Random Wave Propagation Research Articles

Stationary random waves propagation in 3D viscoelastic stratified solid

Propagation of stationary random waves in viscoelastic stratified transverse isotropic materials is investigated. The solid was considered multi-layered and located above the bedrock, which was assumed to be much stiffer than the soil, and the power spectrum density of the stationary random excitation was given at the bedrock. The governing differential equations are derived in frequency and wave-number domains and only a set of ordinary differential equations (ODEs) must be solved. The precise integration algorithm of two-point boundary value problem was applied to solve the ODEs. Thereafter, the recently developed pseudo-excitation method for structural random vibration is extended to the solution of the stratified solid responses.

Modelling directional random wave propagation inshore

Long-term nearshore wave climate prediction requires an efficient solution method for the propagation of directional random waves from offshore. Predictions of laboratory data for waves over submerged shoals indicate that waves with broad, but not narrow, directional spread are well predicted without taking diffraction into account, and an efficient parabolic, ‘ray-tracing’ solver may thus be used. Application to the complex coastal bathymetry off East Anglia indicates that inshore wave climate is generally only slightly affected by the degree of directionality, tidal currents, and wind forcing over the propagation region. It thus seems realistic to define inshore wave climate for a range of offshore conditions, defined only by wave height, period and direction, and of course tidal level, using an efficient parabolic solver.

Modelling directional random wave propagation inshore

Long-term nearshore wave climate prediction requires an efficient solution method for the propagation of directional random waves from offshore. Predictions of laboratory data for waves over submerged shoals indicate that waves with broad, but not narrow, directional spread are well predicted without taking diffraction into account, and an efficient parabolic, ‘ray-tracing’ solver may thus be used. Application to the complex coastal bathymetry off East Anglia indicates that inshore wave climate is generally only slightly affected by the degree of directionality, tidal currents, and wind forcing over the propagation region. It thus seems realistic to define inshore wave climate for a range of offshore conditions, defined only by wave height, period and direction, and of course tidal level, using an efficient parabolic solver.

A precise method for solving wave propagation problems in layered anisotropic media

A precise integration algorithm is used in combination with the extended Wittrick–Williams algorithm to solve the differential equations governing stationary random wave propagation in layered anisotropic material. The natural frequencies and corresponding modes of vibration are calculated to arbitrary precision. Four examples are given, two of which highlight a phenomenon corresponding to beats.

Fully dispersive nonlinear water wave model in curvilinear coordinates

A vertically integrated fully dispersive nonlinear wave model is expressed in curvilinear coordinates with non-orthogonal grids for the simulation of broad-banded nonlinear random water waves in regions of arbitrary geometry. The transformation is performed for both dependent and independent variables, hence an irregular physical domain is converted into a rectangular computational domain with contravariant velocities. Use of contravariant velocity components as dependent variables ensures easy and accurate satisfaction of the wall condition for lateral enclosures surrounding a physical domain, such as a coastal area, channel, or harbor. The numerical scheme is based on finite-difference approximations with staggered grids which results in implicit formulations for the momentum equations and a semi-explicit formulation for the continuity equation. Linear long wave propagation in a channel of varying cross-section and linear random wave propagation in a circular channel are presented as test cases for comparisons with the corresponding analytical solutions. Cnoidal and Stokes waves in a circular channel are also simulated as examples to nonlinear wave propagation within curved walls.

Shoaling of subharmonic gravity waves

In this paper the energy budget of wave group‐induced subharmonic gravity waves in the nearshore region is examined on the basis of the energy equation for long waves in conjunction with analyses of a high‐resolution laboratory data set of one‐dimensional random wave propagation over a barred beach. The emphasis is on the growth of forced subharmonics and the deshoaling of the reflected free waves in the shoaling zone. The incident lower‐frequency subharmonics are nearly fully reflected at the shoreline, but the higher‐frequency components appear to be subject to a significant dissipation in a narrow inshore zone including the swash zone. The previously reported phase lag of the incident forced waves behind the short‐wave groups is confirmed, and its key role in the transfer of energy between the grouped short waves and the shoaling bound waves is highlighted. The cross‐shore variation of the local mean rate of this energy transfer is determined. Using this as a source function in the wave energy balance allows a very accurate prediction of the enhancement of the forced waves in the shoaling zone, where dissipation is insignificant. The phase lag appears to increase with increasing frequency, which is reflected in a frequency‐dependent growth rate, varying very nearly from the free‐wave variation ∼ h−1/4 (Green's law) for the lower frequencies to the shallow‐water equilibrium limit for forced subharmonics ∼h−5/2 for the higher frequencies. This observed frequency dependence is tentatively generalized to a dependence on a normalized bed slope, controlling whether a so‐called mild‐slope regime or a steep‐slope regime prevails, in which enhanced incident forced waves dominate over breakpoint‐generated waves or vice versa.

Stochastic theory of waves and localization in two-dimensional random medium

[1] Random wave field satisfying two-dimensional (2D) wave equation with a random medium is studied using the stochastic representation in the polar coordinate system. The random medium is assumed to be a homogeneous and isotropic Gaussian random field, and its polar spectral representation is effectively employed in the method of solution. The isotropy of a random field implies the invariance under stochastic rotational transformation Dϕ, the operator operating on both angular coordinate and probability parameter, and a random wave field with an eigenvalue eiMϕ under Dϕ, M being an integer, represents an angular mode of random cylindrical wave, and M = 0 corresponding to an isotropic mode; the situation being analogous to the case of cylindrical mode in 2D free space. For a concrete approximate solution, the random medium is assumed to be an omnidirectional random grating with an isotropic narrowband spectrum centered at 2k twice as much as the wave number k, which creates the Bragg scattering in the propagation of random cylindrical waves. The asymptotic form of the random cylindrical wave is obtained by means of the stochastic spectral representation of the random wave, as well as the random medium, and is shown to have an exponential factor , where the average exponent coefficient γ is given in terms of the spectral density of random medium. This exponential behavior due to the index α in the amplitude describes the localized nature of an angular mode of random wave field generated by a source or by resonance. In the case of a narrowband grating with the variance σ2 and the spectral bandwidth kΔ, γ is concretely calculated to be proportional to , which should be compared with the one-dimensional (1D) case where the corresponding exponential coefficient is proportional to the spectral height σ2/Δ.

On the FKPP equation with Gaussian shear advection

A boundary value problem for the Fisher–Kolmogorov–Petrovskii–Piskunov (FKPP) equation with shear random advection is investigated. It is demonstrated that the upper bounds for front propagation in the FKPP equation with Gaussian advection obtained by an analysis of its linear ensemble-averaged upper solution can give too rough predictions. It is shown that the unboundedness of the Gaussian advection affects ensemble-averaged solutions of linear and nonlinear problems in a different way. This analytical prediction is confirmed by direct numerical simulations. The phenomenon of propagation failure due to zero boundary conditions is studied and the critical conditions are found. Numerical procedure for the analysis of random wave propagation is suggested by introduction of mean front position and its variance. Some numerical experiments are presented.

Nonlinear Propagation of Broadband Upper-Hybrid Waves in Collisional Magnetoplasmas

The nonlinear coupling between broadband-upper-hybrid (BBUH) waves and electrostatic ion-cyclotron perturbations in a uniform collisional magnetoplasma is considered, taking into account the combined effects of the UH ponderomotive and thermal nonlinear forces. This coupling is governed by a pair of equations which is employed to derive the nonlinear dispersion relation, which admits modulational instabilities of BBUH waves in some interesting cases. The relevance of our investigation to the nonlinear propagation of random phase upper-hybrid waves in a weakly collisional Earth's ionosphere as well as in low-temperature laboratory discharges is pointed out.

Adiabaticity and random wave propagation

We examine the assumption of adiabaticity that underlies mode-coupling analysis of stochastic phenomena such as roughness and inhomogeneity in passive photonics devices such as couplers, gratings, and simple waveguides. We show that although the adiabaticity condition is formally violated by such phenomena, only the low spatial frequencies actually contribute significantly to such phenomena, thus preserving the validity of the analysis.

Stochastic Benjamin-Ono equation and its application to the dynamics of nonlinear random waves.

The stochastic Benjamin-Ono equation is introduced, which models the propagation of nonlinear random waves in a two-layer fluid system with and without uneven bottom topography. In the case of the flat bottom, the effect of the external random flow field on the evolution of both soliton and periodic wave is investigated. In particular, the mean value and the correlation function of these nonlinear wave fields are calculated exactly under the assumption that the flow obeys the Gaussian stochastic process with a white noise. It is found that in the limit of large time, the mean value of an algebraic soliton approaches a Gaussian wave packet whereas that of a periodic wave is represented by Jacobi's theta function. In the case of the uneven bottom, a perturbation analysis is performed to evaluate the mean value of an algebraic soliton under the influence of random change of bottom topography. The large time asymptotic of the soliton is shown to exhibit a Gaussian wave packet with a small amount of the phase shift caused by the interaction between the soliton and the random bottom topography.

A uniform diffusion limit for random wave propagation with turning point

A random wave propagation problem with turning point is considered for a refractive, layered random medium. The variations of the medium structure are assumed to have two spatial scales; microscopic random fluctuations are superposed upon slowly varying macroscopic variations. An extension of a limit theorem for stochastic differential equations with multiple spatial scales is derived and proved to obtain a uniformly valid diffusion limit for random multiple scattering up to the turning point region. The scale dependence of the infinitesimal generator of the backward Kolmogorov equation provides an insight into the interplay of internal refraction and random scattering as one approaches the turning point.

PROPAGATION OF NON-STATIONARY RANDOM WAVES ALONG SUBSTRUCTURAL CHAINS

Propagation of non-stationary random waves along hysteretically damped repetitive structures is investigated numerically, by extending recently developed theory. It is proved that symplectic mathematics combined with the pseudo-excitation method gives efficient solutions not only for stationary random excitations, but also for non-stationary random excitations. The numerically computed time-dependent variance curves of the displacement and internal force responses at different stations of a substructural chain are given as an example.

Computation of Sea-Wave Direction of Propagation of Random Waves

In the last 20 years, the application of the random wave theory in maritime engineering has been given the utmost importance, especially for evaluating design wave conditions, loads on structures, and sand littoral drift. The aim of the paper is to present a general expression for the evaluation of the local energy flux vector of random waves. This expression can be used for computing a representative direction of propagation of the wave spectrum in the presence of any scattering source. The effectiveness of the presented expression is shown by a numerical experiment. Moreover the expression is compared with the relationship that is normally used in field observations to represent the mean wave direction of a two-dimensional spectrum. The results of the comparison, carried out with a measured directional wave spectrum, are discussed.

Propagation of stationary random waves along substructural chains

Stationary random waves travelling along hysteretically damped repetitive structures are investigated numerically by means of symplectic algebra. Power spectral densities and variances of various responses are computed conveniently by virtue of the “pseudo-excitation method”.

Empirical orthogonal function representations of stochastic sound-speed ensembles

The representation of stochastic ensembles of sound-speed profiles by convenient and efficient means is examined. The procedure is based on using discrete empirical orthogonal functions (EOFs) with random coefficients. The eventual objective is to use the EOF representations in random wave propagation models, for which their efficiency is expected to reduce significantly the computational expense. To illustrate the procedure, two specific model ensembles are considered. The first consists of deep-water (Munk) sound-speed profiles, and the second has shallow-water profiles obtained from temperature profiles generated by a portion of the generalized digital environmental model (GDEM). After a statistical distribution of stochastic environmental input parameters is assumed, EOFs are calculated for ensembles of up to 10 000 profile samples. Relations between the statistics of the input parameters and the sound speed profiles that are obtained from EOF representations are investigated. In each case examined, the number of EOFs needed to account for virtually all of the sample variance is at most four. Additional cases are considered using Continental Shelf data by assuming measured profiles represent ensemble samples. [Work supported by ONR.]

Economical random wave propagation modelling taking into account non-linear amplitude dispersion

Two parabolic computational models for random waves propagating in coastal regions are presented, both based on the mild-slope equation. The first is a linear model in which the number of spectral components used is determined at run-time based on the complexity of the bathymetry. Results show that the linear wave model gives good predictions under a wide range of conditions. The second model includes weak non-linearity efficiently by writing the non-linear source term in an explicit form. It is shown that in shallow water the inclusion of the weak non-linearity provides much better agreement with experimental data than the linear model.

The split-step solution in random wave propagation

This paper considers the accuracy of the split-step solution for wave propagation in random media, both analytically and numerically. The main interest lies in finding the statistical moments of the wavefield, averaged over time of realizations of the random medium. The perturbation error is found explicitly as a function of operators acting on the wavefield. The accuracy of the wave and its moments is then examined, in terms of the error and its autocorrelation function. It is shown that both step-wise and cumulative accuracy of the moments is greater than it is for the wavefield itself. A model is also described in which the random medium may be represented, using a series of correlated phase-screens in arbitrarily fine detail. This is used to examine the convergence of the split-step solution as the step-size is reduced. It is shown that the independent phase-screen model is valid for small scattering when the scale-size of the medium is less than the step-size in the direction of propagation.

DIRECTIONAL NEARSHORE WAVE PROPAGATION AND INDUCED CURRENTS

Recently, a depth and current refraction model has been developed for the computation of directionally spread, random wave propagation in coastal regions (Booij et al., 1985). For the verification of the performance of this model laboratory measurements in a directional, shallow water wave basin were conducted. Specific attention was given to the verification of the new features of the numerical model, viz. the effects of directional spreading and ambient current field on the wave propagation and transformation process, and the change of characteristic spectral wave frequency due to wave dissipation processes.

Wave-kinetic method, phase-space path integrals, and stochastic wave propagation

A phase-space path-integral approach to random wave propagation is introduced as a natural extension of the wave-kinetic method. By virtue of its relationship to the Wigner distribution function, the latter is limited to the computation of second-order statistical moments associated with a nonlocal, complex, stochastic, Schrodinger-like equation. The phase-space path approach, on the other hand, permits the asymptotic evaluation of second- and higher-order statistical observables in a manner analogous to that followed in the recently formulated configuration-space path-integral approaches to the conventional local, complex, stochastic, parabolic equation.