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).
