Integrals with a large parameter: water waves on finite depth due to an impulse

1995 ◽  
Vol 450 (1938) ◽  
pp. 67-87 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Stationary Phase ◽  
Water Waves ◽  
Three Dimensional ◽  
Finite Depth ◽  
Three Dimensions ◽  
Dimensional Problem ◽  
Airy Functions ◽  
Asymptotic Results ◽  
Conventional Solution

The classical Cauchy–Poisson problem, of water waves generated by an impulsive disturbance on the free surface, is treated in three dimensions for finite constant depth. The conventional solution is in the form of a Fourier-Bessel transform. We wish to find its asymptotic behaviour at large distances r and large times t . Difficulties arise at the wavefront, where r/t is equal to the maximum group velocity. In the analogous two-dimensional problem the waves near the front are associated with two nearly coincident points of stationary phase and described asymptotically by Airy functions. In the three-dimensional problem the solution is expressed as a double integral, where there are four nearly coincident points of stationary phase. A systematic procedure is given for successive terms in an asymptotic expansion, involving the square of an Airy function and its derivatives. In both two and three dimensions it is important to use an appropriate transform of the variable(s) of integration, to achieve uniform validity of the asymptotic expansion. Calculations are performed to illustrate the utility of the asymptotic results.

THREE-DIMENSIONAL WAVE-FREE POTENTIALS IN THE THEORY OF WATER WAVES

2013 ◽  
Vol 55 (2) ◽  
pp. 175-195 ◽  
Author(s):  
HARPREET DHILLON ◽  
B. N. MANDAL
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Surface Condition ◽  
Three Dimensional ◽  
Wave Interaction ◽  
Finite Depth ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Thin Elastic Plate ◽  
Linearized Theory

AbstractProblems of wave interaction with a body with arbitrary shape floating or submerged in water are of immense importance in the literature on the linearized theory of water waves. Wave-free potentials are used to construct solutions to these problems involving bodies with circular geometry, such as a submerged or half-immersed long horizontal circular cylinder (in two dimensions) or sphere (in three dimensions). These are singular solutions of Laplace’s equation satisfying the free surface condition and decaying rapidly away from the point of singularity. Wave-free potentials in two and three dimensions for infinitely deep water as well as water of uniform finite depth with a free surface are known in the literature. The method of constructing wave-free potentials in three dimensions is presented here in a systematic manner, neglecting or taking into account the effect of surface tension at the free surface or for water with an ice cover modelled as a thin elastic plate floating on the water. The forms of the wave motion at the upper surface (free surface or ice-covered surface) related to these wave-free potentials are depicted graphically in a number of figures for all the cases considered.

scholarly journals Scattering of gravity waves by a periodically structured ridge of finite extent

2019 ◽  
Vol 871 ◽  
pp. 350-376 ◽  
Author(s):  
Agnès Maurel ◽  
Kim Pham ◽  
Jean-Jacques Marigo
Keyword(s):  
Gravity Waves ◽  
Time Domain ◽  
Water Waves ◽  
Classical Result ◽  
Water Channel ◽  
Three Dimensional ◽  
Dimensional Problem ◽  
Homogenized Model ◽  
The Time Domain ◽  
Finite Extent

We study the propagation of water waves over a ridge structured at the subwavelength scale using homogenization techniques able to account for its finite extent. The calculations are conducted in the time domain considering the full three-dimensional problem to capture the effects of the evanescent field in the water channel over the structured ridge and at its boundaries. This provides an effective two-dimensional wave equation which is a classical result but also non-intuitive transmission conditions between the region of the ridge and the surrounding regions of constant immersion depth. Numerical results provide evidence that the scattering properties of a structured ridge can be strongly influenced by the evanescent fields, a fact which is accurately captured by the homogenized model.

scholarly journals Dynamics of gravity–capillary solitary waves in deep water

2012 ◽  
Vol 708 ◽  
pp. 480-501 ◽  
Author(s):  
Zhan Wang ◽  
Paul A. Milewski
Keyword(s):  
Solitary Waves ◽  
Water Waves ◽  
Periodic Structures ◽  
Time Integration ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Full Potential ◽  
Numerical Time Integration ◽  
The Stability ◽  
Nonlinear Focusing

AbstractThe dynamics of solitary gravity–capillary water waves propagating on the surface of a three-dimensional fluid domain is studied numerically. In order to accurately compute complex time-dependent solutions, we simplify the full potential flow problem by using surface variables and taking a particular cubic truncation possessing a Hamiltonian with desirable properties. This approximation agrees remarkably well with the full equations for the bifurcation curves, wave profiles and the dynamics of solitary waves for a two-dimensional fluid domain, and with higher-order truncations in three dimensions. Fully localized solitary waves are then computed in the three-dimensional problem and the stability and interaction of both line and localized solitary waves are investigated via numerical time integration of the equations. There are many solitary wave branches, indexed by their finite energy as their amplitude tends to zero. The dynamics of the solitary waves is complex, involving nonlinear focusing of wavepackets, quasi-elastic collisions, and the generation of propagating, spatially localized, time-periodic structures akin to breathers.

Diffraction of Electrons by Two and Three Mutually Orthogonal Standing Light Waves

1975 ◽  
Vol 53 (2) ◽  
pp. 157-164 ◽  
Author(s):  
F. Ehlotzky
Keyword(s):  
Electron Scattering ◽  
Light Wave ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Dimensional Problem ◽  
One Dimensional ◽  
Standing Light Wave ◽  
Light Waves ◽  
The One ◽  
Rectangular Lattices

The one-dimensional problem of electron scattering by a standing light wave, known as the Kapitza–Dirac effect, is shown to be easily extendable to two and three dimensions, thus showing all characteristics of diffraction of electrons by simple two- and three-dimensional rectangular lattices.

A contribution to the application of photoelasticity to the micromechanics of composite materials

1969 ◽  
Vol 4 (2) ◽  
pp. 88-94 ◽  
Author(s):  
D E W Stone
Keyword(s):  
Composite Materials ◽  
Three Dimensional ◽  
Maximum Amount ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Casting Technique ◽  
Intensity Measurements ◽  
Photoelastic Model ◽  
Composite Models

Photoelastic-model methods can prove advantageous for the investigation of microstresses in composite materials. Some two-dimensional investigations of this type are discussed and the extension of this work into three dimensions is considered. It is suggested that more than one approach to the three-dimensional problem may be practicable, and special attention is paid to obtaining the maximum amount of information from a sandwiched polariscope by means of light-intensity measurements. A cold-casting technique for the fabrication of composite models is also described.

scholarly journals Three-dimensional surface gravity waves of a broad bandwidth on deep water

2021 ◽  
Vol 926 ◽  
Author(s):  
Yan Li
Keyword(s):  
Deep Water ◽  
Water Waves ◽  
Order Approximation ◽  
Three Dimensional ◽  
Finite Depth ◽  
Second Order ◽  
Leading Order ◽  
Broad Bandwidth ◽  
Stokes Waves ◽  
Deep Water Waves

A new nonlinear Schrödinger equation (NLSE) is presented for ocean surface waves. Earlier derivations of NLSEs that describe the evolution of deep-water waves have been limited to a narrow bandwidth, for which the bound waves at second order in wave steepness are described in leading-order approximations. This work generalizes these earlier works to allow for deep-water waves of a broad bandwidth with large directional spreading. The new NLSE permits simple numerical implementations and can be extended in a straightforward manner in order to account for waves on water of finite depth. For the description of second-order waves, this paper proposes a semianalytical approach that can provide accurate and computationally efficient predictions. With a leading-order approximation to the new NLSE, the instability region and energy growth rate of Stokes waves are investigated. Compared with the exact results based on McLean (J. Fluid Mech., vol. 511, 1982, p. 135), predictions by the new NLSE show better agreement than by Trulsen et al. (Phys. Fluids, vol. 12, 2000, pp. 2432–2437). With numerical implementations of the new NLSE, the effects of wave directionality are investigated by examining the evolution of a directionally spread focused wave group. A downward shift of the spectral peak is observed, owing to the asymmetry in the change rate of energy in a more complex manner than that for uniform Stokes waves. Rapid oblique energy transfers near the group at linear focus are observed, likely arising from the instability of uniform Stokes waves appearing in a narrow spectrum subject to oblique sideband disturbances.

scholarly journals Wave-activity conservation laws for the three-dimensional anelastic and Boussinesq equations with a horizontally homogeneous background flow

2007 ◽  
Vol 594 ◽  
pp. 493-506 ◽  
Author(s):  
TIFFANY A. SHAW ◽  
THEODORE G. SHEPHERD
Keyword(s):  
Conservation Laws ◽  
Large Scale ◽  
Three Dimensional ◽  
Phase Speed ◽  
Wave Activity ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Dimensional Problem ◽  
Background Flow ◽  
Homogeneous Background

Wave-activity conservation laws are key to understanding wave propagation in inhomogeneous environments. Their most general formulation follows from the Hamiltonian structure of geophysical fluid dynamics. For large-scale atmospheric dynamics, the Eliassen–Palm wave activity is a well-known example and is central to theoretical analysis. On the mesoscale, while such conservation laws have been worked out in two dimensions, their application to a horizontally homogeneous background flow in three dimensions fails because of a degeneracy created by the absence of a background potential vorticity gradient. Earlier three-dimensional results based on linear WKB theory considered only Doppler-shifted gravity waves, not waves in a stratified shear flow. Consideration of a background flow depending only on altitude is motivated by the parameterization of subgrid-scales in climate models where there is an imposed separation of horizontal length and time scales, but vertical coupling within each column. Here we show how this degeneracy can be overcome and wave-activity conservation laws derived for three-dimensional disturbances to a horizontally homogeneous background flow. Explicit expressions for pseudoenergy and pseudomomentum in the anelastic and Boussinesq models are derived, and it is shown how the previously derived relations for the two-dimensional problem can be treated as a limiting case of the three-dimensional problem. The results also generalize earlier three-dimensional results in that there is no slowly varying WKB-type requirement on the background flow, and the results are extendable to finite amplitude. The relationship $A^{\cal E}\,{=}\,cA^{\cal P}$ between pseudoenergy $A^{\cal E}$ and pseudomomentum $A^{\cal P}$, where c is the horizontal phase speed in the direction of symmetry associated with $A^{\cal P}$, has important applications to gravity-wave parameterization and provides a generalized statement of the first Eliassen–Palm theorem.

Directed fluid sheets

1976 ◽  
Vol 347 (1651) ◽  
pp. 447-473 ◽  
Keyword(s):  
Water Waves ◽  
Three Dimensional ◽  
Steady Motion ◽  
Finite Depth ◽  
Free Surface Flows ◽  
Inviscid Fluid ◽  
Two Dimensional ◽  
Dimensional Theory ◽  
Inviscid Fluids ◽  
Two Dimensional Flows

This paper is concerned mainly with incompressible inviscid fluid sheets but the incompressible linearly viscous fluid sheet is also considered. Our development is based on a direct formulation using the two dimensional theory of directed media called Cosserat surfaces . The first part of the paper deals with the formulation of appropriate nonlinear equations (which may include the effects of gravity and surface tension) governing the two dimensional motion of incompressible inviscid media for two categories, namely those ( a ) for two dimensional flows confined to a plane perpendicular to a specified direction and ( b ) for propagation of fairly long waves in a stream of variable initial depth. The latter development is a generalization of an earlier direct formulation of a theory of water waves when the fixed bottom of the stream is level (Green, Laws & Naghdi 1974). In the second part of the paper, special attention is given to a demonstration of the relevance and applicability of the present direct formulation to a variety of two dimensional problems of inviscid fluid sheets. These include, among others, the steady motion of a class of two-dimensional flows in a stream of finite depth in which the bed of the stream may change from one constant level to another, the related problem of hydraulic jumps, and a class of exact solutions which characterize the main features of the time-dependent free surface flows in the three dimensional theory of incompressible inviscid fluids.

Trapping of water waves above a round sill

1983 ◽  
Vol 132 ◽  
pp. 105-118 ◽  
Author(s):  
Yuriko Renardy
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Small Amplitude ◽  
Three Dimensional ◽  
Substantial Effect ◽  
Dimensional Problem ◽  
Shallow Water Theory ◽  
Resonant Frequencies ◽  
Wave Trapping ◽  
Linear Shallow Water Theory

The three-dimensional problem of wave trapping above a submerged round sill was first analysed by Longuet-Higgins on the basis of a linear shallow-water theory. The large responses predicted by the theory were, however, not well borne out by the experiments of Barnard, Pritchard & Provis, and this has motivated a more detailed study of the problem. A full linear theory for both inviscid and weakly viscous fluid, without any shallow-water assumptions, is presented here. It reveals important limitations on the use of shallow-water theory and the reasons for them. In particular, while the qualitative features of wave trapping are similar to those of shallow-water theory, the nearly resonant frequencies differ significantly, and, since the resonances are narrow, the observed amplitudes at a given frequency differ greatly. The geometry is strongly indicative of long waves, and the dispersion relation appears quite consistent with that, but the part of the motion at wavenumbers that are not small has, despite the small amplitude, a substantial effect on the response to excitation.

Three-dimensional problem of exponentially stratified flow over bottom roughness in the case of finite depth

1990 ◽  
Vol 25 (3) ◽  
pp. 415-423
Author(s):  
K. A. Bezhanov ◽  
P. G. Zaets ◽  
A. T. Onufriev ◽  
A. M. Ter-Krikorov
Keyword(s):  
Three Dimensional ◽  
Stratified Flow ◽  
Finite Depth ◽  
Dimensional Problem
Sign in / Sign up

Export Citation Format

Share Document