Perturbation analysis of short-crested waves in Lagrangian coordinates

A new second-order asymptotic solution that describes short-crested waves is derived in Lagrangian coordinates. The analytical Lagrangian solution that is uniformly valid satisfies the irrotational condition and there being zero pressure at the free surface, in contrast with the Eulerian solution, in which there is residual pressure at the free surface. The explicit parametric solution highlights the trajectory of a water particle and the wave kinematics above the mean water level. The mass transport velocity and Lagrangian mean level associated with particle displacement can also be obtained directly. In particular, the mean level of the particle motion in a Lagrangian form differs that of the Eulerian form. The new formulation reduces to second-order standing or progressive wave solutions in Lagrangian coordinates at the limiting angles of approach. Expressions for kinematic quantities are also presented.     

On frequency–amplitude dependences for surface and internal standing waves

The analytic approach proposed by Sekerzh-Zenkovich [On the theory of standing waves of finite amplitude, Dokl. Akad. Nauk USSR 58 (1947) 551–554] is developed in the present study of standing waves. Generalizing the solution method, a set of standing wave problems are solved, namely, the infinite- and finite-depth surface standing waves and the infinite- and finite-depth internal standing waves. Two-dimensional wave motion of an irrotational incompressible fluid in a rectangular domain is considered to study weakly nonlinear surface and internal standing waves. The Lagrangian formulation of the problems is used and the fifth-order perturbation solutions are determined. Since most of the approximate analytic solutions to these problems were obtained using the Eulerian formulation, the comparison of the results, as an example the analytic frequency–amplitude dependences, obtained in Lagrangian variables with the corresponding ones known in Eulerian variables has been carried out in the paper. The analytic frequency–amplitude dependences are in complete agreement with previous results known in the literature. Computer algebra procedures were written for the construction of asymptotic solutions. The application of the model constructed in Lagrangian formulation to a set of different problems shows the ability to correctly reproduce and predict a wide range of situations with different characteristics and some advantages of Lagrangian particle models (for example, the bigger radius of convergence of an expansion parameter than in Eulerian variables, simplification of the boundary conditions, parametrization of a free boundary).     

Lagrangian motion of fluid particles in gravity–capillary standing waves

A third-order analytical solution for the gravity–capillary standing wave is derived in Lagrangian coordinates through the Lindstedt–Poincare perturbation method. By numerical computation, the dynamical properties of nonlinear standing waves with surface tension in finite water depth, including particle trajectory and surface profile are investigated. We find that the presence of surface tension leads to a change of the crest form. Moreover, we also find that the particle trajectories near the surface oscillate back and forth along the arcs which will change from concave to convex as the inverse Bond number increases. There is no mass transport of the particles in a wave period.     

Higher-order asymptotic solutions in fluids of various configurations

Applying perturbation methods, symbolic computation, and generalizing the solution method, higher-order asymptotic solutions are constructed in Lagrangian variables for several models describing 2D standing wave motions in fluids of various configurations. Three main parameters of the fluid configuration, depth, capillarity, and stratification layer, are considered. The frequency-amplitude dependences are obtained and compared with those known in the literature in Eulerian and Lagrangian variables. The comparison shows that the analytical frequency-amplitude dependences are in complete agreement with previous results known in the literature and with the results obtained for other models. A generalization allows us to investigate critical phenomena for standing waves in fluids of various configurations. Namely, special attention is focused on critical values of one parameter, the fluid depth. The frequency-amplitude dependences are analyzed from the point of view of critical values: critical points and critical curves are determined for several models describing standing waves in fluids of various configurations.     

Nonlinear waves in two-dimensions generated by variable pressure acting on the free surface of a heavy liquid

The development of nonlinear waves on the free surface of a heavy liquid initially at rest is treated analytically in cases where the external pressure force of limited power is distributed over a large area in the free surface but is otherwise arbitrary. In [1] approximate (up to small terms of higher order) solution of the problem is obtained in the form of functional series. In the present article the convergence theorems for the series are proved. When the pressure varies with time sinusoidally, the sums of the series are found in closed form. By passing to the limit in the solution as time goes to infinity, the form of the nonlinear steady-state wave is found. According to the solution, when the steady-state wave gets away from the variable pressure zone, a long chain of structures develops similar to so called Kelvin-Helmholtz billows. The existence of nonlinear standing waves is discovered, which have a finite number of nodes in the free surface infinite in extent, and the frequency spectrum and the form of these waves are found explicitly.     

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.     

参数激励圆柱形容器中的非线性Faraday波

在柱坐标系下,通过奇异摄动理论的多尺度展开法求解势流方程,研究了垂直强迫激励圆柱形容器中的单一模式水表面驻波模式。假设流体是无粘、不可压且运动是无旋的,在忽略了表面张力的影响下,用两变量时间展开法得到一个具有立方项以及底部驱动项影响的非线性振幅方程。对上述方程进行了数值计算,计算的结果显示了在不同驱动振幅和驱动频率下,会激发不同自由水表面驻波模式,从等高线的图像来看,和以往的实验结果相当吻合。     

ON INTERACTION OF SHOCK AND SOUND WAVE （I）

This paper studies the interaction of shock and gradient wave (sound wave) of solutions to the system of inviscid isentropic gas dynamics as a model for the corresponding problems for nonlinear hyperbolic systems. The problem can be reduced to a boundary value problem in a wedged dormain, By using the method of constructing asymptotic solutions and Newton‘siteration process it is proved that if a weak shock hits a gradient wave, then the grandient wave will split into two gradient waves, while the shock continuses propagating. In this paper the author reduces the problem to a standard form and constructs asymptotic solution of the problem. The existence of the genuine solution will he given in the following paper.     

Resonances and standing waves

Recent investigations on aperiodic waves, which are generated by time-harmonic forces in waveguides, indicate a close relationship between resonances and certain time-harmonic solutions of homogeneous boundary value problems for the wave equation (“standing waves”). This paper is motivated by the observation that, in all known cases, standing waves are connected with resonances if and only if they are subject to suitable asymptotic restrictions as |x| → ∞. These asymptotic properties are used to introduce a class of “admissible” standing waves. We prove that admissible standing waves do not exist in a certain class of local perturbations Ω of the n-dimensional domain Ω₀ bounded by the hyperplanes x_n = 0 and x_n = π. This extends a result of M. Faulhaber on the absence of eigenvalues. As we shall show in a subsequent paper, absence of admissible standing waves implies absence of resonances.     

ON INTERACTION OF SHOCK AND SOUND WAVE (Ⅰ)

1.IntroductionRecentlythestudyofdiscontinuoussolutionforthesystemofconservationlawsinhigherdimensionalspacehasbeenconsiderablydeveloped.In[1,W12]thelocalekistenceof8olutionforsuclisystemwithdiscontinuityinvolvingsingleshock,rarefactionwaveorsoundwav(gradientwave)hasbeenestablished.In[2Jand[14]theproblemsoninteractionoftwoshocksorinteractionofweaksingularitiesarealsoconsidered.Itisnaturaltoaskwhatabouttheresultwhenashockisinteractedbyawavewitliweakersillgularities,particularly,forthenbynsystem…     

Nonlinear Spatial Evolution of Small Disturbances from Boundary Conditions in Flat Inclined-Channel Flow

The asymptotic behavior of small disturbances as they evolve spatially from boundary conditions in a flat inclined channel is determined. These small disturbances develop into traveling waves called roll waves, first discussed by Dressler in 1949. Roll waves exist if the Froude number F exceeds 2, which consist of a periodic pattern of bores, or discontinuities. After confirming the instability condition for   F > 2  for the linearized equations in the boundary value case, the nonlinear boundary value problem for the weakly unstable region of F slightly larger than 2 is studied. Multiple scales and the Fredholm alternative theorem are applied to determine the evolution of the solution in space. It is found that the solution is dominated by the evolution of the disturbance along one characteristic. The shock conditions governing the asymptotic solution are determined and these conditions are used to determine the approximate shape of the resulting traveling wave from the solution. Both asymptotic and numerical results for periodic disturbances are presented.     

A perturbation solution for long wavelength thermoacoustic propagation in dispersions

In thermoacoustic scattering the scattered field consists of a propagating acoustic wave together with a non-propagational “thermal” wave of much shorter wavelength. Although the scattered field may be obtained from a Rayleigh expansion, in cases where the particle radius is small compared with the acoustic wave length, these solutions are ill-conditioned. For this reason asymptotic or perturbation solutions are sought. In many situations the radius of the scatter is comparable to the length of the thermal wave. By exploiting the non-propagational character of the thermal field we obtain an asymptotic solution for long acoustic waves that is valid over a wide range of thermal wavelengths, on both sides of the thermal resonance condition. We show that this solution gives excellent agreement with both the full solution of the coupled Helmholtz equations and experimental measurements. This treatment provides a bridge between perturbation theory approximations in the long wavelength limit and high frequency solutions to the thermal field employing the geometric theory of diffraction.     

Uniform asymptotics of the boundary values of the solution in a linear problem on the run-up of waves on a shallow beach

We consider the Cauchy problem with spatially localized initial data for a two-dimensional wave equation with variable velocity in a domain Ω. The velocity is assumed to degenerate on the boundary ?Ω of the domain as the square root of the distance to ?Ω. In particular, this problems describes the run-up of tsunami waves on a shallow beach in the linear approximation. Further, the problem contains a natural small parameter (the typical source-to-basin size ratio) and hence admits analysis by asymptotic methods. It was shown in the paper “Characteristics with singularities and the boundary values of the asymptotic solution of the Cauchy problem for a degenerate wave equation” [1] that the boundary values of the asymptotic solution of this problem given by a modified Maslov canonical operator on the Lagrangian manifold formed by the nonstandard characteristics associatedwith the problemcan be expressed via the canonical operator on a Lagrangian submanifold of the cotangent bundle of the boundary. However, the problem as to how this restriction is related to the boundary values of the exact solution of the problem remained open. In the present paper, we show that if the initial perturbation is specified by a function rapidly decaying at infinity, then the restriction of such an asymptotic solution to the boundary gives the asymptotics of the boundary values of the exact solution in the uniform norm. To this end, we in particular prove a trace theorem for nonstandard Sobolev type spaces with degeneration at the boundary.     

Solitary Wave Transformation Due to a Change in Polarity

Solitary wave transformation in a zone with sign-variable coefficient for the quadratic nonlinear term is studied for the variable-coefficient Korteweg–de Vries equation. Such a change of sign implies a change in polarity for the solitary wave solutions of this equation. This situation can be realized for internal waves in a stratified ocean, when the pycnocline lies halfway between the seabed and the sea surface. The width of the transition zone of the variable nonlinear coefficient is allowed to vary over a wide range. In the case of a short transition zone it is shown using asymptotic theory that there is no solitary wave generation after passage through the turning point, where the coefficient of the quadratic nonlinear term goes to zero. In the case of a very wide transition zone it is shown that one or more solitary waves of the opposite polarity are generated after passage through the turning point. Here, asymptotic methods are effective only for the first (adiabatic) stage when the solitary wave is approaching the turning point. The results from the asymptotic theories are confirmed by direct numerical simulation. The hypothesis that the pedestal behind the solitary wave approaching the turning point has a significant role on the generation of the terminal solitary wave after the transition zone is examined. It is shown that the pedestal is not the sole contributor to the amplitude of the terminal solitary wave. A negative disturbance at the turning point due to the transformation in the zone of the variable nonlinear coefficient contributes as much to the process of the generation of the terminal solitary waves.