Effect of periodic bottom roughness on gravitational waves in a liquid

The effect of a rigid bottom of periodic form on small periodic oscillations of the free surface of a liquid is considered with the assumption of low amplitude roughness. The methodologically most significant study in this direction, [1], will be utilized. In [1] the steady-state problem for flow over an arbitrarily rough bottom was studied. Other studies have recently appeared on small free oscillations above a rough bottom. Essentially these have considered the effect of underwater obstacles and cavities on surface waves in the shallow-water approximation (for example, [2], [3]). Liquid oscillations in a layer of arbitrary depth slowly varying with length were considered in [4]. However, these results cannot be applied to the study of resonant interaction of gravitational waves with a periodically curved bottom.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 43–48, July–August, 1984.     

EFFECTS OF VARYING BOTTOM ON NONLINEAR SURFACE WAVES

The resonant flow of an incompressible, inviscid fluid with surface tension on varying bottoms was researched. The effects of different bottoms on the nonlinear surface waves were analyzed. The waterfall plots of the wave were drawn with Matlab according to the numerical simulation of the fKdV equation with the pseudo-spectral method. Prom the waterfall plots, the results are obtained as follows: for the convex bottom, the waves system can be viewed as a combination of the effects of forward-step forcing and backward step forcing, and these two wave systems respectively radiate upstream and downstream without mutual interaction. Nevertheless, the result for the concave bottom is contrary to the convex one. For some combined bottoms, the wave systems can be considered as the combination of positive forcing and negative forcing.     

Reflection of long waves from a “nonreflecting” bottom profile

The transformation of long surface waves in a zone of variable depth is investigated within the framework of shallow-water theory. In the particular case of a bottom profile containing a so-called “nonreflecting” relief segment adjacent to an even bottom, expressions for the reflected and transmitted pulse waves are obtained in explicit form. It is shown that waves are reflected from such a profile. The role of distributed and concentrated reflection of a wave propagating above an uneven bottom is discussed.     

Higher order Boussinesq-type equations for water waves on uneven bottom

Higher order Boussinesq-type equations for wave propagation over variable bathymetry were derived. The time dependent free surface boundary conditions were used to compute the change of the free surface in time domain. The free surface velocities and the bottom velocities were connected by the exact solution of the Laplace equation. Taking the velocities on half relative water depth as the fundamental unknowns, terms relating to the gradient of the water depth were retained in the inverse series expansion of the exact solution, with which the problem was closed. With enhancements of the finite order Taylor expansion for the velocity field, the application range of the present model was extended to the slope bottom which is not so mild. For linear properties, some validation computations of linear shoaling and Booij' s tests were carried out. The problems of wave-current interactions were also studied numerically to test the performance of the enhanced Boussinesq equations associated with the effect of currents. All these computational results confirm perfectly to the theoretical solution as well as other numerical solutions of the full potential problem available.     

Influence of broken ice on the propagation of surface waves over a bottom shelf

The influence of drifting broken ice on the propagation of small-amplitude plane surface waves from an infinitely deep region of a basin to a region of finite depth over a bottom shelf is analyzed on the basis of wave source theory. The variations in the characteristics of the reflected and transmitted waves and the fluid surface perturbation profile due to the drifting ice are estimated as functions of the distance from the shelf. Sevastopol. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 106–115, November–December, 1998.     

Open-channel waves generated by propagation of a discontinuous wave over a bottom step

This paper presents the results of theoretical and experimental studies of open-channel waves generated by the propagation of a discontinuous dam-break wave over a bottom step. The cases where the initial tailwater level is higher than the step height (the step is under water) and where this value is smaller than the step height (at the initial time, water is absent on the step) are considered. Exact solutions are constructed using modified first-approximation equations of shallow-water theory, which admit the propagation of discontinuous waves in a dry channel. On the stationary hydraulic jump formed above the bottom step, the total free-stream energy is assumed to be conserved. These solutions agree with experimental data on various parameters (types of waves, wave propagation velocity, asymptotic depths behind the wave fronts). __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 1, pp. 31–44, January–February, 2008.     

Hamiltonian long wave expansions for internal waves over a periodically varying bottom

We derive a Hamiltonian formulation for two-dimensional nonlinear long waves between two bodies of immiscible fluid with a periodic bottom. From the formulation and using the Hamiltonian perturbation theory, we obtain effective Boussinesq equations that describe the motion of bidirectional long waves and unidirectional equations that are similar to the KdV equation for the case in which the bottom possesses short length scale. The computations for these results are performed in the framework of an asymptotic analysis of multiple scale operators.     

Generation of long waves by the bottom contour in a two-layer flow

Novosibirsk. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 3, pp. 34–47, May–June, 1993.     

Precursors for waves in random media

We consider scattering of a pulse propagating through a three-dimensional random media and study the shape of the pulse in the parabolic approximation. We show that, similarly to the one-dimensional O’Doherty–Anstey theory, the pulse undergoes a deterministic broadening. Its amplitude decays only algebraically and not exponentially in time, due to the signal low/midrange frequency component. We also argue that the parabolic approximation captures the front evolution (but not the signal away from the front) correctly even in the fully three-dimensional situation.     

Effects of bottom topography on the standing waves in circular basins

Standing waves in the cylinder basins with inhomogeneous bottom are considered in this paper. We assume that the inviscid, incompressible fluid is in irrotational undulatory motion. For convenience sake, cylindrical coordinates are chosen. The velocity potentials, the wave profiles and the modified frequencies are determined (to the third order) as power series in terms of the amplitude divided by the wavelength. Axisymmetrical analytical solutions are worked out. When ₁=0,the second order frequency are gained. As an example, we assume that cylinder bottom is an axisymmetrical paraboloid. We find out that the uneven bottom has influences on standing waves. In the end, we go into detail on geometric factors.     

Reflection of surface waves by a vertical step at the bottom of a basin

A study is made of the propagation of surface waves in a basin with a vertical step in the bottom. The problem is solved by fitting solutions on the horizontal boundary that continues the bottom of the shallow-water region of the basin; the factorization method is used. It is shown that the reflection of the waves depends on the wavelength of the incident waves and the difference between the two depths in the basin.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 181–185, January–February, 1984.     

Stationary Simple Waves on a Barotropic Liquid Shear Flow

Stationary threedimensional flows of a barotropic liquid in a gravity field are considered. In the shallowwater approximation, the Euler equations are transformed into a system of integrodifferential equations by the EulerLagrange change of coordinates. A system of simplewave equations is obtained, for which the theorem of existence of a solution attached to a given shear flow is proved. As an example, a particular solution analogous to the solution of the problem of a gas flow around a convex angle is given.     

Standing surface waves in a rectangular tank with local wall and bottom irregularities

The results of laboratory experiments on the estimation of the effect of wall and bottom geometry on the frequency, height, and decay rate of standing surface waves in a tank oscillating in the vertical direction are presented. The effect of one or two semi-cylindrical inserts mounted on the face and rear walls of the tank is considered in detail for the cases of a horizontal bottom and a linear shallow on the bottom. The experimental data are interpreted using a mathematical longwave model based on the method of accelerated convergence.     

Barotropic elliptical dipoles in a rotating fluid

Barotropic f-plane dipolar vortices were generated in a rotating fluid and a comparison was made with the so-called supersmooth f-plane solution which—in contrast to the classical Lamb–Chaplygin solution—is marked by an elliptical separatrix and a doubly continuously differentiable vorticity field. Dye-visualization and high-resolution particle-tracking techniques revealed that the observed dipole characteristics (separatrix aspect ratio, cross-sectional vorticity distribution and vorticity versus streamfunction relationship) are in close agreement with those of the supersmooth f-plane solution for the entire lifespan of the dipolar vortex.     

Effects of flexible bottom on radiation of water waves by a sphere submerged beneath an ice-cover

Meccanica - A particular hydro-elastic model is considered to examine a radiation problem involving an immersed sphere in an infinitely extended ice-covered sea, where the lower surface is...     

Effect of turbulent viscosity on the formation and motion of bottom waves

The problem of linear perturbations of the sandy bottom in a rectangular channel with a heavy incompressible fluid is formulated. The turbulent viscosity of the flow is defined as a drag coefficient function, and the hydrodynamic equations are written in the long-wave Boussinesq approximation. In the expression for the hydrostatic pressure, a correction is applied to the Boussinesq approximation that changes the sediment discharge. The problem of the development of bottom perturbations is solved taking into account the modified formula of sediment discharge, resulting in analytical expressions for the velocity of bottom perturbations and the wavelength of the fastest-growing bottom perturbations at small Froude numbers.     

Evolution of a random inhomogeneous field of nonlinear deep-water gravity waves

We have derived an equation governing the evolution of a random field of nonlinear, deep-water, gravity waves by extending the approach used by Zakharov [1] for describing the deterministic system. This equation accounts for both the effects of inhomogeneity and the energy transfer mechanism associated with the homogeneous spectrum. The narrow-band limit of this equation is used to study the stability of a random wavetrain to two-dimensional deterministic perturbations. The effect of randomness is found to reduce the growth rate and the extent of the instability.