## Transcribed Text

(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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