两层流体界面上的孤立波

本文讨论两水平固壁间两层不可压无粘流体界面上的孤立波,计及界面上的表面张力效应.首先建立了适用于这种模型的基本方程组,并在弱色散近似下应用约化摄动法,导得了一阶界面升高所满足的Korteweg-de Vries方程,指出了按该方程系数α和μ的符号的异同,KdV孤立波可能凸向上或凸向下.然后详细讨论了原有近似下非线性效应与色散效应不能平衡的两种临界情形.在采用了适当的近似之后,对第一种临界情形(α=0)得到了修正的KdV方程,并指出,在所考虑的情形中,当μ>0时孤立波不存在,当μ<0时,孤立波仍可能存在,其形式与KdV孤立波不同;对第二种临界情形(μ=0),导得了推广的KdV方程,这时存在振荡型孤立波.文中还对近临界情形作了讨论.本文结果与一些经典结果完全一致,并把它们作了拓广.     

一个二流体系统中两对孤立波的相互作用

本文讨论一个二流体系统中孤立波的相互作用,该系统由水平固壁之上的两层常密度不可压无粘流体组成,上表面为自由面。文中在浅水波假定下,导出了适用于所考虑的模型的基本方程——推广的Boussinesq方程;接着,应用PLK方法和约化摄动法求得了两对表面-界面孤立波迎撞的二阶近似解,给出了碰撞时界面和表面的最大波幅以及碰撞后的非均匀相移,进而表明孤立波迎撞后将发生变形。     

深分层流体中内孤立波的相互作用——Ⅰ.弱相互作用 *

采用Lagrange观点研究分层流体中内孤立波的弱相互作用,它包括不同模式孤立波间的追撞和迎撞,以及相同模式孤立波间的迎撞.分析表明在有限深度情形每个波遵循ILW方程,而在无限深度情形每个波满足Benjamin-Ono方程,相互作用的主要效应体现在相移上.     

三层流体系统中孤立波的产生

采用约化摄动方法导出了三层流体系统中各界面所遵循的KdV方程,讨论了流体深度对孤立波产生的影响,将波分为快模式、中间模式和慢模式波后发现慢模式波的结果与已有的实验结果定性上一致.此外,还发现自由面上可能存在下凹的孤立波,这有待于实验的验证.     

具有扩散项的广义Nizhnik-Novikov-Veselov方程孤立波与周期波

本文研究具有弱向后扩散项的广义Nizhnik-Novikov-Veselov方程;通过运用动力系统方法,特别是几何奇异摄动理论和不变流形理论,得到方程孤立波解与周期波解的存在性;通过计算Abel积分的比值得到波速的单调性.同时给出了方程极限波速的上下界和周期波解的一些性质.     

旋转层结流体中垂直切变基本流、β效应、地形效应和强迫耗散共同作用下的Rossby波

本文运用摄动法和WKB方法(多尺度方法),从位涡守恒方程出发,分析旋转层结大气中基本流有垂直切变以及层结效应对β效应、地形效应和强迫耗散共同作用下的Rossby波的影响,得到一个非标准形式的非线性Schr?dinger方程,而在水平波数小于3时该方程有包络孤立波解;又进一步说明基本流的垂直切变对包络Rossby孤立波的波速的影响;强迫耗散对包络Rossby孤立波稳定度的影响.另外,本文还应用常数变异法求解了非齐次的Bessel方程,得到包络Rossby孤立波的经向结构.     

旋转层结流体中垂直切变基本流、$\beta$效应、地形效应和强迫耗散共同作用下的Rossby波

本文运用摄动法和WKB方法（多尺度方法）, 从位涡守恒方程出发, 分析旋转层结大气中基本流有垂直切变以及层结效应对$\beta$效应、地形效应和强迫耗散共同作用下的Rossby波的影响, 得到一个非标准形式的非线性Schr\"{o}dinger方程,而在水平波数小于3时该方程有包络孤立波解; 又进一步说明基本流的垂直切变对包络Rossby孤立波的波速的影响;强迫耗散对包络Rossby孤立波稳定度的影响.另外, 本 文还应用常数变异法求解了非齐次的Bessel方程, 得到包络Rossby孤立波的经向结构.     

缓变任意截面渠道中的非线性周期波以及孤立波的分裂

本文研究了在流动方向有缓慢变化的任意截面渠道中的非线性周期波、孤立波以及孤立波在这种渠道中的分裂;导出了适用于这种渠道的变系数KdV方程,并求出了该方程的首项近似解;得出了波速、周期、波高和渠道几何尺寸之间的关系,得到了分裂后孤立波个数的判别式及分裂后孤立波波幅的表示式,并应用于矩形渠道和左右对称的三角形渠道。对于矩形渠道的情况,本文的结果和Madsen和Mei,Johnson,Svendsen和Buhr Hansen等人的结果一致。     

血液粘弹性对动脉中脉搏波传播的影响

以Womersley理论为基础,将动脉中脉搏波复波速所满足的频率方程推广到粘弹血液情形,得到了线性粘弹血管和线性粘弹血液情况下脉搏波复波速的一般关系式。应用此关系式,讨论了血液粘弹性对动脉中脉搏波波速和波的衰减的影响。结果表明,血液弹性对脉搏波传播的影响小于血管粘性的影响;对大动脉,血液弹性对脉搏波传播的影响可忽略。     

两层流体中具有β变化和地形影响的Rossby波

本文研究了两层流体中具有变化的Rossby参数和地形Rossby波的问题.利用行波法和摄动的方法,获得了Rossby波振幅满足齐次KdV方程和齐次mKdV方程,推广了Rossby参数和地形对Rossby孤立波的影响.     

Internal Solitary Waves

The expansion procedure introduced by Benney (1966) for weakly nonlinear, planar shallow-water waves is used to provide an alternative derivation of the more general results of Benjamin (1966) for shallow fluid layers possessing arbitrary vertical stratification and horizontal shear. New solutions that include the effects of both shear and stratification are presented. The evolution equation for slowly varying cylindrical solitary waves traveling in a density-stratified fluid is found using two-timing techniques. Not surprisingly, one obtains the same coefficients for the nonlinear and dispersive terms as in the planar case. In the limit for uniform density it is shown that the free-surface evolution equation of Miles (1978) for axisymmetric Boussinesq waves is recovered.     

广义缓坡方程

运用表面波Hamilton方法和缓坡逼近假定,分析缓变三维流场和非平整海底对波浪传播的影响,推导出广义缓坡方程。海底地形由两个分量组成:慢变分量,其水平长度尺度大于表面波的波长;快变分量,其振幅与表面波的波长相比为一小量,但是其频率与表面波频率相当。该广义缓坡方程具有广泛的适用范围,以下著名的缓坡方程成为它的特例:经典的Berkhoff缓坡方程;包含环境流效应的Kirby缓坡方程;描述波状海底特征的Dingemans缓坡方程。另外,同时也得到了描述环境流场和快变海底效应的广义浅水方程。     

Waves in an inhomogeneous fluid in the presence of a dock : PMM vol. 39, n 6, 1975, pp. 1140–1142

We investigate the propagation of waves generated by oscillations of a section of the bottom of a tank through a two-layer fluid, in the presence of a dock. Wave motions in an inhomogeneous fluid generated by displacement of a section of the bottom of a tank were studied in [1] where the upper surface of the fluid was assumed either to be completely free, or completely covered with ice. In the present paper we use the method given in [2] to investigate a similar problem under the assumption that the fluid surface is partly covered with an immovable rigid plate. The expressions obtained for the velocity potential are used to determine the form of the free surface and of the interface. We show that when the fluid is inhomogeneous, the wave amplitude on the free surface increases, while the presence of a plate reduces the amplitude of the surface waves, as well as of the internal waves in the region between the plate and the oscillating section of the bottom.     

A note on the wave propagation in water of variable depth

In the present work, utilizing the two dimensional equations of an incompressible inviscid fluid and the reductive perturbation method we studied the propagation of weakly nonlinear waves in water of variable depth. For the case of slowly varying depth, the evolution equation is obtained as the variable coefficient Korteweg-de Vries (KdV) equation. Due to the difficulties for the analytical solutions, a numerical technics so called “the method of integrating factor” is used and the evolution equation is solved under a given initial condition and the bottom topography. It is observed the parameters of bottom topography causes to the changes in wave amplitude, wave profile and the wave speed.     

A Note on the Theory of Low-Frequency Waves in a Rotating Stratified Channel

It is found that the possible low-frequency, quasigeostrophic motions in a rotating, stratified channel with a wave-maker at one end include: (i) “standing waves” whose amplitudes are damped exponentially away from the forcing, and (ii) baroclinic internal Kelvin waves, trapped to the right-hand wall when facing in the direction of phase propagation. The Kelvin waves are excited only if the wave-maker transfers mean energy to the fluid. The standing waves, on the other hand, carry no energy and thus serve mainly to provide continuity between the wave-maker and the fluid. When the bottom of the channel is inclined to the horizontal by a small angle θ, topographic oscillations are possible. These waves behave like topographic Rossby waves if the forcing frequency is greater than sN and if the ratio HN/fL is small, where s=tanθ, θ is the angle of the bottom slope, L is the width of the channel, H is the mean depth, f is the Coriolis parameter, and N is the Brunt-Vaisala (or buoyancy) frequency. It is determined that topographic Rossby waves cannot exist in the channel if HN/fL?0.65. If the wave-maker frequency is smaller than sN, and if HN/fL~1, the topographic oscillations become bottom-trapped, decaying away from the bottom boundary in a distance ~λf/N, where λ is the horizontal wavelength. The phase and energy of the bottom-trapped wave both move to the left of an observer who is facing shallow water. The Kelvin waves are basically unchanged when the bottom is inclined if their down-channel wavelength is large relative to the width of the channel. The standing oscillations of the flat-bottom case exist as complex-horizontal-wave-number solutions to the topographic wave dispersion relation. Although these waves have propagating phase when s≠0, they are still trapped to the forcing, and do not transfer net energy from the wave-maker to the fluid. All three modes are required to solve the general matching conditions for an arbitrary wave-maker when the channel has a sloping bottom.     

一个二流体系统中非线性水波的Hamilton描述

讨论了一个二流体系统中非线性水波的Hamilton描述,该系统由水平固壁之上的两层常密度不可压无粘流体组成,上表面为自由面.文中将速度势函数展开成垂向坐标的幂级数,在浅水长波的假定下,取下层流体的“动厚度”与上层流体的“折合动厚度”为广义位移、界面上和自由面上的速度势为广义动量,根据Hamilton原理并运用Legendre变换导出该系统的Hamilton正则方程,从而将单层流体情形的结果推广到分层流体的情形.     

变化底部对非线性表面波的影响

考虑表面张力的作用,研究了不可压缩、无粘性流体流过变化壁面时的共振流动,分析了不同的底部壁面变化对非线性表面波的影响．在导出非线性表面波遵循的fKdV方程后,利用拟谱方法进行数值模拟,用Matlab软件绘制瀑布图,由此得出结论:上凸底部上的波可以看成是向前凸台阶和向后凸台阶分别向前后散射发展的结果,二者不发生相互作用;下凹壁面的波形是向前凹台阶和向后凹台阶相互作用的结果;某些组合式底部的波形是上凸和下凹相互作用的结果．     

Nonlinear Cusped Caustics for Dispersive Waves

A multiple-scale perturbation analysis for slowly varying weakly nonlinear dispersive waves predicts that the wave number breaks or folds and becomes triple-valued. This theory has some difficulties, since the wave amplitude becomes infinite. Energy first focuses along a cusped caustic (an envelope of the rays or characteristics). The method of matched asymptotic expansions shows that a thin focusing region with relatively large wave amplitudes, valid near the cusped caustic, is described by the nonlinear Schrödinger equation (NSE). Solutions of the NSE are obtained from an asymptotic expansion of an equivalent linear singular integral equation related to a Riemann-Hilbert problem. In this way connection formulas before and after focusing are derived. We show that a slowly varying nearly monochromatic wave train evolves into a triple-phased slowly varying similarity solution of the NSE. Three weakly nonlinear waves are simultaneously superimposed after focusing, giving meaning to a triple-valued wave number. Nonlinear phase shifts are obtained which reduce to the linear phase shifts previously described by the asymptotic expansion of a Pearcey integral.     

The Superharmonic Instability of Finite-Amplitude Surface Waves on Water of Finite Depth

Saffman's (1985) theory of the superharmonic stability of two-dimensional irrotational waves on fluid of infinite depth has been generalized to solitary and periodic waves of permanent form on finite uniform depth. The frame of reference for the calculation of the Hamiltonian for periodic waves of finite depth is found to be the frame in which the mean horizontal velocity is zero.