Diffraction of acoustic waves by multiple semi-infinite arrays

TL;DR

This work extends the discrete Wiener–Hopf framework to diffraction by an arbitrary set of $oldsymbol{J}$ semi-infinite Dirichlet-cylinder arrays, each with independent geometry and orientation. By solving independent WH problems for each array and coupling them through a full matrix inversion, the authors obtain the scattering coefficients $igrace A^{(j)}_migrace$ without iteration, enabling thousands of scatterers to be included. The methodology yields a block-mentered matrix equation with kernels $K_j(z)$ that are factorized as $K_j(z)=K_j^+(z)K_j^-(z)$ and then solved via inverse Z-transform, producing accurate results that agree with alternative methods in test cases. The paper also discusses numerical optimization (FMM), uniqueness considerations, and the comparative strengths of WH versus LS/ＴMAT approaches, highlighting potential hybrid schemes and future directions in resonance, Bloch waves, and acoustic shielding applications.

Abstract

Analytical methods are fundamental in studying acoustics problems. One of the important tools is the Wiener-Hopf method, which can be used to solve many canonical problems with sharp transitions in boundary conditions on a plane/plate. However, there are some strict limitations to its use, usually the boundary conditions need to be imposed on parallel lines (after a suitable mapping). Such mappings exist for wedges with continuous boundaries, but for discrete boundaries, they have not yet been constructed. In our previous article, we have overcome this limitation and studied the diffraction of acoustic waves by a wedge consisting of point scatterers. Here, the problem is generalised to an arbitrary number of periodic semi-infinite arrays with arbitrary orientations. This is done by constructing several coupled systems of equations (one for every semi-infinite array) which are treated independently. The derived systems of equations are solved using the discrete Wiener-Hopf technique and the resulting matrix equation is inverted using elementary matrix arithmetic. Of course, numerically this matrix needs to be truncated, but we are able to do so such that thousands of scatterers on every array are included in the numerical results. Comparisons with other numerical methods are considered, and their strengths/weaknesses are highlighted.

Figures (6)

• Figure 1: Diagram of a plane wave interacting with multiple arbitrary semi-infinite arrays. For simplicity here, the first array is positioned on the positive $x$-axis (i.e. $R^{(1)}_0=\alpha_1=0$).
• Figure 2: Diagram of the integration contour $C$ on the $\xi$ complex plane. Here, the border of the regions $\Omega_j^\pm$ are shown as blue and red circles respectively and the grey dashed circle is the unit circle $|\xi|=1$. The smaller diagram is the limiting case when the imaginary part of $k$ tends to zero.
• Figure 3: Plot of the absolute value of $\det(\mathcal{M}^{(j,\ell)})$ w.r.t. the truncation $N$, compared with $\lambda_{j,0}^{N+1}$. For all cases, we have $k=5\pi$, $s_j=0.1$ and $a_j=0.001$ for all $j$ so the value of $\lambda_{j,0}$ is unchanged.
• Figure 4: Plots of the computation times to calculate the matrix in \ref{['MSIA-matrixmatrix-system']} for the point scatterer wedge given in FIG. \ref{['fig:test-cases']}. On the left (resp. right) side, these times are plotted w.r.t. the truncation $N$ (resp. $ks$).
• Figure 5: Real part of total field for six different test cases. Here, the incident wave is given by the parameters $k=5\pi$ and $\theta_{\textrm{I}}=\frac{\pi}{4}$ (except for (f) plot which has $k=7.5\pi$), and the array parameters are given in Table \ref{['table:test-cases']}.