2.4. WAVE EQUATION 67

2.4.1. Solution by spherical means.

We began §§2.2.1 and 2.3.1 by searching for certain scaling invariant

solutions of Laplace’s equation and the heat equation. For the wave equation

however we will instead present the (reasonably) elegant method of solving

(1) ﬁrst for n = 1 directly and then for n ≥ 2 by the method of spherical

means.

a. Solution for n = 1, d’Alembert’s formula. We ﬁrst focus our atten-

tion on the initial-value problem for the one-dimensional wave equation in

all of R:

(3)

utt − uxx = 0 in R × (0, ∞)

u = g, ut = h on R × {t = 0},

where g, h are given. We desire to derive a formula for u in terms of g and

h.

Let us ﬁrst note that the PDE in (3) can be “factored”, to read

(4)

∂

∂t

+

∂

∂x

∂

∂t

−

∂

∂x

u = utt − uxx = 0.

Write

(5) v(x, t) :=

∂

∂t

−

∂

∂x

u(x, t).

Then (4) says

vt(x, t) + vx(x, t) = 0 (x ∈ R, t 0).

This is a transport equation with constant coeﬃcients. Applying formula

(3) from §2.1.1 (with n = 1, b = 1), we ﬁnd

(6) v(x, t) = a(x − t)

for a(x) := v(x, 0). Combining now (4)–(6), we obtain

ut(x, t) − ux(x, t) = a(x − t) in R × (0, ∞).

This is a nonhomogeneous transport equation; and so formula (5) from §2.1.2

(with n = 1, b = −1, f(x, t) = a(x − t)) implies for b(x) := u(x, 0) that

(7)

u(x, t) =

t

0

a(x + (t − s) − s) ds + b(x + t)

=

1

2

x+t

x−t

a(y) dy + b(x + t).