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Z x+ct
1 1
u(x, t) = (f(x − ct) + f(x + ct)) + g(ξ) dξ
2 2c x−ct
Z x+ct
1 1
u(x, t) = (u(x − ct, 0) + u(x + ct, 0)) + u t (ξ, 0) dξ
2 2c x−ct

6.4 The Wave Equation for a Semi-Infinite Do-

main

Consider the wave equation for a semi-infinite domain.

2
u tt = c u xx , 0 < x < ∞, t > 0
u(x, 0) = f(x), u t (x, 0) = g(x), u(0, t) = h(t)

Again the solution is the sum of a right-moving and a left-moving wave.

u(x, t) = F(x − ct) + G(x + ct)

For x > ct, the boundary condition at x = 0 does not affect the solution. Thus we know
the solution in this domain from our work on the wave equation in the infinite domain.
Z x+ct
1 1
u(x, t) = (f(x − ct) + f(x + ct)) + g(ξ) dξ, x > ct
2 2c x−ct

From this, F(ξ) and G(ξ) are determined for ξ > 0.
Z
1 1
F(ξ) = f(ξ) − g(ξ) dξ, ξ > 0
2 2c
Z
1 1
G(ξ) = f(ξ) + g(ξ) dξ, ξ > 0
2 2c

In order to determine the solution u(x, t) for x, t > 0 we also need to determine F(ξ)
for ξ < 0. To do this, we substitute the form of the solution into the boundary condition
at x = 0.

u(0, t) = h(t), t > 0
F(−ct) + G(ct) = h(t), t > 0

F(ξ) = −G(−ξ) + h(−ξ/c), ξ < 0
Z −ξ
1 1
F(ξ) = − f(−ξ) − g(ψ) dψ + h(−ξ/c), ξ < 0
2 2c
We determine the solution of the wave equation for x < ct.

u(x, t) = F(x − ct) + G(x + ct)
Z −x+ct
1 1 1
u(x, t) = − f(−x + ct) − g(ξ) dξ + h(t − x/c) + f(x + ct)+
2 2c 2

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