Solve the Neumann problem for the wave equation on the half-line 0 < x < x.

Consider the wave equation

$u_{xx}-c^2{u}_{tt}=0$ for $0<x<\infty$

$$\begin{gathered} u\left(x,0\right)=\varphi \left(x\right) \\ {u}_t\left(0,t\right)=\psi \left(t\right) , x>0 \\ {u}_x\left(0,t\right)=0, t>0 \end{gathered}$$

Using the reflection method with even extensions to reduce the problem to an IVP on the whole line.

Define the even extensions of the initial data

$$\begin{gathered} {\varphi }_{even}=\left\{\begin{array}{l}\varphi \left(x\right) for x\ge 0\\ \varphi \left(-x\right) for x\le 0\end{array}\right. \\ {\psi }_{even}=\left\{\begin{array}{l}\psi \left(x\right) for x\ge 0\\ \psi \left(-x\right) for x\le 0\end{array}\right. \end{gathered}$$

Using D' Alembert formula for the solution we have 

$u\left(x,t\right)=\frac{1}{2}\left[\varphi \left(x+ct\right)+\varphi \left(x-ct\right)\right]+\frac{1}{2c}{\int }_{x-ct}^{x+ct}\psi \left(y\right)dy$