Green's function for the wave equation

[dRic2]

Hi, I'm reading "Wave Physics" by S. Nettel and in chapter 3 he introduces the Green's function for the 1-dimensional wave equation. Using the separation of variables method he restricts his attention to the spatial component only. Let $u(x)$ be the spatial solution to the wave equation and $G(x|x_0)$ the Green's function. The author now states the following:

Let us now suppose that it is $u(x)$ that is specified at an end point. Then we must choose $G(x|x_o)$ to vanish at that end point.

I don't really understand this connection. Can someone help me ?

Thanks
Ric

 

[Orodruin]

You are looking to write down the solution on the form
$$
u(x) = \int G(x|x_0) \kappa(x_0) dx_0 + f(x)
$$
where $\kappa$ is the inhomogeneity in your wave equation and $f(x)$ solves the homogeneous wave equation and satisfies your boundary conditions. If $G(x|x_0)$ does not vanish at the boundary, then the integral will generally not be zero there and your function will not satisfy the boundary conditions.

 

[dRic2]

In the book I'm reading there is a slightly different version of the solution:

(forget about the $\frac 1 {\rho}$)

BTW I also do not understand your answer. If, for example, $k$ vanishes at the end point the integral could also be zero and the boundary condition would be verified.

 

[Orodruin]

If, for example, $k$ vanishes at the end point the integral could also be zero and the boundary condition would be verified.

No, this is incorrect. The point is to evaluate $x$ at the boundary, not $x_0$, which is an integration variable.

 

[dRic2]

Sorry, dumb mistake. Now I get your argument, thanks.