Trapped acoustic waves and raindrops: high-order accurate integral equation method for localized excitation of a periodic staircase
arxiv(2023)
摘要
We present a high-order boundary integral equation (BIE) method for the
frequency-domain acoustic scattering of a point source by a singly-periodic,
infinite, corrugated boundary. We apply it to the accurate numerical study of
acoustic radiation in the neighborhood of a sound-hard two-dimensional
staircase modeled after the El Castillo pyramid. Such staircases support
trapped waves which travel along the surface and decay exponentially away from
it. We use the array scanning method (Floquet–Bloch transform) to recover the
scattered field as an integral over the family of quasiperiodic solutions
parameterized by their on-surface wavenumber. Each such BIE solution requires
the quasiperiodic Green's function, which we evaluate using an efficient
integral representation of lattice sum coefficients. We avoid the singularities
and branch cuts present in the array scanning integral by complex contour
deformation. For each frequency, this enables a solution accurate to around 10
digits in a couple of seconds. We propose a residue method to extract the
limiting powers carried by trapped modes far from the source. Finally, by
computing the trapped mode dispersion relation, we use a simple ray model to
explain an observed acoustic "raindrop" effect (chirp-like time-domain
response).
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