# What is the analytical form of the cylindrical wave appearing on reflection of a plane wave from a corner?

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:

$$\partial_{tt} u(x,y,t)=\partial_{xx}u(x,y,t)+\partial_{yy}u(x,y,t).$$

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?

The keywords to search for this "scattering by a corner reflector" appear to have been "diffraction by a wedge". There's been quite a bit of research on exactly this kind of boundary. Particularly, ref. 1 provides explicit expressions for the solution of the problem that's set up as follows.

A two-dimensional wedge is situated in the space $$0\le r<\infty$$, $$\alpha\le \theta\le 2\pi$$, with $$(r,\theta)$$ being cylindrical coordinates (omitting $$z$$), as shown in the figure below (figure 1 from ref. 1). A plane wave is incident from the direction $$\theta=\theta_0$$ and scattered by the inner part of the wedge, which may be of any opening angle from $$0$$ to at least $$4\pi$$ (the case of hard-soft semi-infinite plane, see ref. 1 p. 277).

The solution of the wave equation, i.e. incident+scattered field, corresponding to $$\alpha=120°$$ is given in $$\S 4$$ as $$G_{2\pi/q}(r,\theta,\theta_0;k)$$ by equation $$(23)$$; in our case $$q=3$$. The expression (tweaked here for presentation) is

$$\begin{multline} G_{2\pi/q}(r,\theta,\theta_0;k)=\sum_{m=0}^{q-1}\sum_{N\in\mathbb Z} \operatorname{H}[\pi-|\Delta\theta_m+4\pi N|]e^{ikr\cos(\Delta\theta_m)}+\\ +\frac{e^{i\pi/4}}{\sqrt\pi}\sum_{m=0}^{q-1} e^{ikr\cos(\Delta\theta_m)} \operatorname{sgn}\left[ \cos\left(\Delta\theta_m/2\right)\right] \intop_{+\infty}^{\left| \cos\left(\Delta\theta_m/2\right) \right| \sqrt{2kr}} e^{-iv^2}\,\mathrm{d}v, \end{multline}$$

where $$\Delta\theta_m=\theta-\theta_0+4\pi m/q,$$ $$\operatorname{H}[\cdot]$$ is the Heaviside step function, and $$\operatorname{sgn}(\cdot)$$ is the signum function. The summation over $$N$$ will pick only a handful of terms, because for others the Heaviside function will vanish. The integral in the second sum is expressible in terms of the Fresnel sine and cosine integrals:

$$\int\limits_{+\infty}^x \exp(-iv^2)\,\mathrm{d}v= \frac{1-i}2 \sqrt{\frac\pi2} \left( (1+i)\operatorname{C}\left(\sqrt{\frac2\pi}x\right) + (1-i)\operatorname{S}\left(\sqrt{\frac2\pi}x\right)-1\right).$$

The cylindrical wave asked about in the OP originates from the second sum with these Fresnel-integral terms.

### References

1. A. D. Rawlins, Plane-wave diffraction by a rational wedge, Proc. R. Soc. A 411, 265-283 (1987).