A Mathieu function boundary spectral method for scattering by multiple variable poro-elastic plates, with applications to metamaterials and acoustics

Many problems in fluid mechanics and acoustics can be modelled by Helmholtz scattering off poro-elastic plates. We develop a boundary spectral method, based on collocation of local Mathieu function expansions, for Helmholtz scattering off multiple variable poro-elastic plates in two dimensions. Such boundary conditions, namely the varying physical parameters and coupled thin-plate equation, present a considerable challenge to current methods. The new method is fast, accurate and flexible, with the ability to compute expansions in thousands (and even tens of thousands) of Mathieu functions, thus making it a favourable method for the considered geometries. Comparisons are made with elastic boundary element methods, where the new method is found to be faster and more accurate. Our solution representation directly provides a sine series approximation of the far-field directivity and can be evaluated near or on the scatterers, meaning that the near field can be computed stably and efficiently. The new method also allows us to examine the effects of varying stiffness along a plate, which is poorly studied due to limitations of other available techniques. We show that a power-law decrease to zero in stiffness parameters gives rise to unexpected scattering and aeroacoustic effects similar to an acoustic black hole metamaterial.

For completeness, as a reference for the reader, and to demonstrate the ease of adopting other types of boundary conditions, we discuss the case of rigid porous plates.
The porous plate impedance boundary condition is given by (0.1) where µ = α H K R /(πR 2 ) is the porosity parameter for a porous plate with evenly-spaced circular apertures of radius R, Rayleigh conductivity of K R = 2R, and fractional open area α H = N πR 2 (where N is the number of apertures per unit area) [4]. As in the main article, φ(x, 0+) and φ(x, 0−) denote the values of the field just above and just below the plate respectively and the jump in φ across the plate is denoted by [φ](x). We also allow the porosity parameter, µ(x), to vary across the plate. The boundary condition (0.1) replaces the thinplate equation and the kinematic condition for the elastic plates considered in the main text.
The solution method is exactly the same, but now we use collocation to solve the boundary condition (0.1). We truncate the Mathieu function expansion to M terms and collocate at points x to obtain the relation For collocation points, we choose which correspond to (rescaled) Chebyshev points in Cartesian coordinates and equally spaced points in elliptic coordinates [1,7]. This gives rise to an M × M linear system. As in the main text, we rescale to ensure that each row of the resulting matrix has a constant l 1 vector norm. The method can also be extended to multiple plates with a mixture of different boundary conditions.

Details for separation of variables for a single plate
We introduce elliptic coordinates via x = d cosh(ν) cos(τ ), y = d sinh(ν) sin(τ ), where, with an abuse of notation, we write functions of (x, y) also as functions of (ν, τ ). The appropriate domain then becomes ν ≥ 0 and τ ∈ [0, π]. The appropriate domain then becomes ν ≥ 0 and τ ∈ [0, π], and the PDE system (now with a homogeneous Dirichlet boundary condition along {(x, y) : To simplify the formulae, we let Q = d 2 k 2 0 /4. Separation of variables for solutions of the form V This Fourier series converges absolutely and uniformly on all compact sets of the complex plane [6]. The eigenfunctions are real and orthogonal, and we choose the normalisation π 0 sem(τ )sen(τ )dτ = π 2 δmn.
We find the coefficients B (m) l via a simple Galerkin method. Namely, we split the eigenfunctions further by symmetry or antisymmetry about τ = π/2 and write sin((2l + 1)τ ).
For the even order solutions, the eigenvalue problem becomes the tridiagonal system These are solved using square n × n truncations of the infinite matrix (also known as the finite section method or Galerkin method). Since the spectrum of the associated (self-adjoint) linear operator is discrete, we do not have to worry about issues such as spectral pollution [3]. The convergence to the eigenvalues and eigenfunctions depends on the parameter Q, in general being slower for larger Q. However, the convergence is exponential, yielding machine precision for small truncation parameter n, even for very large Q [2]. The corresponding V (ν) with the appropriate radiation condition at infinity are given by the Mathieu-Hankel functions Hsem(Q; ν) = Hsem(ν) = Jsem(ν) + iYsem(ν), which can be expanded in a series using Bessel functions as in the main text.