Chapter 12.4, Problem 12E

A model for the motion of a vibrating string whose ends are allowed to slide on frictionless sleeves attached to the vertical axes x = 0 and x = L, is given by the wave equation (1) and the conditions

      $\begin{array}{l}{\frac{\partial u}{\partial x}|}_{x=0}=0,{\frac{\partial u}{\partial x}|}_{x=L}=0,t>0\\ u\left(x,0\right)=f\left(x\right),{\frac{\partial u}{\partial x}|}_{t=0}=g\left(x\right),0<x<L.\end{array}$

See Figure 12.4.8. The boundary conditions indicate that the motion is such that slope of the curve is zero at its ends for t > 0. Find the displacement u(x, t).