Interaction Of Linear Waves Research Articles

Joint Geometry/Frequency Analyticity of Fields Scattered by Periodic Layered Media

The scattering of linear waves by periodic structures is a crucial phenomena in many branches of applied physics and engineering. In this paper we establish rigorous analytic results necessary for the proper numerical analysis of a class of high-order perturbation of surfaces/asymptotic waveform evaluation (HOPS/AWE) methods for numerically simulating scattering returns from periodic diffraction gratings. More specifically, we prove a theorem on existence and uniqueness of solutions to a system of partial differential equations which model the interaction of linear waves with a periodic two-layer structure. Furthermore, we establish joint analyticity of these solutions with respect to both geometry and frequency perturbations. This result provides hypotheses under which a rigorous numerical analysis could be conducted on our recently developed HOPS/AWE algorithm.

Analytical study on an oscillating buoy integrated with a perforated structure with a quadratic pressure drop condition: A case study on a WEC-perforated breakwater integration system

A general analytical method for solving the linear wave interactions with the integration systems comprising an oscillating buoy and perforated structure is presented, with a quadratic pressure drop condition on the perforated surface. Using the method of variable separation and eigenfunction expansion, the motion response of the buoy and total velocity potential can be solved using a two-loop iterative method. The method is validated against the published results. The integration system with a heaving WEC arranged inside a perforated breakwater was investigated particularly. Under the quadratic pressure drop condition, the symmetry of the hydrodynamic coefficients and Haskind relation, are found no longer satisfied, and the dissipation through the perforated wall also contribute to the damping coefficient. Effects of the damping of the power take-off system, front wall porosity, wave amplitude, and position of the WEC are investigated in particular. It was found that the horizontal and vertical wave forces on the WEC increase significantly as wave amplitude increases; however, the hydrodynamic efficiency decreases. When the WEC is near the solid rear wall, the horizontal wave forces on the WEC and solid rear wall increase significantly at the frequencies of piston mode gap resonance and heave motion resonance.

Inter-mode soliton linear-wave scattering in a Kerr microresonator.

Soliton microresonator frequency combs (microcombs) have recently emerged as an attractive new type of optical comb source with a wide range applications proposed and demonstrated. To extend the optical bandwidth of these microresonator sources, several previous studies have proposed and studied the injection of an additional optical probe wave into the resonator. In this case, nonlinear scattering between the injected probe and the original soliton enables the formation of new comb frequencies through a phase-matched cascade of four-wave mixing processes. In this work, we expand the relevant analyses to consider soliton-linear wave interactions when the soliton and the probe fields propagate in different mode families. We obtain an expression for the phase-matched idler locations as a function of the dispersion of the resonator and the phase detuning of the injected probe. We confirm our theoretical predictions in experiments performed in a silica waveguide ring microresonator.

Array of concentric perforated cylindrical systems with torus oscillating bodies integrated on inner cylinders

In this study, a theoretical method was proposed to investigate the linear wave interactions with a square array of hybrid wave energy extraction (HWEE) systems, each constructed by integrating a torus oscillating buoy on a bottom-mounted concentric perforated cylindrical system. The method of variable separation and eigenfunction expansion matching were employed to solve the diffraction and radiation problems. The model was validated against published results. First, the effects of the incident wave angle and distance between two HWEE systems in one column or row on the hydrodynamic performances of the HWEE system array, were investigated without considering the outer cylinders. Subsequently, the effects of arranging arc-shaped perforated cylinders with different porosities, radii, and covering angles around the HWEE system array were investigated. The wave energy extraction performance of the HWEE system array and horizontal wave forces on the inner cylinders were determined as primary concerns. It was found that when incident waves propagate along the side and diagonal of the square array, there exist frequency regions wherein the wave energy extraction efficiency and the wave forces on the inner cylinders were both positively affected by the interactions among the HWEE system array. When the arc-shaped outer cylinders were impermeable, wave energy extraction was significantly improved.

3D hydroelastic modelling of fluid–structure interactions of porous flexible structures

A 3D hydroelastic model subjected to linear wave interaction with horizontal flexible floating and submerged porous structures is developed. The dispersion relation of the aforesaid problem is derived based on Green’s functions approach in water of finite and infinite depths. Three-dimensional Fourier-type expansion formulae for the velocity potentials associated with orthogonal mode-coupling relations are derived in a channel of finite and infinite water depths of finite width. A real physical problem of surface gravity wave interaction with a moored floating and submerged porous plate in finite water depth is analysed by the expansion formula to validate the usefulness of the current method. The numerical accuracy of the present numerical results is checked via convergence test. Further, the effect of different design parameters on the hydroelastic response of floating and a submerged porous plate is studied by analysing the deflection amplitude and structural deformation. It is observed that the present analysis could be useful for a greater insight to design a multi-use flexible floating system.

Hydroelastic behaviour of a submerged horizontal flexible porous structure in three-dimensions

A hydroelastic model is developed for the problem of linear wave interaction with a submerged horizontal flexible porous structure in three-dimensions. The generalized expansion formulae with orthogonal mode-coupling relations are derived by expanding Fourier cosine series in a channel width. Using eigensolutions, various aspects of hydroelastic wave behaviour in a channel with a porous plate submerged in finite water depth is investigated. Further, the usefulness of the expansion is analysed through a real physical problem of wave diffraction by a submerged flexible porous plate connected with mooting lines in three-dimensions. The accurateness of the three-dimensional solution is inspected and the hydroelastic behaviour of the submerged porous structure is analysed for oscillation modes, mooring stiffness, and porous-effect parameters on numerical assessment of reflection, dissipation coefficients, and plate displacements in different cases. The analysis of current hydroelastic behaviour will be beneficial for future design of a device for wave energy absorption in three-dimensions.

The Performance of a Spectral Wave Model at Predicting Wave Farm Impacts

For renewable ocean wave energy to support global energy demands, wave energy converters (WECs) will likely be deployed in large numbers (farms), which will necessarily change the nearshore environment. Wave farm induced changes can be both helpful (e.g., beneficial habitat and coastal protection) and potentially harmful (e.g., degraded habitat, recreational, and commercial use) to existing users of the coastal environment. It is essential to estimate this impact through modeling prior to the development of a farm, and to that end, many researchers have used spectral wave models, such as Simulating WAves Nearshore (SWAN), to assess wave farm impacts. However, the validity of the approaches used within SWAN have not been thoroughly verified or validated. Herein, a version of SWAN, called Sandia National Laboratories (SNL)-SWAN, which has a specialized WEC implementation, is verified by comparing its wave field outputs to those of linear wave interaction theory (LWIT), where LWIT is theoretically more appropriate for modeling wave-body interactions and wave field effects. The focus is on medium-sized arrays of 27 WECs, wave periods, and directional spreading representative of likely conditions, as well as the impact on the nearshore. A quantitative metric, the Mean Squared Skill Score, is used. Results show that the performance of SNL-SWAN as compared to LWIT is “Good” to “Excellent”.

A high-order perturbation of envelopes (HOPE) method for scattering by periodic inhomogeneous media

The interaction of linear waves with periodic structures arises in a broad range of scientific and engineering applications. For such problems it is often mandatory that numerical simulations be rapid, robust, and highly accurate. With such qualities in mind High-Order Spectral methods are often utilized, and in this paper we describe and test a perturbative method which fits into this class. Here we view the inhomogeneous (but laterally periodic) permittivity as a perturbation of a constant value and pursue (regular) perturbation theory. We demonstrate that not only does this lead to a fast and accurate numerical method, but also that the expansion of the field in this geometric parameter is valid for large deformations (up to topological obstruction). Finally, we show that, if the permittivity deformation is spatially analytic, then so is the field scattered by it.

Peculiarities of linear wave interaction with nonlinear media interface

We analyze guided waves in the linear media separated nonlinear interface. The mathematical formulation of the model is a one-dimensional boundary value problem for the nonlinear Schrödinger equation. The Kerr type nonlinearity in the equation is taken into account only inside the waveguide. We show that the existence of nonlinear stationary waves of three types is possible in defined frequency ranges. We derive the frequency of obtained stationary states in explicit form and find the conditions of its existence. We show that it is possible to obtain the total wave transition through a plane defect. We determine the condition for realizing of such a resonance. We obtain the reflection and transition coefficients in the vicinity of the resonance. We establish that complete wave propagation with nonzero defect parameters can occur only when the nonlinear properties of the defect are taken into account.

On analyticity of linear waves scattered by a layered medium

The scattering of linear waves by periodic structures is a crucial phenomena in many branches of applied physics and engineering. In this paper we establish rigorous analytic results necessary for the proper numerical analysis of a class of High-Order Perturbation of Surfaces methods for simulating such waves. More specifically, we prove a theorem on existence and uniqueness of solutions to a system of partial differential equations which model the interaction of linear waves with a multiply layered periodic structure in three dimensions. This result provides hypotheses under which a rigorous numerical analysis could be conducted for recent generalizations to the methods of Operator Expansions, Field Expansions, and Transformed Field Expansions.

Linear Wave Interaction with a Vertical Cylinder of Arbitrary Cross Section: An Asymptotic Approach

AbstractAn asymptotic approach to the linear problem of regular water waves interacting with a vertical cylinder of an arbitrary cross section is presented. The incident regular wave was one-dimens...

Nonlinear dynamics induced by linear wave interactions in multilayered flows

Using simple kinematics, we propose a general theory of linear wave interactions between the interfacial waves of a two-dimensional (2D), inviscid, multilayered fluid system. The strength of our formalism is that one does not have to specify the physics of the waves in advance. Wave interactions may lead to instabilities, which may or may not be of the familiar ‘normal-mode’ type. Contrary to intuition, the underlying dynamical system describing linear wave interactions is found to be nonlinear. Specifically, a saw-tooth jet profile with three interfaces possessing kinematic and geometric symmetry is explored. Fixed points of the system for different ranges of a Froude number like control parameter $\unicode[STIX]{x1D6FE}$ are derived, and their stability evaluated. Depending upon the initial condition and $\unicode[STIX]{x1D6FE}$, the dynamical system may reveal transient growth, weakly positive Lyapunov exponents, as well as different nonlinear phenomena such as the formation of periodic and pseudo-periodic orbits. All these occur for those ranges of $\unicode[STIX]{x1D6FE}$ where normal-mode theory predicts neutral stability. Such rich nonlinear phenomena are not observed in a 2D dynamical system resulting from the two-wave problem, which reveals only stable and unstable nodes.

On M2 tidal amplitude enhancement in the Taiwan Strait and its asymmetry in the cross-strait direction

Enhanced M2 tidal amplitude in the Taiwan Strait (TS) and asymmetric M2 tidal amplitude in the cross-strait direction have been found and reproduced in numerical simulations. In this study, Finite Volume Coastal Ocean Model (FVCOM) is applied to investigate the mechanisms behind these features. Model results show that the linear interaction of waves from the East China Sea (ECS) and the Luzon Strait (LS) can explain the formation of the co-amplitude and co-phase lines of the M2 tide in the nodal point area, while the waves from the ECS dominate the tidal motion in the TS according to a basic linear wave superposition. Model simulation also show that wave reflection and transition occur when the M2 tidal waves from the ECS propagate through the TS and encounter an sharply deepened topography. The interaction of these induced reflection waves and the incident waves from the ECS is the main cause for the enhanced M2 tidal amplitude in the TS. The distribution of the sharply deepened topography, rather than the Coriolis effect, is the main reason for the asymmetry of the M2 tidal amplitude in the cross-strait direction in the TS. These findings provide some references for tidal dynamics in other areas, especially where long waves propagate through the shallow water to the deep sea.

Effect of flux surface curvature on the linear coupling of electron cyclotron waves in tokamak plasmas

Specific features of the linear interaction of ordinary and extraordinary electromagnetic waves in the electron cyclotron frequency range in a nonuniform plasma confined in a toroidal magnetic trap are considered. Reduced wave equations taking into account the curvature of the cut-off surfaces in toroidal geometry are formulated. Using these equations, the distributions of the wave fields in the coupling region are analyzed. A method for calculating quasi-optical beams passed through the region of linear wave interaction is proposed.

Design optimization of floating breakwaters with an interdisciplinary fluid–solid structural problem

The design optimization of floating breakwaters implicates solving an interdisciplinary problem consisting of three models. The first one arises from the interaction of linear waves with a moored floating breakwater with a leeward boundary that is composed of a vertical sidewall representing the quay wall in ports. The second covers the dynamical behaviour of the oscillating structure caused by the incoming waves. These two assemble the hydrodynamic performance of the floating breakwater; while the third concerns its structural mechanics subject to hydrostatic and hydrodynamic forces. The goal of the optimization problem is to design an optimal floating breakwater that can attenuate the waves to the minimum height inside the port and fulfill several constraints related to floating, stability, and structural resistance. The objective function and constraints are expressed in terms of geometrical parameters of the breakwater as mathematical expressions assembled in an optimization algorithm based on the sequential quadratic programming method (SQP). This yields to several optimal structures each corresponding to a specified wave period. Finally, an analysis is performed to determine an optimum structure for a wide range of frequencies.

A coupling method between extended Boussinesq equations and the integral equation method with application to a two-dimensional numerical wave-tank

A coupling method is proposed between extended Boussinesq equations, as formulated by Jamois et al. [2006. A numerical study of nonlinear wave run-up on a vertical plate, Coastal Engineering, 53, 929–945], and the integral equation method, under the assumption of potential flow. In both domains, the same fully nonlinear free surface equations, in Zakharov form, are being used. Implementation is carried out in the case of a two-dimensional numerical wavetank. Application cases concern nonlinear propagation of a steep regular wave, linear wave interaction with a rectangular caisson at the free surface, and wave transformation over varying bathymetry. They demonstrate the robustness and efficiency of the proposed method.

Parametrical and Motion Analysis of a Moored Rectangular Floating Breakwater

Moored floating breakwaters with a leeward boundary, assimilating the port quay walls are described by a large number of coupled variables. This complicates their design and requires a detailed parametrical and motion analysis to assess their hydrodynamic performance. A diffraction-radiation boundary value problem is developed. It arises from the interaction of linear waves on a moored floating breakwater with a leeward boundary described by a partial reflective sidewall. The effects of the sidewall clearance, structural parameters, mooring lines stiffness, and their angle of inclination on the transmission coefficient and the dynamic motion of the floating breakwater are considered. The transmission coefficient is strongly affected by the motion itself and the allowable length change in the mooring lines.

Polarization of Microwave Radio Emission of Flare-Producing Solar Active Regions

On the basis of our multiwavelength observations made with the one-dimensional RATAN-600 radio telescope, we study the inversion of the circular polarization in the solar microwave emission at different frequencies. The inversion is detected in the emission of flare-producing active regions (FPARs) at various stages of their development, starting from the pre-flare stage. During the latest 23rd solar cycle maximum, numerous FPARs revealed spectral inhomogeneities in their polarized microwave radiation (Bogod and Tokhchukova, 2003, Astron. Lett.29, 263). Here, we discuss a particular case of such inhomogeneities, the frequency-dependent double inversion of the sign of circular polarization, which probably reflects some essential processes in FPARs. We consider several mechanisms for the double inversion: linear interaction of waves in the region of a quasitransverse magnetic field, the propagation of waves through a region of zero magnetic field, the scattering of radio waves on waves of high-frequency plasma turbulence, the influence of the current fibrils on the propagation of the radio emission, and the magnetic “dips,” in which the direction of magnetic field lines changes the sign relative to the observer. All of them have shortcomings, but the last mechanism explains the observations the best.

Seismic Halos around Active Regions: A Magnetohydrodynamic Theory

Comprehending the manner in which magnetic fields affect propagating waves is a first step toward constructing accurate helioseismic models of active region subsurface structure and dynamics. Here we present a numerical method for computing the linear interaction of waves with magnetic fields embedded in a solar-like stratified background. The ideal magnetohydrodynamic (MHD) equations are solved in a three-dimensional box that straddles the solar surface, extending from 35 Mm below the photosphere to 1.2 Mm into the atmosphere. One of the challenges in performing these simulations involves generating a magnetohydrostatic (MHS) state wherein the stratification assumes horizontal inhomogeneity in addition to the strong vertical stratification associated with the near-surface layers. Keeping in mind that the aim of this effort is to understand and characterize linear MHD interactions, we discuss a means of computing statically consistent background states. Power maps computed from simulations of waves interacting with thick flux tubes that have peak photospheric field strengths of 600 and 3000 G are presented. Strong modal power reduction in the umbral regions of the flux tube enveloped by a halo of increased wave power is seen in the simulations with the thick flux tubes. These enhancements are also seen in Doppler velocity power maps of active regions observed in the Sun, which leads us to propose that the halo has MHD underpinnings.

Interaction of linear waves with an array of infinitely long horizontal circular cylinders in a two-layer fluid of infinite depth

The linear water wave scattering and radiation by an array of infinitely long horizontal circular cylinders in a two-layer fluid of infinite depth is investigated by use of the multipole expansion method. The diffracted and radiated potentials are expressed as a linear combination of infinite multipoles placed at the centre of each cylinder with unknown coefficients to be determined by the cylinder boundary conditions. Analytical expressions for wave forces, hydrodynamic coefficients, reflection and transmission coefficients and energies are derived. Comparisons are made between the present analytical results and those obtained by the boundary element method, and some examples are presented to illustrate the hydrodynamic behavior of multiple horizontal circular cylinders in a two-layer fluid. It is found that for two submerged circular cylinders the influence of the fluid density ratio on internal-mode wave forces is more appreciable than surface-mode wave forces, and the periodic oscillations of hydrodynamic results occur with the increase of the distance between two cylinders; for four submerged circular cylinders the influence of adding two cylinders on the wave forces of the former cylinders is small in low and high wave frequencies, but the influence is appreciable in intermediate wave frequencies.