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.