Answer to Question #205552 in Differential Equations for Kumar Aditya

Question

 A tightly stretched string of length is fastened at both ends. The midpoint is taken to a height 

   transversely and the string is released from rest in this position. Find the displacement of any point 

  of the string at any subsequent time.

Answer 1

When string is released, then it will vibrate. Let the equation which represents the displacement is given by, $y(x,t)= Asin(\omega t +kx+\phi)$

where $\omega$ is angular velocity, k is wave number, $\phi$ is initial phase.

According to the question, at end points x=0 and x=l, displacement is zero and maximum displacement is at mid point that is h.

So, $A=h$

Now, since string is held at end points so it represents half wavelength of the wave. Then $k=\frac{2\pi}{\lambda}, \lambda =l/2 \implies k=\frac{4\pi}{l}$

For phase, at x=0 and x=l, displacement is zero, so

$0= hsin(\phi)$ and $0 = hsin(\frac{4π}{l}l+\phi)= hsin(4π+\phi)$

so, $\phi=0$

velocity:

$y'_t=\omega Acos(\omega t +kx)$

$y'_t(x,0)=\omega Acos( kx)=0\implies \omega =0$

then:

$y(x,t)= hsin(4\pi x/l)$