Stability and accuracy of spatial approximations for wave equation tidal models

Amplitude and phase characteristics for numerical approximations to the shallow water wave equation are obtained for linear and quadratic finite elements, for finite difference approximations, for non-constant bathemetry, and for uneven node spacing. Stability is shown to require non-zero friction as well as satisfaction of a Courant constraint. Lumping is shown to reduce the Courant constraint for stability while higher order and quadratic finite element approximations require a more restrictive constraint than their second order and linear finite element counterparts. The amplitude of the propagation factor for stable schemes and propagating waves is seen to be independent of the Courant number and type of numerical approximation. Although the higher order and quadratic schemes provide better propagation of the low and moderate frequency waves, the highest frequency waves (2Δx) are better propagated by low order numerical methods.     

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.     

Shallow water rogue wavetrains in nonlinear optical fibers

In addition to deep-water rogue waves which develop from the modulation instability of an optical CW, wave propagation in optical fibers may also produce shallow water rogue waves. These extreme wave events are generated in the modulationally stable normal dispersion regime. A suitable phase or frequency modulation of a CW laser leads to chirp-free and flat-top pulses or flaticons which exhibit a stable self-similar evolution. Upon collision, flaticons at different carrier frequencies, which may also occur in wavelength division multiplexed transmission systems, merge into a single, high-intensity, temporally and spatially localized rogue pulse.     

三层密度分层流体毛细重力波二阶Stokes波解

以小振幅波理论为基础,利用摄动方法研究了三层密度分层流体的毛细重力波,给出了三层成层状态下各层流体速度势的二阶渐近解及毛细重力波波面位移的二阶Stokes波解.结果表明:一阶解及二阶解除了依赖于各层流体的厚度及密度,与表面张力也有很重要的关系.     

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.     

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.     

Ultrasonic surface crack characterization on complex geometries using surface waves

The characterization of surface cracks on complex geometries using surface waves is investigated numerically and experimentally. The specimen geometry is implemented in a finite difference code by approximation of the contour using a Cartesian grid. In the experiments the out-of-plane surface displacement is measured by means of a heterodyne laser interferometer. Good agreement is shown by comparison of the calculated out-of-plane displacement with experimental results for both cracked and non-cracked specimens. The crack depth is measured down to a size of 0.7 times the surface wavelength using a time delay approach. The many Rayleigh pulses propagating after the crack can be separated from the other modes by a filtering procedure based on the surface wave propagation velocity. Only a detailed analysis of the scattering phenomenon using the simulation allows an identification of the transmitted pulse required for crack depth measurement. Application of the method to a specimen with a real fatigue crack shows a systematical error possibly due to the inclined crack profile.     

Rapid and accurate analysis of surface and pseudo-surface waves using adaptive laser ultrasound techniques

A method has been developed to measure the phase velocity of laser generated and detected surface acoustic waves. An optical grating produced by an electronically addressable spatial light modulator (SLM) was imaged onto the sample surface to generate surface acoustic waves whose frequency and wavefront was controlled by the SLM. When the grating period matched the surface acoustic wavelength, the surface wave was strongly excited, thus the wavelength and, thereby the phase velocity was determined. We present results with this method that allows the phase velocity and the angular dispersion of the generalized surface wave as well as the pseudo-surface wave on the (100) nickel to be measured. Measurements on (111) silicon single crystals are also presented. The measurement precision is approximately 0.2%. Methods to further improve the measurement precision are also discussed.     

Pattern transition of two-dimensional Faraday waves at an extremely shallow depth

In this paper, we experimentally investigate the pattern transition of two-dimensional Faraday waves at an extremely shallow depth in a Hele-Shaw cell. Several patterns of Faraday waves are observed, which have some significant differences in wave profile, wave height and wave length. It is found that, in a wide range of the forcing frequency f, there always exists a region of the acceleration amplitude A, in which there exist the so-called hysteretic jumps between different patterns of Faraday waves. All of these experimental observations could enrich our knowledges about the Faraday waves and would be helpful to the further theoretical studies on the related topic in future.     

Pulsed Rayleigh wave scattered at a surface crack

This paper investigates Rayleigh wave interaction with machined slots on flat aluminium blocks to simulate surface breaking cracks. Using a finite element method, Rayleigh wave scattering by narrow slots of varied depth ranging from 0.5 mm to 20 mm is calculated. Pulsed wideband Rayleigh waves with a centre frequency of 590 kHz and -6 dB bandwidth of 520 kHz is considered. Reflection and transmission coefficients are calculated and compare well with the published literature. We and other workers have reported enhancement of the measured amplitude or particle velocity of an apparent Rayleigh wave close to a surface defect. In this paper, it is found that the predicted enhancement of in-plane components of particle velocities close to a crack is significantly higher than that of the out-of-plane components of particle velocities which appears to be mainly due to the mode-converted surface skimming longitudinal wave from the crack that has mainly in-plane components near the sample surface. The enhancement of the in-plane particle velocity will be observed regardless of the type of in-plane sensitive ultrasonic detector used. The explanation of the discrepancy of the reflection and transmission coefficients obtained by pulsed and narrow band or pseudo continuous Rayleigh waves is discussed. The later-arriving Rayleigh waves from reverberation along the inside of the crack surface are observed, as has been previously reported by other workers, and this may also be used to gauge slot depth.     

流体/准饱和多孔介质中伪Scholte波的传播特性

研究流体/多孔介质界面Scholte波的传播特性对于水下勘探、地震工程等领域具有重要意义.本文基于Biot理论和等效流体模型,采用势函数方法,推导了描述有限厚度流体/准饱和多孔半空间远场界面波的特征方程和位移、孔压计算公式.在此基础上,分别以砂岩和松散沉积土为例,研究了流体/硬多孔介质和流体/软多孔介质两种情况下,可压缩流体层厚度和多孔介质饱和度对伪Scholte波传播特性的影响.结果表明：多孔介质软硬程度显著影响界面波的种类、相速度、位移和水压力分布;有限厚度流体/饱和多孔半空间界面处伪Scholte波相速度与界面波波长和流体厚度的比值有关;孔隙水中溶解的少量气体对剪切波的相速度的影响不大,对压缩波相速度、伪Scholte波相速度和孔隙水压力分布影响显著.     

Vertical amplitude phase structure of a low-frequency acoustic field in shallow water

We obtain in integral and analytic form the relations for calculating the amplitude and phase characteristics of an interference structure of orthogonal projections of the oscillation velocity vector in shallow water. For different frequencies and receiver depths, we numerically study the source depth dependences of the effective phase velocities of an equivalent plane wave, the orthogonal projections of the sound pressure phase gradient, and the projections of the oscillation velocity vector. We establish that at low frequencies in zones of interference maxima, independently of source depth, weakly varying effective phase velocity values are observed, which exceed the sound velocity in water by 5–12%. We show that the angles of arrival of the equivalent plane wave and the oscillation velocity vector in the general case differ; however, they virtually coincide in the zone of the interference maximum of the sound pressure under the condition that the horizontal projections of the oscillation velocity appreciably exceed the value of the vertical projection. We give recommendations on using the sound field characteristics in zones with maximum values for solving rangefinding and signal-detection problems.     

Critical surface wave velocity near phase transitions

The temperature dependent Rayleigh wave velocity is discussed for crystals with a surface layer of depth equal to the correlation length, ξ, with special reference to a critical region near phase transitions. For finite qξ the Rayleigh wave velocity reflects the specific critical properties of the surface layer. Under such conditions the temperature dependence of Rayleigh waves cannot be predicted on the temperature dependence of constants. A phenomenological analysis is made for qξ ≦? 1. Experimental results on SrTiO₃ show bulk-dominated critical Rayleigh wave velocities due to the short correlation length in this material.     

Variational derivation of KdV-type models for surface water waves

Using the Hamiltonian formulation of surface waves, we approximate the kinetic energy and restrict the governing generalized action principle to a submanifold of uni-directional waves. Different from the usual method of using a series expansion in parameters related to wave height and wavelength, the variational methods retains the Hamiltonian structure (with consequent energy and momentum conservation) and makes it possible to derive equations for any dispersive approximation. Consequentially, the procedure is valid for waves above finite and above infinite depth, and for any approximation of dispersion, while quadratic terms in the wave height are modeled correctly. For finite depth this leads to higher-order KdV type of equations with terms of different spatial order. For waves above infinite depth, the pseudo-differential operators cannot be approximated by finite differential operators and all quadratic terms are of the same spatial order.     

Decay instability of ion-acoustic wave train (or soliton) and formation of the shock profile

The stability of a periodic ion-acoustic wave of a finite amplitude propagating in a nonisothermal plasma is investigated. It is demonstrated that such a wave is unstable with respect to the splitting into a large number of satellite waves with effective wave numbers different from the wave number of the initial wave. The phase velocity of the satellite waves differs therefore from that of the initial pulse. Hence, the satellite waves with bigger phase velocity will “overtake” the initial pulse and turbulize the upstream plasma. The scattering of ions and electrons by the fluctuations of electric field of turbulent oscillations will cause the energy dissipation of the initial ion-acoustic wave translational motion and produce a collisionless shock wave.     

近表面弹性性质连续变化材料中激光声表面波的理论研究

基于金属材料近表面层弹性性质（杨氏模量）的连续变化，建立了一种激光在基底上的梯度材料中激发声表面波的理论模型，并用有限元方法模拟了脉冲激光作用于材料上表面激发出的超声波及其传播过程.讨论了表面层厚度的变化以及表面层弹性性质的变化对热弹条件下产生的声表面波波形特征的影响，并分别计算了两种情况下的声表面波的相速度色散，得到了两种情况下声表面波的变化规律.     

北极冰-水耦合系统中声波传播特性

利用固体和流体介质中波传播理论,导出了冰-水两层复合结构中导波频散方程。进一步,利用二分法对频散方程进行了数值求解,得到了ω-k频散曲线(ω与k分别为圆频率和波数),以及相速度和群速度频散曲线。结果表明:冰-水两层复合结构中导波由具有相同厚度水层和冰层中导波耦合而成,但与水层和冰层中导波频散曲线相比,复合结构中导波频散曲线除第1阶模式外,其余高阶模式均发生了很大变化。从原水层第1阶模式的截止频率开始,复合结构第2阶模式的相速度曲线被压低,各高阶(大于2阶)模式的相速度曲线出现一个跃变点,群速度曲线出现一个极大和一个极小值。水层越厚,复合结构各高阶模式的截止频率越低,相同频带内导波模式越丰富。水层厚度保持不变时,复合结构各阶模式的相速度和群速度曲线均随冰层厚度的增加而向低频方向移动。另外,还进一步分析了冰-水复合结构的导波波结构,发现第1阶导波模式的能量主要集中在冰层内和海表面附近,而2阶以上高阶导波模式的振动位移幅度随深度方向呈现周期性特征,并且模式阶数越高,振动越复杂。      

Modulational instability in asymmetric coupled wave functions

The evolution of the amplitude of two nonlinearly interacting waves is considered, via a set of coupled nonlinear Schr?dinger-type equations. The dynamical profile is determined by the wave dispersion laws (i.e. the group velocities and the group velocity dispersion terms) and the nonlinearity and coupling coefficients, on which no assumption is made. A generalized dispersion relation is obtained, relating the frequency and wave-number of a small perturbation around a coupled monochromatic (Stokes') wave solution. Explicitly stability criteria are obtained. The analysis reveals a number of possibilities. Two (individually) stable systems may be destabilized due to coupling. Unstable systems may, when coupled, present an enhanced instability growth rate, for an extended wave number range of values. Distinct unstable wavenumber windows may arise simultaneously.     

随时间变化的非平整壁面对液膜表面波演化特性的影响

本文对非平整壁面上的液膜表面波演化过程进行了研究. 针对非平整壁面随时间变化的特性, 采用小参数摄动法对控制方程和边界条件进行求解, 推导出波动壁面上液膜表面波的扰动方程, 采用导数展开法对其进行求解, 并选取简谐波形状的壁面进行数值研究. 对波动壁面下不同参数的影响规律研究可得, 当壁面频率较小时, 静态波与行进波的波长比较相近, 促进表面波之间的合并, 且壁面频率、壁面振幅及Re数的增加, 均会使表面波的振幅明显增大; 对比波动壁面与非平整壁面可得, 在相同位置处, 随着时间的演化, 非平整壁面上表面波呈周期变化, 而波动壁面上表面波呈波长更大的近周期变化; 壁面振幅和壁面频率的降低, 均会使两种壁面结构下的表面波振幅减小, 但所形成的表面波形有所不同, 即波动壁面引起的表面波可看作波动壁面结构与非平整壁面引起的表面波叠加而成.