Page 385 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes
Figure 23.2

y

x

As in the case of the tightly stretched string, let us assume

y , x t e jwt v x ...(7)
Substituting this into (6), we obtain

2 v v 2
x v 0 ...(8)
x 2 x g

This equation resembles Bessel s differential equation. Changing the variable x to Z by the
relation:

4 2 x
Z 2 ...(9)
g
reduces (8) to

2 v v
2
Z 2 Z Z v 0 ...(10)
Z 2 Z
whose general solution is

v AJ Z BY Z ...(11)
0
0
where J Z ,Y Z are Bessel functions of first and second kind.
0
0
In order to satisfy the condition that the displacement of the string y remain finite when x = 0, we
must place
B = 0 ...(12)

Accordingly, in terms of the original variable x, we have the solution

x
v AJ 2 ...(13)
0
g
for the function v.
So far, the value of is undetermined. In order to determine it, we make use of the boundary
condition
v 0 : at x s ...(14)

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