
572 SECOND-ORDER HYPERBOLIC EQUATIONS WITH ONE SPACE VARIABLE
◮ Domain: −∞ < x < ∞. Cauchy problem.
Initial conditions are prescribed:
w = f (x) at t = 0,
∂
t
w = g(x) at t = 0.
Solution:
w(x, t)=
1
2
f(x+at) exp
−
bt
2a
+
1
2
f(x−at) exp
bt
2a
−
σt
2a
exp
bx
2a
2
Z
x+at
x−at
exp
−
bξ
2a
2
J
1
σ
p
t
2
−(x−ξ)
2
/a
2
p
t
2
−(x−ξ)
2
/a
2
f(ξ) dξ
+
1
2a
exp
bx
2a
2
Z
x+at
x−at
exp
−
bξ
2a
2
J
0
σ
p
t
2
−(x−ξ)
2
/a
2
g(ξ) dξ
+
1
2a
Z
t
0
Z
x+a(t−τ)
x−a(t−τ)
exp
b(x−ξ)
2a
2
J
0
σ
p
(t−τ)
2
−(x−ξ)
2
/a
2
Φ(ξ, τ ) dξ dτ,
where J
0
(z) and J
1
(z) are Bessel functions of the first kind, and σ =
1
2
|b|/a.
◮ Domain: 0 ≤ x ≤ l. First boun dary value problem.
The following conditions are prescribed:
w = f
0
(x) at t = 0 (initial condition),
∂
t
w =