For the following wave equation
$
\frac{{\partial ^2 p}}{{\partial ^2 x}} + \frac{{\partial ^2 p}}{{\partial ^2 y}} = A\frac{{\partial ^2 p}}{{\partial ^2 t}} + B\frac{{\partial p}}{{\partial t}}
$
is there a way to show that there are boundary conditions at or near positive and negative infinity, for both non-zero B and B=0 conditions, and for {A,B} as rational numbers? I believe that this should follow from Sommerfeld's condition of radiation, and should perhaps be similar to conditions for the ordinary wave equation. What are these boundary conditions? Ideally, I think that the boundary conditions should involve both time and spatial derivatives.
By "positive and negative infinity" I mean that I am interested in what happens when $x \to \pm \infty
$ and $y \to \pm \infty$. I've been working on a problem where I would like to computationally solve the wave equation with boundary conditions that approximate infinity. So I suppose that this would be an imposed compatibility condition.
Best Answer
First of all, can you be more precise in your question? You are asking about boundary conditions at infinity, and this might make sense, but... for what purpose? do you need a set of conditions that imply existence and uniqueness of a global solution? or, do you need to classify solutions of the standard Cauchy problem (with data at t=0) according to their behaviour at infinity?
Anyway, there are a couple general tools that might help you at least to clarify what you are looking for exactly:
1) If you need a tool to classify solutions according to their behaviour at infinity, then scattering theory (mentioned by Willie in his comment) might be helpful. However, its main purpose is to compare two different equations, i.e., use the solutions of a simpler equation to classify the solutions of a 'more difficult' equation. So I do not think this is what you actually need.
2) If you need to understand what might be reasonable 'data at infinity' for a Cauchy problem, then the Kelvin transform might be of use. This is a space-time change of coordinates that transforms a wave equation into a wave equation, and exchanges infinity with t=0. Playing with it might give you some insight into what kind of conditions you might impose at infinity on your solution. There is also a much more sofisticated transform with a similar effect, the Penrose transform, but this might be overkill in your case.