Page 384 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 23: Wave and Diffusion Equations by Separation of Variable

Notes
y 2 y
T m ...(vii)
x x t 2
Now if the stretching force is constant throughout the string then we can write

2 y 2 y
T 2 m 2 ...(viii)
x t

2 y 1 2 y
or 2 2 2 ...(ix)
x c t

T
where c ...(x)
m

This equation (ix) is known as one dimensional wave equation and is a special case of the
general wave equation.

The Oscillations of a Hanging Chain

Let us consider the small coplanar oscillations of a uniform flexible string or chain hanging
from a support under the action of gravity as shown in Figure 23.2. We consider only small
deviations y from the equilibrium position; x is measured from the free end of the chain. Let it
be required to determine the position of the chain

y y , x t ...(1)

where at t = 0 we give the chain an arbitrary displacement
y y x ...(2)
0
In this case the tension T of the chain is variable, and hence eq. governing the displacement of the
chain at any instant is given by

y 2 y
T m ...(3)
x x t 2
where m is the mass per unit length of the chain. In this case the tension T is given by

T mgx ...(4)
Hence we have

y 2 y
mgx m ...(5)
x x t 2
Or, differentiating and dividing both members by the common factor m, we have

2 y y 1 y
2
x ...(6)
x 2 x g t 2

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