
Chapter 7
Second-Order
Hyperbolic Equations
with Two Space Variables
7.1 Wave Equation
∂
2
w
∂t
2
= a
2
∆
2
w
7.1.1 Problems in Cartesian Coordinates
The wave equation with two space variables in the rectangular Cartesian system of coordi-
nates has the form
∂
2
w
∂t
2
= a
2
∂
2
w
∂x
2
+
∂
2
w
∂y
2
.
◮ Particular solutions and some relations.
1
◦
. Particular solutions:
w(x, y, t) = A exp
k
1
x + k
2
y ± at
p
k
2
1
+ k
2
2
,
w(x, y, t) = A sin(k
1
x + C
1
) sin(k
2
y + C
2
) sin
at
p
k
2
1
+ k
2
2
,
w(x, y, t) = A sin(k
1
x + C
1
) sin(k
2
y + C
2
) cos
at
p
k
2
1
+ k
2
2
,
w(x, y, t) = A sinh(k
1
x + C
1
) sinh(k
2
y + C
2
) sinh
at
p
k
2
1
+ k
2
2
,
w(x, y, t) = A sinh(k
1
x + C
1
) sinh(k
2
y + C
2
) cosh
at
p
k
2
1
+ k
2
2
,
w(x, y, t) = ϕ(x sin β