On an infinite string the wave equation has a beautifully explicit solution. Because
u_{tt} - c^2 u_{xx} = 0 factors like a difference of squares —
(\partial_t - c\partial_x)(\partial_t + c\partial_x)u = 0 — its general
solution is a sum of two
travelling
shapes:
u(x, t) = F(x - ct) + G(x + ct).
One profile F moves right at speed c, the
other G moves left. The wave equation is just two transport equations
bundled together, one for each direction.
Matching the initial data
Given the initial shape u(x, 0) = f(x) and initial velocity
u_t(x, 0) = g(x), solving for F and
G yields d'Alembert's formula:
u(x, t) = \tfrac{1}{2}\big[f(x - ct) + f(x + ct)\big] + \frac{1}{2c}\int_{x - ct}^{x + ct} g(s)\,ds.
With zero initial velocity (g = 0) it says something vivid: the initial
shape splits in half, and the two halves glide apart in opposite directions at
speed c.
The formula also reveals the domain of dependence: the value at
(x, t) depends only on the initial data in the interval
[x - ct, x + ct]. Information travels at finite speed
c — nothing outside that interval can yet have reached the point. This
is the sharpest contrast with the heat equation, whose influence is instantaneous.
- General solution: u = F(x - ct) + G(x + ct) — a right-mover plus a left-mover.
- With data u(x,0)=f, u_t(x,0)=g: the boxed formula above.
- Finite speed: u(x,t) depends only on the data in [x-ct, x+ct] (its domain of dependence).