Announcements 10/10/11 Prayer Exams graded, see email In HW 17-5b: be very careful to track the

correct peak when plotting it for t = 0.1 s and t = 0.5 s, and when calculating the velocity of the peak.

Dispersion Review Any wave that isn’t 100% sinusoidal contains

more than one frequencies. To localize a wave in space or time, you need

lots of frequencies--spatial (k values) or angular ( values), respectively. Really an infinite number of frequencies spaced infinitely closely together.

A dispersive medium: velocity is different for different frequencies.

Two Different Velocities What happens if a wave pulse is sent through a

dispersive medium? Nondispersive? Dispersive wave example:

a. f(x,t) = cos(x-4t) + cos(2 (x-5t)) – What is “v”?– What is v for =4? What is v for =10?

What does that wave look like as time progresses? (next slide)

Mathematica

0.7 seconds 1.3 seconds

0.1 seconds What if the two velocities had been the same?

Time Evolution of Time Evolution of DispersiveDispersive Pulse PulseCredit: Dr. Durfee

Wave moving in time

Peak moves at about 13 m/s (on my office computer)

How much energy is contained in each frequency component

Power spectrumNote: frequencies are infinitely close together

Phase and Group VelocityPhase and Group Velocity

evaluatedat

velocity of "envelope"ave

gk

dv

dk

velocity of "wiggles"pv k

Credit: Dr. Durfee

Can be different for each frequency component that makes up the wave

A property of the wave as a whole

Window is moving along with the peak of the pulse

13 m/s

12.5 m/s, for dominant component

(peak)

From Wikipedia Example where vphase > vgroup

http://en.wikipedia.org/wiki/Group_velocity

One of my contributions to Wikipedia Example where vphase is negative!

http://en.wikipedia.org/wiki/Group_velocity

Thought 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

Reading Quiz 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)

ms343

293KsoundT

v

ms343

293KsoundT

v

Intensity Intensity: power/area

a. “Spherical waves”b. Non-spherical waves?

Question: you measure the sound intensity produced by a spherically-emitting speaker to be 10 W/m2 at a distance of 2 meters. What will be the intensity at 8 meters away?

Question: What is the total sound power (watts) being produced by the speaker?

Reading Quiz How do we calculate the sound level in

decibels?a. β = 10 log( I / Io )b. β = 10 ( I / Io )c. β = 10 ( I - Io )d. β = 10 e( I / Io )

e. β = e10 ( I / Io )

add 10 to 10 to I

Decibels

Threshold of hearing 0 dB 10-12 W/m2

Whisper 30 dB 10-9 W/m2

Vacuum cleaner 70 dB 10-5 W/m2

Rock Concert 120 dB 1 W/m2

Nearby jet airplane 150 dB 1000 W/m2

Logarithm Review Log10(x) is the inverse of 10y

→ if x = 10y then y = log10(x)

a. I.e. “10 to the what equals 22?” answer: 1.3424 calculator:

log10(22) Review of “Laws of Logs”:

– 1. log(ab) = log(a) + log(b)– 2. log(an) = n log(a)

log10(100) = ? Translation: 10 to what equals 100?

ln(100) = ? (“ln” = loge = log2.71828…)

Translation: e to what number =100? (4.605…)

Ambiguity: “log(100)”…could be either log10 or ln

Question: log10(1,000,000) = ? Question: If log(3) = 0.477, what is log(300)?