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We present a high-order boundary integral equation (BIE) method for thefrequency-domain acoustic scattering of a point source by a singly-periodic,infinite, corrugated boundary. We apply it to the accurate numerical study ofacoustic radiation in the neighborhood of a sound-hard two-dimensionalstaircase modeled after the El Castillo pyramid. Such staircases supporttrapped waves which travel along the surface and decay exponentially away fromit. We use the array scanning method (Floquet--Bloch transform) to recover thescattered field as an integral over the family of quasiperiodic solutionsparameterized by their on-surface wavenumber. Each such BIE solution requiresthe quasiperiodic Green's function, which we evaluate using an efficientintegral representation of lattice sum coefficients. We avoid the singularitiesand branch cuts present in the array scanning integral by complex contourdeformation. For each frequency, this enables a solution accurate to around 10digits in a couple of seconds. We propose a residue method to extract thelimiting powers carried by trapped modes far from the source. Finally, bycomputing the trapped mode dispersion relation, we use a simple ray model toexplain an observed acoustic "raindrop" effect (chirp-like time-domainresponse).

Trapped acoustic waves and raindrops: high-order accurate integral equation method for localized excitation of a periodic staircase