
582 SECOND-ORDER HYPERBOLIC EQUATIONS WITH ONE SPACE VARIABLE
◮ Reduction to a constant coefficient equation.
The substitution u(r, t) = rw(r, t) leads to the constant coefficient equation
∂
2
u
∂t
2
= a
2
∂
2
u
∂r
2
,
which is discussed in Section 6.1.1.
◮ Domain: 0 ≤ r < ∞. Cauchy problem.
Initial conditions are prescribed:
w = f (r) at t = 0,
∂
t
w = g(r) at t = 0.
Solution:
w(r, t) =
1
2r
(r − at)f
|r − at|
+ (r + at)f
|r + at|
+
1
2ar
Z
r+at
r−at
ξg
|ξ|
dξ.
Solution at the center r = 0:
w(0, t) = atf
′
(at) + f(at) + tg(at).
⊙ Literature: B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
◮ Domain: 0 ≤ r ≤ R. First boundary value problem.
The following conditions are