Efficient computation of steady solitary gravity waves

An efficient numerical method to compute solitary wave solutions to the free surface Euler equations is reported. It is based on the conformal mapping technique combined with an efficient Fourier pseudo-spectral method. The resulting nonlinear equation is solved via the Petviashvili iterative scheme. The computational results are compared to some existing approaches, such as Tanaka’s method and Fenton’s high-order asymptotic expansion. Several important integral quantities are computed for a large range of amplitudes. The integral representation of the velocity and acceleration fields in the bulk of the fluid is also provided.     

Generation of unsteady waves by concentrated disturbances in an inviscid fluid with an inertial surface

The surface waves generated by unsteady concentrated disturbances in an initially quiescent fluid of infinite depth with an inertial surface are analytically investigated for two- and three-dimensional cases. The fluid is assumed to be inviscid, incompressible and homogenous. The inertial surface represents the effect of a thin uniform distribution of non-interacting floating matter. Four types of unsteady concentrated disturbances and two kinds of initial values are considered, namely an instantaneous/oscillating mass source immersed in the fluid, an instantaneous/oscillating impulse on the surface, an initial impulse on the surface of the fluid, and an initial displacement of the surface. The linearized initial-boundary-value problem is formulated within the framework of potential flow. The solutions in integral form for the surface elevation are obtained by means of a joint Laplace-Fourier transform. The asymptotic representations of the wave motion for large time with a fixed distance- to-time ratio are derived by using the method of stationary phase. The effect of the presence of an inertial surface on the wave motion is analyzed. It is found that the wavelengths of the transient dispersive waves increase while those of the steady-state progressive waves decrease. All the wave amplitudes decrease in comparison with those of conventional free-surface waves. The explicit expressions for the freesurface gravity waves can readily be recovered by the present results as the inertial surface disappears.     

The role of buoyancy–frequency oscillations in the generation of mountain gravity waves

There is evidence from balloon measurements that atmospheric buoyancy–frequency profiles, apart from a sharp increase (roughly by a factor of two) at the tropopause, often feature appreciable oscillations (typical wavelength 1–2 km) with altitude. It is argued here that such short-scale oscillatory variations of the background buoyancy frequency, which usually are ignored in theoretical models, can have a profound effect on the generation of mountain waves owing to a resonance mechanism that comes into play at certain wind speeds depending on the dominant oscillation wavelength. A simple linear model assuming small sinusoidal buoyancy–frequency oscillations suggests, and numerical solutions of the Euler equations for more realistic flow conditions confirm, that under resonant conditions the induced gravity-wave activity is significantly increased above and upstream of the mountain, similarly to resonant flow of finite depth over topography.      

Dispersion of linear gravity waves on a viscoelastic fluid in an horizontal canal

A characteristic equation is derived that describes the spatial decay of linear surface gravity waves on Maxwell fluids. Except at small frequencies, the derived dispersion relation is different from the temporal decay dispersion relation which is normally studied within fluid mechanics. The implications for waves on viscous Newtonian fluids using the two different dispersion relations is briefly discussed. The wave number is measured experimentally as function of the frequency in a horizontal canal. Seven Newtonian fluids and four viscoelastic liquids with constant viscosity have been used in the experiments. The spatial decay theory for Newtonian fluids fits well to the experimental data. The model and experiments are used to determine limits for the Maxwell fluid time numbers for the four viscoelastic liquids. As a result of low viscosity it was not possible within this study to obtain these time numbers from oscillatory experiments. Therefore, a comparison of surface gravity wave experiments with theory is applicable as a method to evaluate memory times of low viscosity viscoelastic fluids.     

Reconciling different formulations of viscous water waves and their mass conservation

The viscosity of water induces a vorticity near the free surface boundary. The resulting rotational component of the fluid velocity vector greatly complicates the water wave system. Several approaches to close this system have been proposed. Our analysis compares three common sets of model equations. The first set has a rotational kinematic boundary condition at the surface. In the second set, a gauge choice for the velocity vector is made that cancels the rotational contribution in the kinematic boundary condition, at the cost of rotational velocity in the bulk and a rotational pressure. The third set circumvents the problem by introducing two domains: the irrotational bulk and the vortical boundary layer. This comparison puts forward the link between rotational pressure on the surface and vorticity in the boundary layer, addresses the existence of nonlinear vorticity terms, and shows where approximations have been used in the models. Furthermore, we examine the conservation of mass for the three systems, and how this can be compared to the irrotational case.     

The stability of finite-amplitude interfacial solitary waves

The linear stability of finite-amplitude interfacial gravity solitary waves propagating in a two-layer fluid is investigated analytically focusing on the occurrence of an exchange of stability. We make an asymptotic analysis for small growth rates of infinitesimal disturbances, and explicitly obtain their growth rates near an exchange of stability. The result indicates that an exchange of stability occurs at every stationary value of the total energy of the solitary waves. It also gives us information whether the number of growing modes increases or decreases after experiencing the exchange of stability. We apply these analytical results to specific interfacial solitary waves, and find various features on their stability that are not seen in the case of surface solitary waves.     

Diffraction of unsteady gravity waves on a wedge

We consider the occurrence and diffraction on a wedge of gravity waves which appear on the surface of an ideal incompressible liquid as a result of the activation of a periodically acting source. An exact solution is obtained. The time of the transient response is determined. Asymptotic formulas are established for the height of the free liquid surface, which are applicable for any instant of time.The author wishes to thank S. S. Voit for his guidance and assistance in carrying out this study.     

The attenuation of monochromatic surface waves due to the presence of an inextensible cover

The attenuation of surface gravity waves is an important process associated with air–sea and wave–current interactions. Here we investigate experimentally the attenuation of monochromatic surface gravity waves due to the presence of various surface covers. The surface covers are fixed in space such that they do not advect with the wave motion and are selected such that the bending modulus is negligible for the wave frequencies used in the experiment in order to minimize any flexural effects. Wave attenuation rates are found to be independent of wave steepness and the type of cover used over the tested parameter range. Results are consistent with the theoretical attenuation rate for an inextensible surface cover.     

Fluid particle dynamics and stokes drift in gravity and capillary waves generated by the faraday instability

In this paper we study particle motions in nearly square containers due to gravity and capillary waves generated by vertical, periodic oscillation of the container. The method of second order partial averaging is used to decompose the particle motions into periodic oscillations and a slow Stokes drift. In the case of gravity waves, it is shown that long distance (several wavelengths) particle transport is possible. In the case of capillary waves, it is shown that, in agreement with experimental observations of Ramshankaret al., particle trajectories can be chaotic even when the wave pattern is regular so long as the pattern is spatially modulated.Dedicated to Professor P. R. Sethna on the Occasion of His 70th BirthdayThis research was partially supported by an NSF Presidential Young Investigator Award and an ONR Grant No. N00014-89-J-3023.     

Surface waves in an exponentially graded, general anisotropic elastic material under the influence of gravity

In a recent paper Destrade [1] studied surface waves in an exponentially graded orthotropic elastic material. He showed that the quartic equation for the Stroh eigenvalue p is, after properly modified, a quadratic equation in p² with real coefficients. He also showed that the displacement and the stress decay at different rates with the depth x₂ of the half-space. Vinh and Seriani [2] considered the same problem and added the influence of gravity on surface waves. In this paper we generalize the problem to exponentially graded general anisotropic elastic materials. We prove that the coefficients of the sextic equation for p remain real and that the different decay rates for the displacement and the stress hold also for general anisotropic materials. A surface wave exists in the graded material under the influence of gravity if a surface wave can propagate in the homogeneous material without the influence of gravity in which the material parameters are taken at the surface of the graded half-space. As the wave number k → ∞, the surface wave speed approaches the surface wave speed for the homogeneous material. A new matrix differential equation for surface waves in an arbitrarily graded anisotropic elastic material under the influence of gravity is presented. Finally we discuss the existence of one-component surface waves in the exponentially graded anisotropic elastic material with or without the influence of gravity.     

Nonlinear dynamics of hydrostatic internal gravity waves

Stratified hydrostatic fluids have linear internal gravity waves with different phase speeds and vertical profiles. Here a simplified set of partial differential equations (PDE) is derived to represent the nonlinear dynamics of waves with different vertical profiles. The equations are derived by projecting the full nonlinear equations onto the vertical modes of two gravity waves, and the resulting equations are thus referred to here as the two-mode shallow water equations (2MSWE). A key aspect of the nonlinearities of the 2MSWE is that they allow for interactions between a background wind shear and propagating waves. This is important in the tropical atmosphere where horizontally propagating gravity waves interact together with wind shear and have source terms due to convection. It is shown here that the 2MSWE have nonlinear internal bore solutions, and the behavior of the nonlinear waves is investigated for different background wind shears. When a background shear is included, there is an asymmetry between the east- and westward propagating waves. This could be an important effect for the large-scale organization of tropical convection, since the convection is often not isotropic but organized on large scales by waves. An idealized illustration of this asymmetry is given for a background shear from the westerly wind burst phase of the Madden–Julian oscillation; the potential for organized convection is increased to the west of the existing convection by the propagating nonlinear gravity waves, which agrees qualitatively with actual observations. The ideas here should be useful for other physical applications as well. Moreover, the 2MSWE have several interesting mathematical properties: they are a system of nonconservative PDE with a conserved energy, they are conditionally hyperbolic, and they are neither genuinely nonlinear nor linearly degenerate over all of state space. Theory and numerics are developed to illustrate these features, and these features are important in designing the numerical scheme. A numerical method is designed with simplicity and minimal computational cost as the main design principles. Numerical tests demonstrate that no catastrophic effects are introduced when hyperbolicity is lost, and the scheme can represent propagating discontinuities without introducing spurious oscillations.      

Fourier analysis of a class of upwind schemes in shallow water systems for gravity and Rossby waves

A Fourier analysis has been performed for a class of upwind finite volume schemes, including the study of phase speed, group velocity, damping and dispersion. In the first part, pure gravity waves are investigated. As expected, most upwind schemes lead to a significant damping, but they exhibit a better phase behavior than most centered schemes. In the second part, the Coriolis parameter is considered and the Rossby modes are studied. In this case, all selected upwind schemes lead to a severe damping. The numerical results are also compared with those obtained by using a slope limiter approach. It is concluded that most upwind schemes with or without slope limiters present poor results for an accurate calculation of the Rossby modes. Copyright © 2008 John Wiley & Sons, Ltd.     

Effect of viscosity on dispersion of capillary-gravity waves

The effect of viscosity on dispersion of capillary-gravity waves becomes significant when the attenuation coefficient is greater than about 2.5% of the wave number. For low viscosity fluids such as water this condition is met at frequencies greater than about 5 kHz in which case direct measurement of wavelength is difficult. For higher viscosity fluids the effect appears at much lower frequencies but direct measurement of wavelength becomes difficult since viscosity causes severe attenuation of surface waves. We have overcome the measurement difficulties by using a new miniature laser interferometer, which directly measures the wavelength of standing capillary waves with the requisite precision to yield reliable dispersion data for viscous fluids. Here we review the effect of viscosity on the dispersion relation and present new experimental data on dispersion of capillary waves in several water-glycerol mixtures. Our data provides direct experimental verification of the theoretical analysis.     

Oblique water entry of a wedge into waves with gravity effect

The hydrodynamic problem of a two dimensional wedge entering waves with gravity effect is analysed based on the incompressible velocity potential theory. The problem is solved through the boundary element method in the time domain. The stretched coordinate system in the spatial domain, which is based on the ratio of the Cartesian system in the physic space to the vertical distance the wedge has travelled into the water, is adopted based on the consideration that the decay of the effect of the impact away from the body is proportional to this ratio. The solution is sought for the total potential which includes both the incident and disturbed potentials, and decays towards the incident potential away from the body. A separate treatment at initial stage is used, in which the solution for the disturbed potential is sought to avoid the very large incident potential amplified by dividing the small travelled vertical distance of the wedge. The auxiliary function method is used to calculate the pressure on the body surface. Detailed results through the free surface elevation and the pressure distribution are provided to show the effect of the gravity and the wave, and their physical implications are discussed.     

Mesh effects on the computation of non-linear gravity waves of arbitrary discharge

Numerical computations of non-linear gravity waves are presented and the effects of mesh variations on the results are discussed. Waves are regarded as two-parameter families (λ,A)_Q of arbitrary discharge Q, and computations are carried out using a new Kantorovich algorithm. Mesh effects are to a large extent dependent on the particular wave region under consideration. Three such regions are identified and typical examples are computed and discussed.     

Numerical simulation of internal gravity waves using a lattice gas model

Internal waves are modelled in two different circumstances: in a continuously stratified fluid and at the interface between two immiscible fluids. This is done using the lattice gas approach. The standard single phase model and an immiscible two-phase model are both modified to incorporate gravitational interactions. Standing internal waves are set up in both models and are seen to oscillate under the action of the gravitational interaction. The results obtained suggest that the lattice gas approach can be a useful tool in the modelling of such phenomena. © 1998 John Wiley & Sons, Ltd.     

New families of pure gravity waves in water of infinite depth

Nonlinear periodic gravity waves propagating at a constant velocity at the surface of a fluid of infinite depth are considered. The fluid is assumed to be inviscid and incompressible and the flow to be irrotational. It is known that there are both regular waves (for which all the crests are at the same height) and irregular waves (for which not all the crests are at the same height). We show numerically the existence of new branches of irregular waves which bifurcate from the branch of regular waves. Our results suggest there are an infinite number of such branches. In addition we found additional new branches of irregular waves which bifurcate from the previously calculated branches of irregular waves.     

Generation of nonstationary waves by a thick-walled piezoceramic cylinder excited by electric signals

Coupled electroelasticity theory, acoustic approximation, and two-wire transmission line theory are used to study the generation of waves by a submerged cylindrical piezoelectric transducer connected by a cable to a source of nonstationary electric signals. The problem is reduced to a system of integral Volterra equations using the Laplace transform and analytical inversion of boundary conditions. The results of calculations for different cable lengths are presented __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 11, pp. 127–136, November 2005.     

Weakly nonlinear long waves in a prestretched Blatz-Ko cylinder: Solitary, kink and periodic waves

This paper studies nonlinear waves in a prestretched cylinder composed of a Blatz-Ko material. Starting from the three-dimensional field equations, two coupled PDEs for modeling weakly nonlinear long waves are derived by using the method of coupled series and asymptotic expansions. Comparing with some other existing models in literature, an important feature of these equations is that they are consistent with traction-free surface conditions asymptotically. Also, the material nonlinearity is kept to the third order. As these two PDEs are quite complicated, the attention is focused on traveling waves, for which a first-order system of ODEs are obtained. We use the technique of dynamical systems to carry out the analysis. It turns out that the system is three parameters (the prestretch, the propagating speed and an integration constant) dependent and there are totally seven types of phase planes which contain trajectories representing bounded traveling waves. The parametric conditions for each phase plane are established. A variety of solitary and periodic waves are found. An important finding is that kink waves can propagate in a Blatz-Ko cylinder. We also find that one type of periodic waves has an interesting feature in the profile slope. Analytical expressions for all bounded traveling waves are obtained.