Question

7. (a) Solve the wave equation in three dimensions for t > 0 with the
initial conditions φ(x) = A for |x| < ρ, φ(x) = 0 for |x| > ρ, and
ψ|x| ≡ 0, where A is a constant. (This is somewhat like the plucked
string.) (Hint: Differentiate the solution in Exercise 6(b).)

((b) Solve the wave equation in three dimensions for t > 0 with the
initial conditions φ(x) ≡ 0,ψ(x) = A for |x| < ρ, and ψ(x) = 0
for |x| > ρ, where A is a constant. Sketch the regions in space-
time that illustrate your answer. (This is like the hammer blow of
Section 2.1.))

(b) Sketch the regions in space-time that illustrate your answer.Where
does the solution have jump discontinuities?
(c) Let |x0| < ρ. Ride the wave along a light ray emanating from
(x0,0). That is, look at u(x0+ tv, t ) where |v| = c. Prove that
t · u(x0+ tv, t ) converges as t → ∞.

Answer 1

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