Page 387 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes That is, we must expand the arbitrary displacement y (x) into a series of Bessel functions to
0
zeroth order. To do this, we can make use of the results of unit 13. It is shown there that an
arbitrary function of F(x) may be expanded in a series of the form
n
F x A J u x ...(25)
n
0
n
n 1
where the quantities u are successive positive roots of the equation
n
J u 0 ...(26)
n
The coefficient A are then given by the equation
n
2 1
A n zJ u z F z dz
0
n
2
J u n 0 ...(27)
1
To make use of this result to obtain the coefficients of the expansion (24), it is necessary to
introduce the variable
x
z ...(28)
s
In view of (17) and (18), eq. (24) becomes

n
y x y sz 2 F z A J u z
0 0 n 0 n ...(29)
n 1
This is the form (25), and the arbitrary constants are determined by (27).
The determination of the possible frequencies and modes of oscillation of a hanging chain is of
historical interest. It appears to have been the first instance where the various normal modes of
a continuous system were determined by Daniel Bernoulli (1732).

Self Assessment

1. Find the relations between l, m , n and k so that

y
z
V x , , ,t A exp i lx my nz kct B exp i lx my nz kct
is the solution of wave equation
1 2 V
2
V
c 2 t 2
23.1.1 Solution of One Dimensional Wave Equation

We shall now solve one dimensional wave equation under some boundary conditions. Let f(x)
and g(x) be the initial deflection and initial velocity of the string and the string is stretched
between two points (0, 0), (L, 0). Hence for the wave equation

2 2
u 1 u 0
x 2 c 2 t 2 ...(i)
u(0, t) = 0,
and u(L, t) = 0, for all t, and initial conditions ...(ii)

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