )x t A t

Complex numbers & traveling waves

Traveling wave: A cos(kx – t + )

Write as:

Often:

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

the amplitude of which represents the amplitude of the wave

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 A

Clicker 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: Use #3, Aei(-kx-t)

b. Reasons: – (1) All terms will then have same e-it factor. – (2) Whether you have kx then indicates the direction

the wave is traveling.c. “Wavevector”

ˆk k i

From warmup What was wrong with the first solution that was

tried in the reading today (PpP section 3.2)? What assumption did it start with and how could Dr. Durfee tell that that assumption was wrong?

a. it started by assuming that the wave passed straight from one rope to the next and was wrong because that would lead to the wave having the same velocity on both ropes.

How did the next guess (section 3.3) build on the first?

a. He then guessed that a wave was partially reflected, instead of solely transmitted

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….

Reflected & Transmitted 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

Clicker question: A wave at frequency ω traveling from a string to a

rope. At the junction, 80% of the power is reflected. How much power would be reflected if the wave was going from the rope to the string instead?

a. Much less than 80%b. A little less than 80%c. About 80%d. More than 80%e. It depends on the color of the rope.

2 1

1 2

R

I

A v vr

v vA

2

1 2

2T

I

A vt

v vA

2R r 1T R

Demo Reflection at a boundary. Measure v1 and

v2.

2 1

1 2

v vrv v

2

1 2

2vtv v

Now, on to sound!

Clicker question: Sound waves are typically fastest in:

a. solidsb. liquidsc. gases

Sound Waves What type of wave? What is waving? Demo: Sound in a vacuum Demo: tuning fork Demo: Singing rod Sinusoidal?

a. Demo: musical disk

Speed of sound Speed of sound…

a. in gases: ~300-1200 m/sb. in liquids: ~1000-1900 m/sc. in solids: ~2000-6000 m/s

v = sqrt(B/) compare to v = sqrt(T/)

Speed of sound in aira. 343 m/s for air at 20Cb. Dependence on temperature (eqn in

book and also given on exam)

ms343293KsoundT

v

ms343293KsoundT

v