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Handbook of Linear Partial Differential Equations for Engineers and Scientists, 2nd Edition
book

Handbook of Linear Partial Differential Equations for Engineers and Scientists, 2nd Edition

by Andrei D. Polyanin, Vladimir E. Nazaikinskii
December 2015
Intermediate to advanced content levelIntermediate to advanced
1643 pages
59h 33m
English
Chapman and Hall/CRC
Content preview from Handbook of Linear Partial Differential Equations for Engineers and Scientists, 2nd Edition
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
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Publisher Resources

ISBN: 9781466581494