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

TL;DR

This work addresses accurate 2D acoustic scattering from a singly-periodic corrugated boundary, focusing on surface-trapped modes that influence far-field behavior. It introduces a high-order boundary integral equation framework paired with an array scanning (Floquet–Bloch) transform and contour deformation to resolve trapped modes and Wood anomalies efficiently. Key contributions include efficient evaluation of the quasiperiodic Green's function via lattice sums, corner-refined Nyström discretization, a Fredholm-determinant method to locate trapped modes, and a residue-based approach to quantify power in left- and right-going surface waves, plus a dispersion-informed ray model for the observed raindrop chirp on stairs. The results illuminate how trapped surface waves govern energy transport along a staircase and connect dispersion to time-domain chirp phenomena, with clear paths to 3D extension and optimization.

Abstract

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).

Figures (11)

• Figure 1: Left: Photograph of El Castillo at Chichen Itza, Mexico. Center: 2D infinite staircase geometry with coordinates used. Right: a single period $\Gamma$ of the boundary, now shown with $x_1$ horizontal (the orientation used throughout).
• Figure 2: Location of Wood anomalies (black lines) and the first Brillouin zone (red shaded region of the $\kappa$ axis) in the $\omega$-$\kappa$ plane, for the case of spatial periodicity $d=1$.
• Figure 3: Discretization nodes on a single unit cell $\Gamma$ of the boundary. The underlying coarse discretization has $8$ equal panels on each straight line. Panels touching corners have then been subdivided dyadically $10$ times, with shrinkage ratio $r = 2$. Each resulting panel was populated with $16$ Gauss--Legendre quadrature nodes. An inset shows the result of the refinement.
• Figure 4: Convergence test via flux conservation for the case of an incident plane wave. The figure shows the net flux exiting the central unit cell, which is analytically zero, as a function of both the number of times the corner-adjacent quadrature panel has been subdivided on the boundary, and the size of the resulting linear system. During refinement the panels have been split in a $1:2$ ratio ($r = 3$).
• Figure 5: Left: numerically computed band structure (dispersion relation) for evanescent trapped modes of the $\pi/4$-slope staircase with period $d=1$. Only the right (positive) half of the Brillouin zone is shown. Dots show the band structure $\omega_\mathrm{tr}(\kappa)$, and the line shows the values $\omega=\kappa$. Radiation into the upper half plane is only possible when $\omega>|\kappa|$. Right: group velocity $v_g := d\omega_\mathrm{tr}(\kappa)/d\kappa$ plotted over the same domain.

Theorems & Definitions (6)

• Remark 1
• Remark 2
• Remark 3
• Theorem 4
• proof
• Remark 5