preexam6a

# preexam6a - 1 Consider the following standing wave with...

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1. Consider the following standing wave, with period T = 2 π/ω and wave- length λ = 2 π/k , y 1 = A cos kx sin ωt a) Suppose this solution describes a standing wave on a string that has one end at x = 0 and the other at x = L . The end at x = L is tied down so that y = 0 at that point for all times. Find the relation between λ and L for the three lowest-frequency standing waves (i.e. the ﬁrst three normal modes of vibration). b) Plot the 1st and 2nd normal modes from a) at time t = T/ 4 on the axis provided below, labeling each clearly. c) Now consider a second standing wave solution, y 2 = A sin kx cos ωt . Is the superposition, y = y 1 + y 2 a standing wave or a traveling wave? You will ﬁnd it useful to rewrite y using one of the trig. identities below and argue by inspection. sin( x ± y ) = sin x cos y ± cos x sin y cos( x ± y ) = cos x cos y sin x sin y Solution: a) We must have: y ( L,t ) = A cos kL sin ωt = 0. A cannot be zero because there would then be no standing wave. sin

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preexam6a - 1 Consider the following standing wave with...

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