Quasi-potential Method to Study Nonlinear Surface Shallow Water Waves

By using the potential method and the perturbation method under the condition of small amplitude and shallow water waves, we analytically get the KdV-type equation for a viscous shallow water. It indicates that for one soliton-like solution, its amplitude will decrease as it propagates away due to the viscous effects of water.     

KdV-Burgers Equation for a Viscous Flowing Shallow Water Waves

Korteweg, de Vries-Burges equation is obtained for an incompressible and viscous fluid which is flowing in one direction for the shallow water. We assume that the wave amplitude is small but finite, the viscosity of the fluid is also small enough.     

Effects of viscosity on shock-induced damping of an initial sinusoidal disturbance

A lack of reliable data treatment method has been for several decades the bottleneck of viscosity measurement by disturbance amplitude damping method of shock waves.In this work the finite difference method is firstly applied to obtain the numerical solutions for disturbance amplitude damping behavior of sinusoidal shock front in inviscid and viscous flow.When water shocked to 15 GPa is taken as an example,the main results are as follows:(1) For inviscid and lower viscous flows the numerical method gives results in good agreement with the analytic solutions under the condition of small disturbance(a0/λ=0.02);(2) For the flow of viscosity beyond 200 Pa s(η=κ) the analytic solution is found to overestimate obviously the effects of viscosity.It is attributed to the unreal pre-conditions of analytic solution by Miller and Ahrens;(3) The present numerical method provides an effective tool with more confidence to overcome the bottleneck of data treatment when the effects of higher viscosity in experiments of Sakharov and flyer impact are expected to be analyzed,because it can in principle simulate the development of shock waves in flows with larger disturbance amplitude,higher viscosity,and complicated initial flow.     

空气中快速运动声源水下声场的波数积分模型

本文采用二维波数积分，对空气中高速运动声源激发的水下声场进行建模。针对二维波数积分计算声场时域解计算量大的问题，提出一种快速计算方法。用本文提出的方法，对深海和浅海情况下，空气中高速运动单频点声源激发的水下声场进行了计算和仿真。计算结果表明：在深海，水下接收信号的幅度和瞬时频率随时间发生变化；接收器深度、接收器与声源运动轨迹的最小距离对接收信号的变化快慢有较大影响，而声源高度的影响较小；在浅海中，接收信号呈现快速的幅度起伏，明显的多普勒频移和大的频率展宽效应。与简正波方法相比，本文方法主要适用于近场计算，而简正波方法适用于远场。另外，当声源频率较高时，二维波数积分方法的计算量将迅速增大。     

Stable gravity wave of arbitrary amplitude in finite depth

A combination and modification of two existing methods, which involves balancing static and dynamic pressure differences between points along the surface and conserving mass through cross sections below the surface in the reference frame moving with the phase velocity, is applied to surface gravity waves of arbitrary amplitude in water of finite depth. For a given still water depth and wave height the method determines in closed form the phase velocity, wavelength, and wave profile of the stable wave. The main assumption is that the horizontal component of the fluid velocity be independent of depth. The motion is not assumed to be irrotational. The wavelength of the stable wave is found to be about 3.6 times the still water depth for infinitesimal amplitude, and at finite amplitude the wavelength decreases as the amplitude increases. Therefore, shallow water waves are concluded to be unstable even at infinitesimal amplitude, for which the assumption is accurate. Previously it has been argued that only at finite amplitude will shallow water waves change form as they propagate. The wave profile is found to be sinusoidal for infinitesimal amplitude and to be asymmetric at finite amplitude, the crests being higher and narrower and the troughs shallower and broader. These results are consistent with well-known theoretical work and laboratory measurements.     

Eddy viscosity for time reversing waves in a dissipative environment

We present new results for the time reversal of weakly nonlinear pulses traveling in a random dissipative environment. Also we describe a new theory for calculating the eddy viscosity for weakly nonlinear waves propagating over a random surface. The turbulent viscosity is calculated from first principles, namely, without imposing any stress-strain hypothesis. A viscous shallow water model is considered and its effective viscosity characterized. We also show that weakly nonlinear waves can still be time reversed under weak dissipation. Incoherently scattered signals are recompressed, both for time reversal in transmission as well as in reflection. Under the weakly nonlinear, weakly dissipative regime, dissipation only affects the refocused pulse profile regarding its amplitude, but its shape is not corrupted. Numerical experiments are presented.     

Existence of Global Weak Solutions for a 2D Viscous Shallow Water Equations and Convergence to the Quasi-Geostrophic Model

 We consider a two dimensional viscous shallow water model with friction term. Existence of global weak solutions is obtained and convergence to the strong solution of the viscous quasi-geostrophic equation with free surface term is proven in the well prepared case. The ill prepared data case is also discussed. Received: 4 October 2002 / Accepted: 22 January 2003 Published online: 28 May 2003 Communicated by P. Constantin     

Non-dispersive traveling waves in inclined shallow water channels

Existence of traveling waves propagating without internal reflection in inclined water channels of arbitrary slope is demonstrated. It is shown that traveling non-monochromatic waves exist in both linear and nonlinear shallow water theories in the case of a uniformly inclined channel with a parabolic cross-section. The properties of these waves are studied. It is shown that linear traveling waves should have a sign-variable shape. The amplitude of linear traveling waves in a channel satisfies the same Green's law, which is usually derived from the energy flux conservation for smoothly inhomogeneous media. Amplitudes of nonlinear traveling waves deviate from the linear Green's law, and the behavior of positive and negative amplitudes are different. Negative amplitude grows faster than positive amplitude in shallow water. The phase of nonlinear waves (travel time) is described well by the linear WKB approach. It is shown that nonlinear traveling waves of any amplitude always break near the shoreline if the boundary condition of the full absorption is applied.     

浅海近程混响的振荡现象*

研究浅海近程混响特性对于评估和提高主动声纳性能具有重要意义。多次浅海混响实验显示,近程混响强度存在稳定的振荡现象,脉宽基本对振荡的幅度和周期没有影响。为解释这一现象,本文基于射线理论和小斜率近似给出了浅海近程混响模型,仿真与实测数据结果基本吻合。数值仿真结果表明：海底反射声场对单站声纳接收到回声信号的贡献远小于海底近垂向大掠射角散射声场的作用;混响强度振荡现象是海底近程散射声场的多途现象造成的,并由此给出了振荡周期与海深及收发深度的关系。     

Symmetric waves are traveling waves for a shallow water equation modeling surface waves of moderate amplitude

Following a general principle introduced by Ehrnström, Holden and Raynaud in 2009, we prove that for an equation modeling the free surface evolution of moderate amplitude waves in shallow water, all symmetric waves are traveling waves.     

Stability analysis of solutions for the sixth-order nonlinear Boussinesq water wave equations in two-dimensions and its applications

In the present study, we are concerned with the generalized Boussinesq equation including the singular sixth-order Boussinesq equation, which describes the bi-directional propagation of small amplitude and long capillary-gravity waves on the surface of shallow water for bond number less than but very close to 1/3. By using the extended auxiliary equation method, we obtained some new soliton like solutions for the two-dimensional sixth-order nonlinear Boussinesq equation with constant coefficients. These solutions include symmetrical, non-symmetrical kink solutions, solitary pattern solutions, Jacobi and Weierstrass elliptic function solutions and triangular function solutions. The stability analysis for these solutions is discussed.     

Shallow water sound propagation with surface waves

The theory of wavefront modeling in underwater acoustics is extended to allow rapid range dependence of the boundaries such as occurs in shallow water with surface waves. The theory allows for multiple reflections at surface and bottom as well as focusing and defocusing due to reflection from surface waves. The phase and amplitude of the field are calculated directly and used to model pulse propagation in the time domain. Pulse waveforms are obtained directly for all wavefront arrivals including both insonified and shadow regions near caustics. Calculated waveforms agree well with a reference solution and data obtained in a near-shore shallow water experiment with surface waves over a sloping bottom.     

Properties of underwater acoustic communication channels in shallow water

Underwater acoustic channels are band-limited and reverberant, posing many obstacles to reliable, phase-coherent acoustic communications. While many high frequency communication experiments have been conducted in shallow water, few have carried out systematic studies on the channel properties at a time scale relevant for communications. To aid communication system design, this paper analyzes at-sea data collected in shallow water under various conditions to illustrate how the ocean environments (sea surface waves and random ocean medium) can affect the signal properties. Channel properties studied include amplitude and phase variations, and temporal coherence of individual paths as well as the temporal and spatial coherence of multipaths at different time scales. Reasons for the coherence loss are hypothesized.     

黏性液滴变形过程的核梯度修正光滑粒子动力学模拟

本文提出了一种核梯度改进光滑粒子动力学(KGC-SPH)方法,模拟了黏性液滴形变自由表面问题.首先,通过模拟等温黏性液滴拉伸和旋转变形,验证了KGC-SPH法较SPH法具有较高精度和更好稳定性,且能很好地保持总角动量守恒.其次,基于非等温van der Waals模型对平衡态圆形液滴的形成过程进行数值研究,观察到小幅度振荡现象,并给出了一种新的克服张力不稳定性的方法和一种适合KGC-SPH方法的新的表面张力处理技术.最后,研究了van der Waals液滴的周期性振荡现象,讨论了初始椭圆形液滴长短半轴比 关键词： 光滑粒子动力学 黏性液滴 van der Waals模型 表面张力     

Time reversing solitary waves

We present new results for the time reversal of nonlinear pulses traveling in a random medium, in particular for solitary waves. We consider long water waves propagating in the presence of a spatially random depth. Both hyperbolic and dispersive regimes are considered. We demonstrate that in the presence of properly scaled stochastic forcing the solution to the nonlinear (shallow water) conservation law is regularized leading to a viscous shock profile. This enables time-reversal experiments beyond the critical time for shock formation. Furthermore, we present numerical experiments for the time-reversed refocusing of solitary waves in a regime where theory is not yet available. Solitary wave refocusing simulations are performed with a new Boussinesq model, both in transmission and in reflection.     

Influence of resonator shape on nonlinear acoustic field in a thermoacoustic engine

The influence of the resonator shape on nonlinear acoustic field in a thermoacoustic engine is studied.The resonator of themoacoustic engine is boundary driving by a piston at one end,and the other end of it is rigid closed.A one-dimensional wave equation that accounts for gas dynamic nonlinearities and viscous dissipation in the resonator is established based on the governing equations of viscous hydromechanics.The nonlinear wave equation is solved using approximate Galerkin method.The nonlinear acoustic field in four different types of shaped resonators including hyperbolical,exponential,conical and sinusoidal are obtained and compared with that of a cylindrical resonator.It is found that the amplitude and waveform of the pressure are strongly affected by the resonator shape,the driving amplitude and the oscillation frequency of the piston.Waveform distortion,resonance frequency shift and hysteresis are observed,when the piston oscillation amplitude is large enough.The advantages of shaped resonator for thermoacoustic engine lie in inhibition of higher order harmonics and improvement of pressure ratio,etc.     

谐振管形状对热声发动机内非线性声场的影响

研究了谐振管一端受活塞声源激励,另一端刚性封闭条件下,管道形状对热声发动机谐振管内部非线性声场的影响。基于流体力学基本方程建立了渐变截面谐振管内一维非线性声场的模型,考虑了黏性耗散及非线性效应的影响。利用伽辽金法数值求解了该模型的速度势方程,分析了谐振管形状、活塞振动速度及激励频率对管内声场的影响。将双曲形、指数形、锥形、正弦形等四种变截面谐振管内的非线性声场与圆柱形直管的情况进行了比较。结果反映了谐振管内声场的压力波动受活塞振动速度及谐振管形状的影响;显示了当活塞振动幅度较大时,谐振管内出现的波形畸变、频率曲线偏移、共振频率滞后等非线性现象;揭示了变截面谐振管在抑制管内的高阶谐波及提高压比等方面的优越性。      

具有随机起伏表面的正声速梯度浅海中的简正波声场

本文考虑了随机起伏表面对正声速梯度浅海中简谐点源的声场的影响,获得了简正波的极点方程和振幅函数的近似表达式。结果指出,文献[1]中的结果为本文的极限结果。     

Nonlinear gravitational waves on the surface of viscous fluid: Lagrangian approach

The problems of the asymptotic theory of weakly nonlinear surface waves in viscous fluid are discussed. For standing waves on deep water, the solutions obtained in the first- and second-order approximations in a small parameter—wave steepness—are analyzed. The evolution equation for the amplitude of wave packet envelope is obtained where the inverse Reynolds number is equal to the squared steepness. It is shown that this is a nonlinear Schrödinger equation with linear dissipation.     

Investigation into ferrofluid magnetoviscous effects under an oscillating shear flow

The use of ferrofluid seals in mechanical systems can lead to viscous damping that affects their dynamic behavior. This paper describes an investigation into local viscous properties in the case of an axial harmonic force. The influence of magnetic field level, shear stress amplitude and frequency are studied. Even for ferrofluid particles in a highly saturated magnetic field, it is shown that viscosity increases with magnetic intensity, decreases with the frequency of harmonic excitation and is not sensitive to shear rate amplitude. Viscosity is lower for oscillatory flows than for steady flows.