Announcements 2/16/11 Prayer Exams… hopefully graded by Friday 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 Remember in HW 17-5b to 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.

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

Dispersion Summary A dispersive medium: velocity is different for

different frequencies Any wave that isn’t 100% sinusoidal contains

multiple frequencies. To localize a wave in space or time, you need

lots of frequencies… really an infinite number of frequencies spaced infinitely closely together.

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

dispersive medium? Nondispersive? Dispersive wave example:

a. s(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

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)

On Wikipedia Example where vphase > vgroup

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

Not yet on Wikipedia Example where vphase is negative!

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/s (343 m/s for air at 20C)

b. in liquids: ~1000-1900 m/sc. in solids: ~2000-6000 m/s

General form: v = sqrt(xxx/yyy)a. What are “xxx” and “yyy”?

Speed of sound in aira. Dependence on Temperature (eqn in

book)

Intensity Intensity: power/area

a. Spherical Wavesb. 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

Thought Question

A 3 dB increase in intensity is just about a factor of 2. How many dB represents a factor of 4 increase in intensity?

a. 2b. 4c. 6d. 8e. 9

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

Translation: e to what number =100? (4.605…) If the problem just says log(100)…could be either log10 or ln Question: log10(1,000,000) = ? Question: If log(3) = 0.477, what is log(300)?

Power and Intensity Scales

Power or Intensity

dB β = 10 log(I/I0) I0 = 10-12 W/m2

dBWβ = 10 log(P/P0) P0 = 1 W

dBmβ = 10 log(P/P0) P0 = 1 mW

What if you need to solve for I?