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Analytic and Numerical Solutions to the Wave Equation

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a) We have

(1) D_tt = D_xx

with boundary conditions

D(0,t) = D(1,t) = 0

and initial conditions

D(x,0) = 0


D_t(x,0) = 4 sin(2 pi x).

We solve (1) by separation of variables. We write

D(x,t) = X(x)T(t).

From (1) we have

XT'' = X''T


T''/T = X''/X = -k^2

for some constant k. Thus we have

X(x) = A cos kx + B sin kx


T(t) = M cos kt + N sin kt.

From the boundary conditions, we have

X(0) = X(1) = 0,

whence A = 0 and k = n pi

for some integer n. Thus we have

X_n(x) = B_n sin(n pi x)


T_n(t) = M_n cos(n pi t) + N_n sin(n pi t),


D(x,t) = sum_{n=0}^infinity(X_n(x) T_n(t))
= sum_{n=0}^infinity(sin(n pi x)(P_n cos(n pi t) + Q_n sin(n pi t))).

From the first initial condition, we have

D(x,0) = sum_{n=0}^infinity(P_n sin(n pi x)) = 0,

whence P_n = 0 for all n.

Thus we have

D(x,t) = ...

Solution Summary

We solve a particular case of the wave equation both analytically and numerically and compare the results.