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Question

A wave propagates on a string in the positive x -direction at a velocity v. The shape of the string at t=t0 is given by g(x,t0)=Asin(x/a). Write the wave equation for a general time t .

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Solution

At t=t0,g(x,t0)=Asin(x/a) ___(1)
For a wave traveling in the positive x-direction, the general equation is given by
y=f(xatT)
Putting t=t0 and comparing with equation (1), we get
g(x,0)=Asin{(x/a)+(t0/T)}
g(x,t)=Asin{(x/a)+(t0/T)(t/T)}
As T=a/v (a = wave length, v = speed of the wave)
y=Asin(xa+t0(a/v)t(a/v))
=Asin(x+v(t0t)a)
y=Asin[xv(tt0)a]

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