1. Dispersion and beats. Consider two waves f 1 (x,t) = Acos(k 1 x 1 t) and f 2

Question

1. Dispersion and beats.

Consider two waves f1(x,t) = Acos(k1x 1t) and f2(x,t) = Acos(k2x 2t) that propagate
through the same medium. Assume k = k1 k2 is much smaller than k1 and k2. We saw in
Lecture 6 that the resulting wave can be written as a product of two traveling waves, one
oscillating at high frequency / short wavelength and one oscillating at low frequency / long
wavelength. If the medium in which the waves propagate has a dispersion relation (k) = ck2 ,
where c is a constant, what approximately are the wave velocities of the high frequency and low
frequency part of the resulting wave? Are they the same? Show your work, and use a sketch of
(k) to explain your answer.

Explanation / Answer

given two wavees
f1 = Acos(k1x - w1t)
f2 = Acos(k2x - w2t)
then f1 + f2 = A[cos(k1x - w1t) + cos(k2x - w2t)]
ugin Cos A + Cos B = 2cos([A + B]/2)cos([A - B]/2)
f1 + f2 = 2Acos([x(k1 + k2) - t(w1 + w2)]/2)cos([x(k1 - k2) - t(w1 - w2)]/2)

Now wave velocity is given by v = w/k
w(k) = ck^2
so, v = w/k = c k

so, wave velocity of high frequency part = v1 = c*(k1 + k2)/2
wave velocity of low frequency part, v2 = c(k1 - k2)/2
hence they both are different with higher frequency travelling faster

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