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).
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