ps06 - Physics 2214 Problem Set #6 (Due 11:15 am, Thursday...

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Page 1 of 2 Physics 2214 Problem Set #6 (Due 11:15 am, Thursday 10/16/08) 1. Consider the superposition of two harmonic sound waves with equal amplitudes but slightly different k ’s and ω ’s: 11 2 2 ()( ) (,) ikx t t sxt Ae + + =+ where k 1 > k 2 and 1 > 2 . The average wavenumber and angular frequency are ( ) ( ) 12 1 2 22 and kk k ωω = + (a) Show that s ( x,t ) can be written in the form ( ) t Axte + = and find A ( x , t ). Interpretation: ( ) t e + represents a carrier wave with the average wavenumber and frequency and A ( x , t )—the envelope —is an amplitude that depends on position and time. Express A ( x , t ) in terms of Δ k = k 1 k 2 and Δ = 1 2 , where Δ k ا k and Δ ا . (b) At what speed does a point of constant phase—e.g. a peak of the carrier wave—travel? This is called the phase speed v ph . (c) At what speed does the envelope travel? This is called the group speed v gr . (d) Show that in a dispersionless medium (which means v ph is independent of k ), v ph = v gr . To see what happens in a dispersive medium, use the applet at http://galileo.phys.virginia.edu/classes/109N/more_stuff/Applets/sines/GroupVelocity.html
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ps06 - Physics 2214 Problem Set #6 (Due 11:15 am, Thursday...

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