
Chapter 8
Second-Order
Hyperbolic Equations with
Three or More Space Variables
8.1 Wave Equation
∂
2
w
∂t
2
= a
2
∆
3
w
8.1.1 Problems in Cartesian Coordinates
The wave equation with three space variables in the rectangular Cartesian coordinate sys-
tem has the form
∂
2
w
∂t
2
= a
2
∂
2
w
∂x
2
+
∂
2
w
∂y
2
+
∂
2
w
∂z
2
.
This equation is of fundamental importance in sound propagation theory, the propagation
of electromagnetic fields theory, and a number of other areas of physics and mechanics.
◮ Particular solutions and their properties.
1
◦
. Particular solutions:
w(x, y, z, t) = A exp
k
1
x+k
2
y+k
3
z±at
p
k
2
1
+k
2
2
+k
2
3
,
w(x, y, z, t) = A sin(k
1
x+C
1
) sin(k
2
y+C
2
) sin(k
3
z+C
3
) sin
at
p
k
2
1
+k
2
2
+k
2
3