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).
Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2010 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.
