Fundamental Solution of Traveling Wave

Question

So given the stationary equation for a traveling wave with wave-number k, $-\dfrac{d^2y}{dx^2}-{k^2g}=\delta(x-\xi)$ with $0<x,\xi<1$, how can I find a causal fundamental solution to this? I assume we can start by using the heuristic approach which after calculations would yield a causal solution E$(x,\xi)=u_{\xi}(x)H(x-\xi)$ where then we first find the solution $u_{\xi}$ to get E$(x,\xi)$. Also, since this is a causal solution we also know that E$(x,\xi)=0$ for $x<\xi$. This is pretty much what I can provide for my attempt for finding E$(x,\xi)$ but I am still new to this concept and I honestly don't know where to begin. It is also asked that a verification through differentiation (in the distributional sense) of the causal fundamental solution E$(x,\xi)$ can satisfy the original equation. Any help would be greatly appreciated, thanks in advance!

Answer 1

Note that in distribution,

$$y''(x)=-k^2g$$

for $x\ne\xi$. So, we have

$$y(x)=\begin{cases} -\frac12 k^2g x^2+C_1x+C_2&, x<\xi\\\\ -\frac12 k^2g x^2+C_3 x+C_4&,x>\xi \end{cases}$$

Now enforce continuity of $y(x)$ at $x=\xi$ and

$$y'(\xi^+)-y'(\xi^-)=-1$$