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16.2 Wave Motion on an Interval 575

0.4

0.2

0
0 0.5 1 1.5 2 2.5 3
x
–0.2

–0.4

–0.6

FIGURE 16.7 Wave profiles in Example 16.6,
decreasing as increases.

16.2.5 Wave Motion with a Forcing Term
Separation of variables may fail if the partial differential equation contains terms allowing for
some type of external forcing, or if the boundary conditions are nonhomogeneous. In such a case
it may be possible to transform the initial-boundary value problem to one that we know how to
solve.

EXAMPLE 16.7
We will solve the problem
2
∂ y ∂y 2
= + Ax for 0 < x < L,t > 0,
∂t 2 ∂x 2
y(0,t) = y(L,t) = 0for t ≥ 0,
and
∂y
y(x,0) = 0, (x,0) = 1for 0 < x < L.
∂t
A is a positive constant and the term Ax in the wave equation represents an external force having
magnitude Ax at x.Wehavelet c = 1 in this problem.
If we put y(x,t) = X(x)T (t) into the partial differential equation we obtain
XT = X T + Ax,

and there is no way to separate the t dependency on one side of an equation and the x dependency
on the other. One strategy in such a case is to try to transform this problem into one to which
separation of variables applies. Let
y(x,t) = Y(x,t) + ψ(x).
The idea is to choose ψ to obtain a problem for Y that we can solve. Substitute y(x,t) into the
partial differential equation to get
2
2
∂ Y ∂ Y

= + ψ (x) + Ax.
∂t 2 ∂x 2

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October 14, 2010 15:23 THM/NEIL Page-575 27410_16_ch16_p563-610