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(2) Consider the modified wave equation = Uxx, 0 < x < L, , t>0, u(0,t) = u(L,t) = 0, t > 0, u(x,0) = f(x), ut(x,0) = 0, 0 < x < L. (ii) By using trigonometric identities, rewrite the solution as 8 u(x, t) = 1 NTT (x + ant) Cn sin + 2 L sin NT L - )] n=1 Determine an, the speed of wave progagation. (iii) Observe that an found in part (ii), depends on n. This means that components of different wave lengths (or frequencies) are prop- agated at different speeds, resulting in a distortion of the original wave form over time (called dispersion). Find the condition under which an is independent of n. (3) Consider the situation in problem (2) with a = 1, L = 10, and x - 4, 4 < x < 5, f(x) = 6 - x, 5 < x V 6, 0, otherwise. (i) Determine the coefficients of Cn in the solution of problem 2 (i). (4) Consider the damped wave equation describing vibrations on the infinitely long string: Utt - a² Uxx + rut = 0 where r > 0, is constant. Show that the energy is decreases.

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