Scattering from periodic boundaries

Periodic surfaces have long been used to manipulate wave propagation in applications such as antennae, radars, diffraction gratings, solar cells, sonic and photonic crystals, and in buildings like amphitheatres. Scattering from periodic boundaries, however, is notoriously difficult to compute. Challenges arise from:

  • The infinite extent of the domain (for both exterior and interior problems) and the scattering surface (in 3D) or boundary (in 2D),
  • The fact that the infinite domain cannot be truncated without introducing $\mathcal{O}(1)$ errors – this is precisely because of the wave-guiding property of the surface, leading to power being transported a long way along the surface,
  • If the surface is ragged, corners introduce singularities in the computation.

Here, I focus on acoustic scattering problems in 2D, and the boundary is assumed to extend to infinity in the 3rd idmension. The unknown scalar field $u$ corresponds to acoustic pressure. It is obtained by solving a partial differential equation (PDE) called the Helmholtz equation with Neumann boundary conditions (which arise because I assume the surface is sound-hard), \[\label{eq:helm-pde} - (\Delta + \omega^2)u = \delta_{\mathbf{x}_0}, \quad \text{in } \Omega, \] \[\label{eq:helm-bc} \mathbf{n}\cdot \nabla u = 0, \quad \text{on } \partial\Omega. \]

I use high-order integral equation methods to solve scattering problems by periodic surfaces and tackle the above challenges. This involves rewriting the relevant PDE as a boundary integral, then discretizing the boundary to convert it to a dense linear system, which is then solved. This reduces the dimensionality (and therefore computational complexity) of the problem.

…More coming soon!