Wave equation with Neumann BDC

[BustedBreaks]

The problem statement is:

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

Here is what I have

$$U_{tt}=c^{2}U_{xx}$$
Initial conditions
$$U(x,0)=\phi(x)$$
$$U_{t}(x,0)=\psi(x)$$
Neumann BC
$$U_{x}(0,t)=0$$

So I extend $$\phi(x)$$ and $$\psi(x)$$ evenly and get:


$$\phi_{even}(x)={^{\phi(x) for x\geq 0}_{\phi(-x) for x\leq 0}$$
$$\psi_{even}(x)={^{\psi(x) for x\geq 0}_{\psi(-x) for x\leq 0}$$


I'm not sure what to do from here. I know the d'Alembert equation, but just plugging in phi even and psi even doesn't make much sense to me as the solution seems too general.

I'm also not too comfortable with the meaning behind the boundary conditions, so if you can dumb down your answers, I'd be appreciative.

Thanks!

 

[mighty2000]

Thank you for your interest in solving the Neumann problem for the wave equation on the half line. It seems like you have a good understanding of the initial conditions and boundary conditions for this problem. However, I can see that you are struggling with how to proceed with finding the solution. Let me try to break it down for you in simpler terms.

The d'Alembert equation you mentioned is indeed the key to solving this problem. It states that the solution to the wave equation can be expressed as the sum of two functions, one representing a forward-moving wave and the other representing a backward-moving wave. These functions can be written as U(x,t) = F(x-ct) + G(x+ct), where F and G are arbitrary functions that satisfy the initial conditions and boundary conditions.

Now, in order to apply this equation to the Neumann problem, we need to find appropriate expressions for the functions F and G. This is where your even extensions of the initial conditions come in. By extending the functions evenly, we are essentially creating a continuous function that satisfies the boundary conditions at x=0. This is because the derivative of an even function is always symmetric about the y-axis, and thus will be equal to 0 at x=0.

So, using the even extensions of \phi(x) and \psi(x), we can write the solution as:

U(x,t) = F(x-ct) + G(x+ct) = \frac{1}{2}[\phi_{even}(x-ct) + \phi_{even}(x+ct)] + \frac{1}{2c}[\psi_{even}(x-ct) - \psi_{even}(x+ct)]

This is the most general solution to the Neumann problem on the half line. You can further simplify it by choosing specific forms for F and G, depending on the initial conditions given. I hope this helps clarify the solution for you. Let me know if you have any further questions. Good luck!