Solve the boundary-value problem a2 02 u 0x 2 02 u 0t 2 , 0 , x , L, t . 0 u(0, t) 0, E

Chapter 13, Problem 6

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Solve the boundary-value problem

\(a^{2} \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial t^{2}}\),  0<x<L,  t>0

u(0, t)=0,  \(\left.E \frac{\partial u}{\partial x}\right|_{x=L}=F_{0}\),  t>0

u(x, 0)=0,  \(\left.\frac{\partial u}{\partial t}\right|_{t=0}=g(x)\),  0<x<L

The solution u(x, t) represents the longitudinal displacement of a vibrating elastic bar that is anchored at its left end and is subjected to a constant force \(F_{0}\) at its right end. See Figure 13.4.7 on page 723. E is called the modulus of elasticity.

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