Chapter 12.4, Problem 17E

The transverse displacement u(x, t) of a vibrating beam of length L is determined from a fourth-order partial differential equation

$a^2\frac{{\partial }^{4}u}{\partial x^4}+\frac{{\partial }^{4}u}{\partial t^2}=0,0<x<L,t>0$

If the beam is simply supported, as shown in Figure 12.4.9, the boundary and initial conditions are

$\begin{array}{l}u\left(0,t\right)=0,u\left(L,t\right)=0,t>0\\ {\frac{{\partial }^{2}u}{\partial x^2}|}_{x=0}=0,{\frac{{\partial }^{2}u}{\partial x^2}|}_{x=L}=0,t>0\\ u\left(x,0\right)=f\left(x\right),{\frac{\partial u}{\partial t}|}_{t=0}=g\left(x\right),0<x<L.\end{array}$

Solve for u(x, t). [Hint: For convenience use λ = α⁴ when separating variables.]