Integral equations for flexural scattering problems with periodic boundaries

Abstract

We develop a method for computing the scattering of flexural waves off of a periodic wall or a periodic line of scatterers. These waves model the fluctuations of thin plates with periodic clamped, supported, or free edges. We use the Floquet-Bloch transform to convert the problem into a collection of uncoupled quasi-periodic problems. We then solve each quasi-periodic problem efficiently and accurately using a novel integral equation based on the quasi-periodic flexural Green's function. Finally, we show how the proposed method can be used to simulate scattering from junctions of semi-infinite lines of scatterers.

Figures (7)

• Figure 1: These figures show $\log_{10} |u+u^{\mathrm{in}}|/\max|u^{\mathrm{in}}|$ for $\boldsymbol{x}_0=(0,-2)$ with $k=7,$ and $d=2$.
• Figure 2: The imaginary parts of fields generated by a point source at $(1,1.5)$ with $k=7,$ and $d=2$.
• Figure 3: The dispersion relations for the free plate modes in the region shown in \ref{['fig:free_modes']}. The red x indicates a $\xi$ where the solver failed to find a mode with adaptive refinement tolerance $10^{-4}$. The location was obtained by linear interpolation from the adjacent $k$-s. The ratio of the first and last singular value at that $(\xi,k)$ pair was $1.9\times 10^{-7}$, which is only slightly larger than the comparable to the ratio of $1.3\times 10^{-8}$ at the adjacent pair on the left and indicates the presence of a nearby mode.
• Figure 4: This figure shows the imaginary part of the trapped modes with $\xi=\pi/d$ and $k\in(0.3,\pi/d)$ when free plate boundary conditions are imposed on the shown $\gamma$. The frequencies are $k =$ 0.642700 (top left), 1.011460 (top right), 1.367986 (bottom left), and 1.515168 (bottom right).
• Figure 5: The figure shows the imaginary part of the trapped modes for the Helmholtz equation with Neumann boundary conditions with $\xi = \pi/d$ and $k\in(0.3,\pi/d)$. The frequencies used are $k = 0.533608$ (left), $k=0.9136840$ (right), and $k=1.383154$ (bottom).