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Unformatted text preview: ASSIGNMENT 3 Physics 218 Due Feb 15 Spring 2008 Reading: Pain Ch 5. Labs start next week! Prelabs due at beginning of lab session. 1. Group and phase velocities I: Don’t just give a one-word answer: justify what you write! (a) One way to measure the velocity of sound in air is to clap your hands and determine the time delay between the clap and an echo from a known reflector. Another way is to measure the length of a mailing tube that resonates at a known frequency (and correct for end effects.) Do these methods determine the phase or the group velocity? (b) One way to measure the velocity of light is to send a chopped light beam through the air from one distant mountain peak to another, reflect it from a mirror, and time the round trip. Another way is to find the length of a resonant cavity oscillating in a known mode at a known frequency. Do these methods determine the phase velocity or the group velocity? 2. Group and phase velocities II: The dispersion relation for sinusoidal electromagnetic waves in the ionosphere is given by ω 2 = ω 2 p + c 2 k 2 , for frequencies above a cutoff frequency of ν p = ω p / (2 π ) ≈ 20 MHz (where c is the speed of light here.) Calculate the group velocity v g and the phase velocity v p . How do they compare to the speed of light? Is this consistent with your understanding of special relativity? Explain! 3. String with a massive bead: A uniform string of linear mass density λ and under a tension τ has a small bead of mass m attached to it at x = 0. Find expressions for the reflection and transmission coefficients for sinusoidal waves brought about by the mass discontinuity at the origin. Do these coefficients hold for a wave of arbitrary shape? Why or why not? (See also Pain Problem 5.5 for hints.) 4. Pain 5.26 - Doppler effect due to thermal broadening. 5. Continuity equation: Any physical quantity that is conserved must satisfy the so-called equation of continuity , which states that the rate of increase of the physical quantity per unit volume must be equal to the rate at which the quantity enters the unit volume. When applied to the energy density in an elastic medium, the equation takes the form ∂E ∂t =-∇ · ~ P, where δE = δK + δV and ∇ · ~ P is the divergence of the directed power flow per unit area. (a) Show that the power for a one-dimensional elastic wave is given by P =- T ∂y ( x,t ) ∂t ∂y ( x,t ) ∂x . (b) Show that given the expressions for power above and energy derived in class, the continuity equation is satisfied for elastic media (such as strings.) ( Nota bene : you can treat this as a 1d problem.) (Continued on reverse.) 6. String Waves: There are a series of six graphs below. Based on the appearance of the first, you will answer questions about the other graphs....
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This note was uploaded on 02/24/2008 for the course PHYS 2218 taught by Professor Wittich,p during the Spring '08 term at Cornell University (Engineering School).
- Spring '08