This is my first question/contribution even though I have been following hot topics for a while, I have been confronted a to a problem recently and I don't really know how to address it.
This comes from acoustics, there is a consequent amount of literature around solving the wave equation under homogeneous Neumann condition on some boundaries. This represent the case of a sound wave propagating in a room with perfectly reflecting walls. One method for addressing this problem is the image source method where we construct a lattice from the source by reflecting though a list of walls. This was done for an arbitrary polyhedral room in 1984 in a paper named "Extension of the image model to arbitrary polyhedra" by Jeffrey Borish.
Unfortunately he don't give proofs of his argument, just some intuition. I am currently researching on this and I would like to be able to prove his results, but I would like to also generalize the method for non closed rooms
So long story short, I end up been stuck solving the following equation for the pressure $$\nabla^2 p({\bf x}) + k^2 p({\bf x}) = 0$$ Subject to $$\nabla p({\bf y}) = {\bf q}$$ Where we know $k$, ${\bf q}$ and ${\bf y}$ (${\bf x}$ and ${\bf y}$ are vector in $\mathbb{R}^n$) Of course as is, there might even not be a solution to this problem, I have been reading through section 6 of the book "Methods of Theoretical physics" which gave me some useful insight, in the end of section 6.2 he states the cases when the solution exists and about uniqueness. For my case, Hyperbolic equation with open boundaries, they state that we have insufficient information about the problem. So I may actually have a solution.
In the sane book section 6.3, he solve a lot of these kinds of equations but the boundaries condition always seems to be homogeneous. They are using an eigen decomposition of the problem, maybe it is possible to adapt this reasoning to my problem. Finally I went through a substantial amount of literature on the subject but the boundaries conditions where almost always homogeneous. I also could not find a similar question here or somewhere else on the web.
So if any one have an idea about this I would be glad to discuss it. I hope that this is clear enough. Thanks in advance.
EDIT : I forgot to mention that in I also have boundaries conditions at infinite distance from the origin, the pressure is supposed to be zero.
EDIT 2 : I think the solution may actually be unique. The uniqueness should not depend on the choice of ${\bf q}$ or ${\bf y}$ and so we can probably just look at ${\bf y} = {\bf q} = {\bf 0}$. In this case if we use a eigen decomposition argument, we get that our eigen functions would be of the form $\cosh(a_{k,i} x_i)$ for $x_i$ the i'th coordinate of ${\bf x}$ and for any $a_{k,i}$, then we can express our pressure as $$p({\bf x}) = \sum_k \sum_i A_{k,i} \cosh(a_{k,i} x_i)$$ and $$\nabla^2 p({\bf x}) = \sum_k \sum_i A_{k,i} a_{k,i}^2 \cosh(a_{k,i} x_i)$$ Now the next step is a bit unclear but I think that in order to have $\nabla^2 p({\bf x}) + k^2 p({\bf x}) = 0$, then we need to have $A_{k,i} a_{k,i}^2 + A_{k,i} k^2 =0$ for all $k$ and $i$. This can only be true if for all $k$ and $i$, $A_{k,i}=0$. Does this reasoning makes sense ? And is way to generalize to any ${\bf q}$ ? Then generalizing to any ${\bf y}$ is trivial.