Page 382 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 23: Wave and Diffusion Equations by Separation of Variable

Notes
v v

R T
2 2 2 2
v v R 2 v . T v . T
r 2 R 2 r R T r T 2 r

2 v 2 v 2 v
2 2 2
R R T T
v v R v T
.
r R t T t

r v
( e ) e
R T

2 2 2 2
v e v R 2e 2 v e 2 v
r 2 R 2 t R T T 2

2 v 2 v 2 v
e 2 2 2 2
R R T T
Substituting in (D) we have

2 2 2 2 2 2 2
v 2 v v c v 2 v v
R 2 R T T 2 a 2 R 2 R T T 2

2 v
or 0 ...(E)
R T
Integrating with respect to T we have

v
F R ...(F)
R
where F(R) is a constant as far as T is concerned.
Integrating (F) we have

T
R
v F ( )dR G ( )
T
R
H ( ) G ( )
or v H r ct G r ct
This is known as D, Alemberts, solution of the wave equation.

The Transverse Vibrations of a Stretched String

Consider a perfectly flexible string that is stretched between two points having a constant
tension T which is large enough so that the gravity may be neglected. Let the string be uniform
and have a mass per unit length equal to m.

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