This is a cross-post from Math.SE, where no answer was given after 3 months.
Consider a plane 2D wavelet moving towards a corner reflector with 120° opening angle with infinitely extended sides. The surface of the reflector has homogeneous Dirichlet boundary conditions imposed, and the wave obeys the usual hyperbolic wave equation:
The solution, with the initially propagating part and the reflections, can be easily constructed by rotating the initial wavelet, changing its sign and putting the resulting wavelet next to the initial one so as to satisfy the boundary conditions by canceling the wave function at the boundaries. The result will look like this (ignore the small-wavelength artifacts, they are due to numeric errors in the simulation):
But as the points of the slanted reflected waves come close to the corner, there appears a problem: simply “sliding” the reflected part no longer works, since the shape of the reflecting boundary abruptly changes. Moreover, numerical simulation (see below) shows that the reflection from the corner doesn’t produce a backwards-propagating plane wave (that we’d get from a 90° corner): instead the original slanted reflections continue their paths, and a new, cylindrical, wave originates from the corner. This cylindrical wave appears to cancel out the values of the slanted reflections on the boundary to satisfy the boundary conditions.
My question is: what is the analytical form of this cylindrical wave? It doesn’t seem to be a Bessel function, because Bessel functions don’t have constant amplitude nor constant wavelength (they change with radius). So what is it then? Does it have a closed form? Or is it at least explicitly expressible as an integral or a series?