How can I solve the wave equation with the D'alambert solution in a finite domain with one end of the string clamped and the other end free to move vertically? i.e.
$u_{tt}=c^2u_{xx}, \quad 0<x<L,\quad t>0$
$u(0,x)=f(x), \quad u_t(0,x)=0, \quad 0\le x\le L$
$u_x(t,0)=u(t,L)=0, \quad t>0.$
What will be the extension of $f(x)$?