A High-Accuracy Mode Solver for Acoustic Scattering by a Periodic Array of Axially Symmetric Obstacles

This paper is concerned with guided modes of an acoustic wave propagation problem on a periodic array of axially symmetric obstacles. A guided mode refers to a quasi-periodic eigenfield that propagates along the obstacles but decays exponentially away from them in the absence of incidences. Thus, the problem can be studied in an unbound unit cell due to the quasi-periodicity. We truncate the unit cell onto a cylinder enclosing the interior obstacle in terms of utilizing Rayleigh’s expansion to design an exact condition on the lateral boundary. We derive a new boundary integral equation (BIE) only involving the free-space Green function on the boundary of each homogeneous region within the cylinder. Due to the axial symmetry of the boundaries, each BIE is decoupled via the Fourier transform to curve BIEs and they are discretized with high-accuracy quadratures. With the lateral boundary condition and the side quasi-periodic condition, the discretized BIEs lead to a homogeneous linear system governing the propagation constant of a guided mode at a given frequency. The propagation constant is determined by enforcing that the coefficient matrix is singular. The accuracy of the proposed method is demonstrated by a number of examples even when the obstacles have sharp edges or corners.