2.5. PROBLEMS 89 for arbitrary functions F, G. (b) Using the change of variables ξ = x + t, η = x − t, show utt − uxx = 0 if and only if uξη = 0. (c) Use (a) and (b) to rederive d’Alembert’s formula. (d) Under what conditions on the initial data g, h is the solution u a right-moving wave? A left-moving wave? 20. Assume that for some attenuation function α = α(r) and delay func- tion β = β(r) ≥ 0, there exist for all profiles φ solutions of the wave equation in (Rn − {0}) × R having the form u(x, t) = α(r)φ(t − β(r)). Here r = |x| and we assume β(0) = 0. Show that this is possible only if n = 1 or 3, and compute the form of the functions α, β. (T. Morley, SIAM Review 27 (1985), 69–71) 21. (a) Assume E = (E1,E2,E3) and B = (B1,B2,B3) solve Maxwell’s equations Et = curl B, Bt = − curl E div B = div E = 0. Show Ett − ΔE = 0, Btt − ΔB = 0. (b) Assume that u = (u1,u2,u3) solves the evolution equations of linear elasticity utt − μΔu − (λ + μ)D(div u) = 0 in R3 × (0, ∞). Show w := div u and w := curl u each solve wave equations, but with differing speeds of propagation. 22. Let u denote the density of particles moving to the right with speed one along the real line and let v denote the density of particles moving to the left with speed one. If at rate d 0 right-moving particles randomly become left-moving, and vice versa, we have the system of PDE ut + ux = d(v − u) vt − vx = d(u − v). Show that both w := u and w := v solve the telegraph equation wtt + 2dwt − wxx = 0.

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