Chapter 13.4, Problem 18E

(a)

To determine

To show: The wave equation can be written as $\frac{{\partial }^{2}u}{\partial \eta \partial \xi }=0$.

(b)

To determine

To show: That $G\left(x\right)=\frac{1}{2}f\left(x\right)-\frac{1}{2a}{\displaystyle \underset{x_0}{\overset{x}{\int }}g\left(s\right)ds}-c$ and $F\left(x\right)=\frac{1}{2}f\left(x\right)+\frac{1}{2a}{\displaystyle \underset{x_0}{\overset{x}{\int }}g\left(s\right)ds}+c$ by the integration of the partial differential equation $\frac{{\partial }^{2}u}{\partial \eta \partial \xi }=0$.

(c)

To determine

To show: The given solution of d’Alembert’solution of the wave equation is $u\left(x,t\right)=\frac{1}{2}\left[f\left(x+at\right)+f\left(x-at\right)\right]+\frac{1}{2a}{\displaystyle \underset{x-at}{\overset{x+at}{\int }}g\left(s\right)ds}$.