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.
