Question

A string is stretched and fastened to two points distance l apart. Motion is started by displacing the string into the form y=k(lx-x^2),from which it is released at time t=0.write the governing differential equation and initial and boundary condition for this problem

Answer 1

The displacement y(x,t) is given by the equation,

$\frac{d^2y}{dt^2}=a^2\frac{d^2y}{dt^2}$

The boundary condition are:-

i ) y(0,t)= 0 , for t$\ge0$

ii ) y(l,t)= 0 , for t$\ge0$

iii) y(x,0)= 0 , for $0\le x\le l$

iv) $[\frac{dy}{dt}]_{t=0}=kx(l-x), for 0\le x\le l$