Page 386 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 23: Wave and Diffusion Equations by Separation of Variable

This leads to the equation Notes
s
0 AJ 0 2 ...(15)
g

Now, for a non-trivial solution, A cannot be equal to zero, and hence we have

s
J 2 0
0 ...(16)
g
If we let

s
u 2
g ...(17)
we must find the roots of the equation
J u 0 ...(18)
0
If we consult a table of Bessel functions, we find that the first three zeros of the Bessel function
J u are given by the values
0
2.405, 5.52, 8.654
Accordingly the various possible values of are given by
2.405 g 5.52 g 8.654 g
1 2 3 etc. ...(19)
2 s 2 s 2 s
To each value of we associate a characteristic function or eigenfunction v of the form
n
x
v n A J 0 2 n ...(20)
n
g
Since the real and imaginary parts of the assumed solution (7) are solutions of the original
differential equation, we can construct a general solution of (6) satisfying the boundary conditions
by summing the particular solutions corresponding to the various possible values of n in the
manner
n x
y , x t J 2 A cos t B sin t
0 n n n n n ...(21)
n 1 g
where the quantities A and B are arbitrary constants to be determined from the boundary
n n
conditions of the problem. In the case under consideration there is no initial velocity imparted
to the chain; hence
y
0 ...(22)
t t 0
This leads to the condition

B n 0 ...(23)
At t = 0 we have

n x
y x A J 2
0 n 0 n g ...(24)
n 1

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