Announcements 10/10/11 Prayer Exams graded, see 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. 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 seconds1.3 seconds 0.1 seconds What if the two velocities had been the same?

Time Evolution of Dispersive Pulse Credit: 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 spectrum Note: frequencies are infinitely close together

Phase and Group Velocity 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 v phase > v group

One of my contributions to Wikipedia Example where v phase is negative!

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

Demo Reflection at a boundary. Measure v 1 and v 2.

Reading Quiz Sound waves are typically fastest in: a. a.solids b. b.liquids c. c.gases

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

Speed of sound Speed of sound… a. a.in gases: ~ m/s b. b.in liquids: ~ m/s c. c.in solids: ~ m/s v = sqrt(B/  ) compare to v = sqrt(T/  ) Speed of sound in air a. a.343 m/s for air at 20  C b. b.Dependence on Temperature (eqn in book and also given on exam)

Intensity Intensity: power/area a. a.“Spherical waves” b. b.Non-spherical waves? Question: you measure the sound intensity produced by a spherically-emitting speaker to be 10 W/m 2 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. a.β = 10 log( I / Io ) b. b.β = 10 ( I / Io ) c. c.β = 10 ( I - Io ) d. d.β = 10 e ( I / Io ) e. e.β = e 10 ( I / Io ) add 10 to    10 to I

Decibels Threshold of hearing0 dB W/m 2 Whisper30 dB10 -9 W/m 2 Vacuum cleaner70 dB10 -5 W/m 2 Rock Concert120 dB1 W/m 2 Nearby jet airplane150 dB1000 W/m 2

Logarithm Review Log 10 (x) is the inverse of 10 y → if x = 10 y then y = log 10 (x) a. a.I.e. “10 to the what equals 22?” answer: calculator: log 10 (22) Review of “Laws of Logs”: – – 1. log(ab) = log(a) + log(b) – – 2. log(a n ) = n log(a) log 10 (100) = ? Translation: 10 to what equals 100? ln(100) = ? (“ln” = log e = log … ) Translation: e to what number =100? (4.605…) Ambiguity: “log(100)”…could be either log 10 or ln Question: log 10 (1,000,000) = ? Question: If log(3) = 0.477, what is log(300)?