The rainbow reflection effect describes the broadband spatial separation of wave spectral components caused by a spatially graded array of resonators. Although mainly studied in optics and acoustics, this phenomenon has recently been demonstrated both theoretically and experimentally for water waves travelling through an array of vertical cylinders. The scattering of two-dimensional linear water waves by an array of vertical, surface-piercing barriers is considered here, in which both the submergence and spacing between the barriers are spatially graded. The rainbow reflection effect arises naturally as wave energy temporarily becomes amplified at different locations depending on frequency. Band diagram calculations are used to demonstrate that this is a consequence of the wave gradually slowing down throughout the array. The wave/barriers scattering problem is then augmented by positioning heave-restricted, rectangular floating bodies equipped with a linear damping mechanism between each adjacent pair of barriers. A solution to the resulting boundary value problem is obtained using an integral equation/Galerkin method. Using constrained optimisation, passive rainbow absorbers are designed that achieve near-perfect absorption (i) over a discrete set of frequencies and (ii) over an octave. This suggests potential applications of rainbow absorbers in the design of smart coastal technologies.

Time-domain wave scattering in an unbounded two-dimensional acoustic medium by sound-hard scatterers is considered. Two canonical geometries, namely a split-ring resonator (SRR) and an array of cylinders, are used to highlight the theory, which generalizes to arbitrary scatterer geometries. The problem is solved using the generalized eigenfunction expansion method (GEM), which expresses the time-domain solution in terms of the frequency-domain solutions. A discrete GEM is proposed to numerically approximate the time-domain solution. It relies on quadrature approximations of continuous integrals and can be thought of as a generalization of the discrete Fourier transform. The solution then takes a simple form in terms of direct matrix multiplications. In parallel to the GEM, the singularity expansion method (SEM) is also presented and applied to the two aforementioned geometries. It expands the time-domain solution over a discrete set of unforced, complex resonant modes of the scatterer. Although the coefficients of this expansion are divergent integrals, we introduce a method of regularizing them using analytic continuation. The results show that while the SEM is usually inaccurate at t = 0 , it converges rapidly to the GEM solution at all spatial points in the computational domain, with the most rapid convergence occurring inside the resonant cavity.

Wave interaction with graded metamaterials exhibits the phenomenon of rainbow reflection, in which broadband wave signals slow down and separate into their frequency components before being reflected. This phenomenon has been qualitatively understood by describing the wave field in the metamaterial using the local Bloch wave approximation (LBWA), which locally represents the wave field as a superposition of propagating wave solutions in the cognate infinite periodic media (so-called Bloch waves). We evaluate the performance of the LBWA quantitatively in the context of two-dimensional linear water-wave scattering by graded arrays of surface-piercing vertical barriers. To do this, we implement the LBWA numerically so that the Bloch waves in one region of the graded array are coupled to Bloch waves in adjacent regions. This coupling is computed by solving the scattering of Bloch waves across the interface between two semi-infinite arrays of vertical barriers, where the barriers in each semi-infinite array can have different submergence depths. Our results suggest that the LBWA accurately predicts the free surface amplitude across a wide range of frequencies, except those just above the cutoff frequencies associated with each of the vertical barriers in the array. This highlights the importance of decaying Bloch modes above the cutoff in rainbow reflection.

The convergence of two numerical methods for solving linear water wave scattering problems, namely the eigenfunction matching method (EMM) and the singularity-respecting Galerkin approximation (SRGA), is examined. To do so, the methods are applied to two simple problems, namely scattering by a partially submerged vertical barrier and scattering in a parallel walled channel with a step change in width. These problems contain corner singularities in the velocity potential of order $-1/2$ and $-1/3$, respectively, which the SRGA accounts for but EMMs do not. The results presented to compare the methods show that SRGA solutions are consistently more accurate than EMM solutions for the same amount of computing time. The results also show that the EMM solution for the channel problem is more accurate than the EMM solution for the vertical barrier problem due to the weaker singularity. Nevertheless, the EMM for the barrier is shown to still converge beyond three decimal places if a sufficiently large matrix is used—slower computation may be a worthwhile trade-off in certain situations because the EMM is usually considered to be more straightforward to implement. Our results serve as a practical guide for researchers selecting between the numerical methods.