Single Incoming Wave Research Articles

Uniqueness in determining rectangular grating profiles with a single incoming wave (part II): TM polarization case

This paper is concerned with an inverse transmission problem for recovering the shape of a penetrable rectangular grating sitting on a perfectly conducting plate. We consider a general transmission problem with the coefficient λ ≠ 1 which covers the transverse magnetic (TM) polarization case. It is proved that a rectangular grating profile can be uniquely determined by the near-field observation data incited by a single plane wave and measured on a line segment above the grating. In comparison with the transverse electric (TE) case (λ = 1), the wave field cannot lie in H 2 around each corner point, bringing essential difficulties in proving uniqueness with one plane wave. Our approach relies on singularity analysis for Helmholtz transmission problems in a right-corner domain and also provides an alternative idea for treating the TE transmission conditions which were considered in the authors’ previous work (Xiang and Hu 2023 Inverse Problems 39 055004).

Uniqueness in determining rectangular grating profiles with a single incoming wave (Part I): TE polarization case

We investigate inverse diffraction problems for penetrable gratings in a piecewise constant medium. In the TE polarization case, it is proved that a rectangular grating profile together with the refractive index beneath it can be uniquely determined by the near-field observation data incited by a single plane wave and measured on a line segment above the grating. Our approach relies on the expansion of solutions to the Helmholtz equation and the corner singularity analysis of solutions to the inhomogeneous Laplace equation with a piecewise continuous source term in a sector. This paper also contributes to corner scattering theory for the Helmholtz equation in a special non-convex domain.

Uniqueness to inverse acoustic scattering from coated polygonal obstacles with a single incoming wave

It is proved that a connected polygonal obstacle coated by thin layers together with its surface impedance function can be determined uniquely from the far field pattern of a single incident plane wave. As a by-product, we prove that the wave field cannot be real-analytic on each corner point lying on the convex hull of the scatterer. Our arguments are based on the Schwarz reflection principle for the Helmholtz equation satisfying the impedance boundary condition on a flat boundary.

Uniqueness and factorization method for inverse elastic scattering with a single incoming wave**Dedicated to the memory of Armin Lechleiter.

The first part of this paper is concerned with the uniqueness to inverse time-harmonic elastic scattering from bounded rigid obstacles in two dimensions. It is proved that a connected polygonal obstacle can be uniquely identified by the far-field pattern corresponding to a single incident plane wave. Our approach is based on a new reflection principle for the first boundary value problem of the Navier equation. In the second part, we propose a revisited factorization method to recover a rigid elastic body with a single far-field pattern.

Unique determination of a penetrable scatterer of rectangular type for inverse Maxwell equations by a single incoming wave

This work is concerned with an inverse electromagnetic scattering problem in two dimensions. We prove that in the TE polarization case, the knowledge of the electric far-field pattern incited by a single incoming wave is sufficient to uniquely determine the shape of a penetrable scatterer of rectangular type. As a by-product, the uniqueness is also confirmed to inverse transmission problems modelled by scalar Helmholtz equations with discontinuous normal derivatives at the scattering interface.

On the Generation of Isolated Green Water Events Using Wet Dam-Break

Green water occurs when an incoming wave exceeds the freeboard and propagates onto the deck of naval/offshore structures, such as floating production storage and offloading units and platforms. This water can affect the integrity of facilities and equipment that are installed on the deck, compromise the safety of the crew, and affect the dynamic stability of the structure. Traditionally, wave trains have been used to study the green water problem, which is a good approach to analyzing consecutive green water events. However, to carry out systematic studies that allow local details to be identified for different types of green water, an alternative method is to study isolated events generated by a single incoming wave. The purpose of this paper was to experimentally investigate the generation of different types of isolated green water events using the wet dam-break (DB) approach as an alternative to generating the incoming wave. Tests were carried out in a rectangular tank with a fixed internal structure. Different freeboard conditions were tested for two aspect ratios of the wet DB (h0/h1=0.40 and 0.6). Conventional wave probes were used to measure the water levels in the tank, and a high-speed camera was set to capture details of the generated green water events. The results demonstrated the ability of this approach to represent different types of green water, similar to those obtained with unbroken regular waves in barge-shaped fixed structures, including DB, plunging-dam-break (PDB) and hammer-fist (HF).

Acoustic Scattering from Corners, Edges and Circular Cones

Consider the time-harmonic acoustic scattering from a bounded penetrable obstacle imbedded in an isotropic homogeneous medium. The obstacle is supposed to possess a circular conic point or an edge point on the boundary in three dimensions and a planar corner point in two dimensions. The opening angles of cones and edges are allowed to be any number in $(0,2\pi)\backslash\{\pi\}$. We prove that such an obstacle scatters any incoming wave non-trivially (i.e., the far field patterns cannot vanish identically), leading to the absence of real non-scattering wavenumbers. Local and global uniqueness results for the inverse problem of recovering the shape of a penetrable scatterers are also obtained using a single incoming wave. Our approach relies on the singularity analysis of the inhomogeneous Laplace equation in a cone.

The Determination of Anisotropic Surface Impedance in Electromagnetic Scattering

We consider the inverse scattering problem of determining the anisotropic surface impedance of a bounded obstacle from far eld measurements of the electromagnetic scattered eld due to incident plane waves. Such an anisotropic boundary condition can arise from surfaces covered with patterns of conducting and insulating patches. We show that the anisotropic impedance is uniquely determined if sucient data is available, and characterize the non-uniqueness present if a single incoming wave is used. We derive an integral equation for the surface impedance in terms of solutions of a certain interior impedance boundary value problem. These solutions can be reconstructed from far eld data using the Herglotz theory underlying the Linear Sampling Method. We complete the paper with preliminary numerical results.

Uniqueness in determining polyhedral sound-hard obstacles with a single incoming wave

We consider the inverse acoustic scattering problem of determining a sound-hard obstacle by far-field measurements. It is proved that a polyhedral scatterer in , consisting of finitely many solid polyhedra, is uniquely determined by a single incoming plane wave.

Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers

This paper addresses the uniqueness for an inverse acoustic obstacle scattering problem. It is proved that a general sound-hard polyhedral scatterer in , possibly consisting of finitely many solid polyhedra and subsets of (N − 1)-dimensional hyperplanes, is uniquely determined by N far-field measurements corresponding to N incident plane waves given by a fixed wave number and N linearly independent incident directions. A simple proof, which is quite different from that in Alessandrini and Rondi (2005 Proc. Am. Math. Soc. 6 1685–91), is also provided for the unique determination of a general sound-soft polyhedral scatterer by a single incoming wave.

Uniqueness in determining polygonal sound-hard obstacles with a single incoming wave

We consider the two-dimensional inverse scattering problem of determining a sound-hard obstacle by the far field pattern. We establish the uniqueness within the class of polygonal domains by a single incoming plane wave.

A Three-Dimensional Numerical Model of Surface Waves in the Surf Zone and Longshore Current Generation over a Plane Beach

A three-dimensional numerical model is developed for the propagation of shallow-water short-period surface waves in the surf zone and longshore current generation over a plane beach topography. This model, which is based on Reynolds-averaged non-linear shallow-water (NSW) equations and, hence, includesimplicitlythe classical radiation stress concept, resolves time- and space-dependence of the sea surface elevation and the velocity fields during one wave cycle (short-wave-resolving). The generation of turbulence by wave breaking and vertical fluid shear above the beach is parameterized by the application of a generalized turbulence energy closure scheme. The instantaneous position of the moving shoreline is determined from the model equations during the simulated propagation process. In the case of a single incoming wave train, the wave amplitude, wave period and angle of incidence are prescribed at an offshore open boundary by application of a forced radiation condition. For uniform alongshore topographic conditions, when cyclic boundary conditions are appropriate at alongshore open boundaries whose positions are determined by the alongshore component of wavelength in an incoming single wave train, the model is used to determine the (mean) longshore current during one wave cycle. It is shown that the maximum longshore depth-averaged current occurs at an approximate offshore position where the generation of turbulence energy through wave breaking is a maximum. It is further shown that the cross-shore gradient of the longshore momentum flux is of predominant importance in generating longshore currents. Experiments are described that determine the dependence of the computed longshore current on the bottom roughness and the length scale prescription in that part of the turbulence closure scheme pertaining to the parameterization of the wave breaking process. The implications of the model results are discussed in the context of the longshore bedload transport of sedimentary material. Finally, a comparison is made between the model predictions and observational data on longshore currents and wave heights.