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9 Wave Equations
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In one spatial dimension the linear wave equation reads:
2
u tt = v u xx , x ∈ R , t ∈ R , (9.7)
where v is a positive parameter. This equation is particularly easy to solve. We
introduce characteristic coordinates r = x − vt, s = x + vt and rewrite (9.7) as:

u rs = 0 . (9.8)

The general solution of (9.8) is the sum of a function of r and afunctionof s
such that after back-transformation we obtain:
u(x, t) = f (x + vt)+ g(x − vt) (9.9)

for the general solution of (9.7), where f and g are arbitrary smooth functions.
Thus, the general solution of the one dimensional linear wave equation is the
sum of two travelling waves, one travelling to the left and the other travelling
to the right. Consider now the one dimensional wave equation (9.7) with initial
data given by a point source with vanishing initial velocity:

u(x, t) = δ(x), u t (x, t = 0) = 0, x ∈ R .

Fig. 9.1. Circular Waves in a Kyoto Zen Garden