Solved: Dispersive Waves. Consider the modified wave equation a2 utt + 2 u = uxx, 0 < x

Chapter 10, Problem 23

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Dispersive Waves. Consider the modified wave equation a2 utt + 2 u = uxx, 0 < x < L, t > 0 (i) with the boundary conditions u(0, t) = 0, u(L, t) = 0, t > 0 (ii) and the initial conditions u(x, 0) = f(x), ut(x, 0) = 0, 0 < x < L. (iii) (a) Show that the solution can be written as u(x, t) = n=1 cn cosn22 L2 + 2 at sin nx L , where cn = 2 L L 0 f(x)sin nx L dx. (b) By using trigonometric identities, rewrite the solution as u(x, t) = 1 2 n=1 cn ) sin n L (x + ant) + sin n L (x ant) * . Determine an, the speed of wave propagation. (c) Observe that an, found in part (b), depends on n. This means that components of different wave lengths (or frequencies) are propagated at different speeds, resulting in a distortion of the original wave form over time. This phenomenon is called dispersion. Find the condition under which an is independent of nthat is, there is no dispersion.