Influence of bottom topography on the propagation of linear shallow water waves: an exact approach based on the invariant imbedding method

The wave equation for linear shallow water waves propagating over a varying bottom topography has the same form as that for p-polarized electromagnetic waves in inhomogeneous dielectric media. The role played by the dielectric permittivity in the case of electromagnetic waves is played by the inverse water depth. We apply the invariant imbedding theory of wave propagation, which has been developed mainly to study the electromagnetic wave propagation, to linear shallow water waves in the special case where the depth depends on only one coordinate. By comparing the numerical result obtained using our method, when the depth profile is linear, with an exact analytical formula, we demonstrate that our method is numerically reliable. The invariant imbedding method can be used in studying the influence of complicated bottom topography on the propagation of shallow water waves, in a numerically exact manner. We illustrate this by considering the case where a periodic modulation is added to a linear depth profile. Bragg scattering due to the periodic component competes with the tsunami effect due to the linear depth variation. This competition is seen to generate interesting physical effects. We also consider a ridge-type bottom topography and examine the resonant transmission phenomenon associated with the Fabry–Perot effect.     

Propagation of p-polarized electromagnetic waves obliquely incident on stratified random media: Random phase approximation

We consider the propagation of p-polarized electromagnetic waves obliquely incident on stratified random dielectric media. Using the invariant imbedding method generalized to random media and applying the random phase approximation, we derive a simple analytical expression of the localization length and calculate the disorder-averaged reflectance and transmittance and the fluctuations of the localization length and the reflectance as functions of the incident angle. We also calculate the disorder-averaged intensity profile of the magnetic field inside the random medium. We find that within the random phase approximation, the p wave can be delocalized and transmitted completely at a certain critical incident angle, which is bigger than the Brewster angle in the uniform case.     

Berkhoff approximation in a problem on surface gravity wave propagation in a basin with bottom irregularities

The problem of the propagation of small-amplitude surface gravity waves in a basin of constant mean depth with one- and two-dimensional bottom roughness is solved in the framework of the Berkhoff model by a mean-field method. In both cases the solutions obtained are compared with the solutions of sets of exact linearized equations of the hydrodynamics of an incompressible fluid. The comparison of the exact and approximate mean-field attenuation coefficients has shown that the Berkhoff approximation is appropriate for the solution of this problem in the case of shallow water for an arbitrary correlation length of bottom irregularities and in the case of arbitrary depth and large-scale irregularities. An explanation is given for the limits of applicability of the Berkhoff approximation which are connected with the weak variability of the vertical structure of the wave field in shallow water and in a basin with large-scale depth fluctuations. The mean-field attenuation coefficients reach their maximum values in the region k_oh_o≥1 (where k_o is the wavenumber of the surface gravity wave in a basin of constant depth h_o). The location of these maxima is practically independent of the correlation length of the bottom irregularities. For the case of one-dimensional irregularities the effect of bottom roughness on the surface gravity wave velocity is investigated. It is shown that the surface wave in a basin with an uneven bottom may propagate more slowly, as well as faster than the wave in a basin with an even bottom, depending on the relations between the wavelength, depth and correlation length of the bottom imperfections.     

Experimental and numerical study of the propagation of focused wave groups in the nearshore zone

The propagation of focused wave groups in intermediate water depth and the shoaling zone is experimentally and numerically considered in this paper. The experiments are carried out in a two-dimensional wave flume and wave trains derived from Pierson-Moskowitz and JONSWAP spectrum are generated. The peak frequency does not change during the wave train propagation for Pierson-Moskowitz waves; however, a downshift of this peak is observed for JONSWAP waves. An energy partitioning is performed in order to track the spatial evolution of energy. Four energy regions are defined for each spectrum type. A nonlinear energy transfer between different spectral regions as the wave train propagates is demonstrated and quantified. Numerical simulations are conducted using a modified Boussinesq model for long waves in shallow waters of varying depth. Experimental results are in satisfactory agreement with numerical predictions, especially in the case of wave trains derived from JONSWAP spectrum.     

Effect of shallow water internal waves on ocean acoustic striation patterns

Contour plots of underwater acoustic intensity, mapped in range and frequency, often exhibit striations. It has been claimed that a scalar parameter 'beta', defined in terms of the slope of the striations, is invariant to the details of the acoustic waveguide. In shallow water, the canonical value is β=1. In the present paper, the waveguide invariant is modelled as a distribution rather than a scalar. The effects of shallow water internal waves on the distribution are studied by numerical simulation. Realizations of time-evolving shallow water internal wave fields are synthesized and acoustic propagation simulated using the parabolic equation method. The waveguide invariant distribution is tracked as the internal wave field evolves in time. Both random background internal waves and more event-like solitary internal waves are considered.     

Effect of shallow water internal waves on ocean acoustic striation patterns

Abstract

Contour plots of underwater acoustic intensity, mapped in range and frequency, often exhibit striations. It has been claimed that a scalar parameter ‘beta’, defined in terms of the slope of the striations, is invariant to the details of the acoustic waveguide. In shallow water, the canonical value is β=1. In the present paper, the waveguide invariant is modelled as a distribution rather than a scalar. The effects of shallow water internal waves on the distribution are studied by numerical simulation. Realizations of time-evolving shallow water internal wave fields are synthesized and acoustic propagation simulated using the parabolic equation method. The waveguide invariant distribution is tracked as the internal wave field evolves in time. Both random background internal waves and more event-like solitary internal waves are considered.     

Propagation of surface plasmon polaritons through gradient index and periodic structures

We study propagation of surface electromagnetic waves along a metallic surface covered by various layered dielectric structures. We show that strong radiative losses, typical for scattering of a surface wave, can be considerably suppressed when a single dielectric step is substituted by gradient index or periodic layered structure.     

Nonlinear shoaling of shallow water waves: perspective in terms of the inverse scattering transform

Summary We study nonlinear interactions in measured surface wave trains obtained in the Northern Adriatic Sea about 16 kilometres from Venice, Italy.Nonlinear Fourier analysis is discussed in terms of the exact spectral solution to the Korteweg-deVries (KdV) equation as given by theinverse scattering transform (IST). For the periodic and/or quasi-periodic boundary conditions assumed herein, the approach may be viewed as a nonlinear, broad-banded generalization of the ordinary, linear Fourier transform. In particular, we study solition interactions, their properties and the nonlinear dynamics of the radiation (or oscillation) modes as found from the inverse scattering transform analysis. We also conduct a number of computer experiments in which measured wave trains are numerically propagated forward in time toward shallow water and backward in time into deep water in order to assess how the nonlinear wave dynamics are influenced by propagation over variable bathymetry. On this basis we develop a scenario for the evolution of nonlinear wave trains, initially far offshore in deep water, as they propagate into shallow water regions. The deep-water waves have a small Ursell number and are hence not very nonlinear; as they propagate toward shallow water, the Ursell number gradually increases in the numerical experiments by about an order of magnitude. A useful parameterization of nonlinearity in these studies is the ?spectral modulus,? a number between 0 and 1, which is associated with each IST spectral frequency. Small values of the modulus mean that a particular spectral component is linear (a sine wave); large values of the modulus (≈1) indicate that the component is nonlinear (a soliton). There is a systematic increase of the modulus as the waves propagate into shallow water where nonlinear effects predominate; we describe how the modulus varies as a function of spectral frequency during this shoaling process. The results suggest that the effect of increasing nonlinearity ?saturates? the IST spectrum (i.e. the modulus ≈1 for all frequencies) to that virtually all spectral components become solitons in sufficiently shallow water.     

Unstable Surface Waves in Running Water

We consider the stability of periodic gravity free-surface water waves traveling downstream at a constant speed over a shear flow of finite depth. In case the free surface is flat, a sharp criterion of linear instability is established for a general class of shear flows with inflection points and the maximal unstable wave number is found. Comparison to the rigid-wall setting testifies that the free surface has a destabilizing effect. For a class of unstable shear flows, the bifurcation of nontrivial periodic traveling waves is demonstrated at all wave numbers. We show the linear instability of small nontrivial waves that appear after bifurcation at an unstable wave number of the background shear flow. The proof uses a new formulation of the linearized water-wave problem and a perturbation argument. An example of the background shear flow of unstable small-amplitude periodic traveling waves is constructed for an arbitrary vorticity strength and for an arbitrary depth, illustrating that vorticity has a subtle influence on the stability of free-surface water waves.     

Surface-to-volume wave conversion in shallow water with a corrugated bottom

Acoustic transmission between points onshore or in very shallow water and points in deep water is strongly influenced by the shear rigidity of marine sediments, which control the parameters and the very existence of seismoacoustic surface waves. Previously, it was found that coupling between acoustic modes and the seismoacoustic surface waves is normally weak, although not negligible in the case of a gently sloping seafloor and soft sediments. In this paper, the previous work is extended by accounting for the small-scale roughness of the seafloor. The significant role of roughness in coupling between volume and surface waves is demonstrated. The combined effect of bottom topography, roughness, and wave attenuation in soft marine sediments on the sound propagation between points in shallow and deep water is discussed. Published in Russian in Akusticheskiĭ Zhurnal, 2008, Vol. 54, No. 3, pp. 400–407. The article was translated by the author.     

Two-dimensional solitons in shallow water of variable depth

In this letter we study the propagation of two-dimensional solitons in shallow water of variable depth, in case a relation between wave surfaces and variable depth exists. The existence of N-solitons and cnoidal waves is proved. Finally energetic considerations are presented.     

�Ǿ��ȷֲ�����������Ӿ�����Ӵ�϶����

用时域有限差分法研究了电磁波在等离子体光子晶体中的传播特性。数值模拟中使用完全匹配层吸收边界条件,计算了电磁波通过等离子体光子晶体的反射和透射系数。讨论了等离子体密度、等离子体温度、介电常数比和引入缺陷层对等离子体光子晶体光子带隙的影响。     

Existence of gaps in the spectrum of periodic dielectric structures on a lattice

We consider the finite-difference counterpart, i.e., the true lattice analog, of Maxwell's equations and equations that govern the propagation of acoustic waves in a medium with a periodic dielectric structure. In particular, the vector nature of electromagnetic waves is fully taken into account. The existence of true gaps for these lattice models is proved for a two-component medium for which the dielectric constant is everywhere real and positive, and the dielectric constant of the background is essentially larger than the one corresponding to the embedded component.     

垂直入射电磁波在大气压非平衡等离子体层中的传播特性

用数值模拟的方法对大气压非平衡等离子体薄层中，不同的电子密度分布对微波反射、吸收和透射的影响进行了研究。所采用的理论分析方法是分层模型和镶嵌不变原理。计算中考虑了微波在子层间的多次反射和吸收。数值结果表明，对于电磁波的吸收来说，等离子体中具有二次分布的电子密度，其效果要高于线性分布10%左右；而对于反射来说，线性分布效率更高。功率反射系数随波长的增大而增大，功率吸收系数A也不是单调的，当电子密度不变时，A存在一个峰值，随着电磁波波长的增加而增加，达到最大值后，缓慢降低。     

一维等离子体光子晶体的带隙研究

采用时域有限差分方法(FDTD),结合等离子体计算中的分段线性电流密度卷积技术(PLJERC)对一维等离子体光子晶体(1D-PPC)进行了数值模拟,给出了微分高斯脉冲在一维等离子体光子晶体中的传播过程。计算得到的带隙结构与K-P模型方法的结果一致。计算并分析了等离子体频率、介质介电常数、等离子体-介质层的厚度比以及周期厚度对一维等离子体光子晶体带隙结构的影响。     

Theoretical analysis and experimental results of interference striation pattern of underwater target radiated noise in shallow water waveguide

Waveguide invariant in shallow water is an attractive topic in recent three decades. The interference phenomena of direct wave of radiated noise of underwater target and reflection wave from sea surface and sea bottom can be considered as a typical case of shallow water waveguide.The interference striation pattern of direct wave and its reflection is the effective and comprehensive figure for better understanding the essence of shallow water waveguide invariant.The theoretical analysis of interference phenomena generated by direct wave of radiated noise of underwater target and its reflection wave from sea surface and sea bottom is presented in this paper.It is shown that the interference wave resulted by sea surface reflection will produce striation pattern centered at high frequency band.But the interference of nulling frequency resulted by sea bottom reflection may be at low frequency or high frequency, it strongly depends on the acoustic behavior of sea bottom.The relationship between main parameters of interference striation pattern and target,receiver,and environment is derived.It is shown that the interference striation have the shape of hyperbolic curve.The equation set of the hyperbolic curve and its asymptotic line is presented.The at sea experiment carried out in South China sea shows some interesting results.A part of data processing results are illustrated in this paper.The results expressed in this paper show that the interference striation pattern can be used,in some conditions,as a potential means for target recognition.     

Transverse Wave Propagation in Relativistic Two-Fluid Plasmas around Schwarzschild-de Sitter Black Hole

We investigate transverse electromagnetic waves propagating in a plasma influenced by the gravitational field of the Schwarzschild-de Sitter black hole. Applying 3+1 spacetime split we derive the relativistic two-fluid equations to take account of gravitational effects due to the event horizon and describe the set of simultaneous linear equations for the perturbations. We use a local approximation to investigate the one-dimensional radial propagation of Alfvén and high frequency electromagnetic waves. We derive the dispersion relation for these waves and solve it for the wave number k numerically.     

Multipath pulse shapes in shallow water: theory and simulation

In shallow water propagation the steeper ray angles are weakened most by boundary losses. Regarding the sound intensity as a continuous function of angle it can be converted into a function of travel time to reveal the multipath pulse shape received from a remote source (one-way path) or a target (two-way path). The closed-form isovelocity pulse shape is extended here to the case of upward or downward refraction. The envelope of the earliest arrivals is roughly trapezoidal with a delayed peak corresponding to the slowest, near horizontal refracted paths. The tail of the pulse falls off exponentially (linearly in decibels) with a decay constant that depends only on the bottom reflection properties and water depth, irrespective of travel time, a useful property for geoacoustic inversion and for sonar design. The nontrivial analytical problem of inverting explicit functions of angle into explicit functions of time is solved by numerical interpolation. Thus exact solutions can be calculated numerically. Explicit closed-form approximations are given for one-way paths. Two-way paths are calculated by numerical convolution. Using the wave model C-SNAP in several broadband cases of interest it is demonstrated that these solutions correspond roughly to a depth average of multipath arrivals.     

NETWORK CHARACTERIZATION OF STEP DISCONTINUITY WITH GENERALIZED CONSERVATION OF COMPLEX POWER TECHNIQUE AND ITS APPLICATION TO ANALYZING PERIODIC DIELECTRIC WAVEGUIDES

The propagation of electromagnetic waves along open periodic, dielectric waveguides is formulated in the case that surface wave is guided and propagates normally to the corrugation. Our approximate analysis with the propagation characteristics is to consider a corresponding bounded waveguide problem in which perfect electric or magnetic walls are introduced, and the periodic corrugation is regarded as consisting of step discontinuities connected by a length of uniform slab waveguide. By properly taking into account of both surface modes and only a few non-surface-modes, a novel network approach is proposed for characterizing step discontinuity based on the generalized conservation of complex power technique (GCCPT). Employing solution selection rule (SSR), we can readily derive propagation characteristics in the Bragg interaction region. A number of numerical results are shown to demonstrate the usefulness of our approach.