Page 381 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes propagated from the source of the sound. The electrical oscillations of a radio antenna generate
electromagnetic waves that are propagated through space. All these entities are governed by a
certain differential equation, called a wave equation. This equation has the form

2 2 2 2
2 u u u 1 u
u ...(1)
x 2 y 2 z 2 c 2 t 2
Where c is a constant having dimension of velocity, t is the time, x, y, z are the co-ordinates of a
certain reference frame and u is the entity under consideration, whether it be a mechanical
displacement of components of electromagnetic wave or currents or potentials of an electrical
transmission line.
In finding the solution of equation (1) we some times also employ cylindrical co-ordinate
system or spherical polar co-ordinate system.

In cylindrical co-ordinate system, wave equation is given by
2 2 2
u 2 u 1 sin u 1 u 1 u ...(A)
r 2 r r r 2 sin 2 r 2 sin 2 2 c 2 t 2

where as in cylindrical co-ordinate system r, , z the wave equation becomes
2 2 2 2
u 1 u 1 u u 1 u
...(B)
r 2 r r r 2 2 z 2 c 2 t 2

Example: Solution of wave equation symmetric in all directions about the origin, i.e.
independent of and .
In this case u is independent of and . So from equation (A) we have
2 2
u 2 u 1 u
...(C)
r 2 r r c 2 t 2
Putting
v ru

v u
r u
r r
v 2 u u
r 2
r t 2 r
so from (C)

2 2
v 1 u
...(D)
r 2 c 2 t 2
Putting
R r ct
T r ct
gives

v v R v T
r R r T r

374 LOVELY PROFESSIONAL UNIVERSITY