Gravity waves in a circular well

2002 ◽  
Vol 460 ◽  
pp. 177-180
Author(s):  
JOHN MILES
Keyword(s):  
Free Surface ◽  
Half Space ◽  
Gravity Waves ◽  
Natural Frequencies ◽  
Single Mode ◽  
Two Dimensional ◽  
Characteristic Determinant ◽  
Parallel Argument ◽  
Single Mode Approximation ◽  
Infinite Reservoir

The natural frequencies of gravity waves in a circular well that is bounded above by a free surface and below by a semi-infinite reservoir are approximated by neglecting the off-diagonal terms of the characteristic determinant (single-mode approximation) and invoking the known results for an aperture in a half-space (well of zero depth). A parallel argument yields the corresponding results for a two-dimensional well (a slot). Comparison with Molin's (2001) numerical results for a slot suggests that the error in the single-mode approximation is [lsim ] 1%.

scholarly journals New exact relations for steady irrotational two-dimensional gravity and capillary surface waves

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170220 ◽  
Author(s):  
Didier Clamond
Keyword(s):  
Free Surface ◽  
Gravity Waves ◽  
Water Waves ◽  
Two Dimensional ◽  
Physical Plane ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Dimensional Gravity ◽  
Irrotational Motion ◽  
Exact Relations

Steady two-dimensional surface capillary–gravity waves in irrotational motion are considered on constant depth. By exploiting the holomorphic properties in the physical plane and introducing some transformations of the boundary conditions at the free surface, new exact relations and equations for the free surface only are derived. In particular, a physical plane counterpart of the Babenko equation is obtained. This article is part of the theme issue ‘Nonlinear water waves’.

Stationary gravity waves on non-uniform free streams: jet-like streams

1975 ◽  
Vol 77 (2) ◽  
pp. 415-438 ◽  
Author(s):  
D. H. Peregrine ◽  
Ronald Smith
Keyword(s):  
Free Surface ◽  
Gravity Waves ◽  
Small Amplitude ◽  
Wave Number ◽  
Explicit Solutions ◽  
Stationary Waves ◽  
Two Dimensional ◽  
Large Wave ◽  
Surface Gravity Waves ◽  
Large Wave Number

AbstractThe basic state considered in this paper is a parallel flow of a jet-like character with the centre of the jet being at or near a free surface which is horizontal. Stationary surface gravity waves may exist on such a flow, and a number of examples are looked at for small amplitude waves. Explicit solutions are given for ‘top-hat’ profile jets and for two-dimensional flows. Asymptotic solutions are developed for stationary waves of large wave-number.

scholarly journals On bounds and non-existence in the problem of steady waves with vorticity

2015 ◽  
Vol 765 ◽  
Author(s):  
V. Kozlov ◽  
N. Kuznetsov ◽  
E. Lokharu
Keyword(s):  
Free Surface ◽  
Gravity Waves ◽  
Surface Profile ◽  
Finite Depth ◽  
Two Dimensional ◽  
Vorticity Distribution ◽  
Unidirectional Flow ◽  
Wave Solutions ◽  
Free Surface Profile ◽  
Parallel Shear Flow

AbstractFor the problem describing steady gravity waves with vorticity on a two-dimensional unidirectional flow of finite depth the following results are obtained. (i) Bounds are found for the free-surface profile and for Bernoulli’s constant. (ii) If only one parallel shear flow exists for a given value of Bernoulli’s constant, then there are no wave solutions provided the vorticity distribution is subject to a certain condition.

scholarly journals A two-dimensional numerical and experimental study of piston and sloshing resonance in moonpools with recess

2019 ◽  
Vol 877 ◽  
pp. 142-166 ◽  
Author(s):  
Senthuran Ravinthrakumar ◽  
Trygve Kristiansen ◽  
Bernard Molin ◽  
Babak Ommani
Keyword(s):  
Free Surface ◽  
Numerical Methods ◽  
Eigenfunction Expansion ◽  
Natural Frequencies ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Surface Conditions ◽  
Body Boundary ◽  
Sloshing Mode ◽  
The Mean

The piston and first sloshing modes of two-dimensional moonpools with recess are investigated. Dedicated forced heave experiments are carried out. Different recess lengths are tested from $1/4$ to $1/2$ of the length of the moonpool at the mean waterline. A theoretical model to calculate the natural frequencies is developed based on linearized potential flow theory and eigenfunction expansion. Two numerical methods are implemented: a boundary element method (BEM) and a Navier–Stokes solver (CFD). Both the BEM and CFD have linearized free-surface and body-boundary conditions. As expected, the BEM over-predicts the moonpool response significantly, in particular at the first sloshing mode. The CFD is in general able to predict the maximum moonpool response adequately, both at the piston and first sloshing modes. Both numerical methods fail to predict the Duffing-type behaviour at the first sloshing mode, due to the linearized free-surface conditions. The Duffing behaviour is more pronounced for the largest recess. The main source of damping in the proximity of the first sloshing mode is discussed.

Two-dimensional linear and nonlinear wave propagation in a half-space

1999 ◽  
Vol 89 (4) ◽  
pp. 903-917 ◽  
Author(s):  
Heming Xu ◽  
Steven M. Day ◽  
Jean-Bernard H. Minster
Keyword(s):  
Wave Propagation ◽  
Free Surface ◽  
Phase Velocity ◽  
Rayleigh Wave ◽  
Half Space ◽  
Frequency Band ◽  
Pseudospectral Method ◽  
Two Dimensional ◽  
Wave Phase ◽  
Wave Phase Velocity

Abstract We examine a staggered pseudospectral method to solve a two-dimensional wave propagation problem with arbitrary nonlinear constitutive equations, and evaluate a general image method to simulate the traction-free boundary condition at the surface. This implementation employs a stress-velocity formulation and satisfies the free surface condition by explicitly setting surface shear stress to zero and making the normal stress antisymmetric about the free surface. Satisfactory agreement with analytical solutions to Lamb's problem is achieved for both vertical point force and explosion sources, and with perturbation solutions for nonlinearly elastic wave propagation within the domain of validity of such solutions. The Rayleigh wave, however, suffers much more severe numerical dispersion than do body waves. At four grids per wavelength, the relative error in the Rayleigh-wave phase velocity is 25 times greater than the corresponding error in the body-wave phase velocity. Thus for the Rayleigh wave, the pseudospectral method performs no better than a low-order finite difference method. A substantial merit of the image approach is that it does not assume any particular rheology, the method is readily applicable even when stresses are not analytically related to kinematic variables, as is the case for most nonlinear models. We use this scheme to investigate the response of a nonlinear half-space with endochronic rheology, which has been fit to quasi-static and dynamic observations. We find that harmonics of a monochromatic source are generated and evolve with epicentral range, and energy is transferred from low to higher frequencies for a broadband source. This energy redistribution characteristic of the propagation is strain-amplitude dependent, consistent with laboratory experiments. Compared with the linear response, the nonlinear response of an endochronic layer near the surface shows a deamplification effect in the intermediate-frequency band and an amplification effect in the higher-frequency band. The computational method, with modifications to accommodate realistic nonlinear soil characteristics, could be applied to estimate earthquake strong ground motions and path effects.

Theoretical seismograms for the two welded quarter-planes

1972 ◽  
Vol 62 (2) ◽  
pp. 619-630
Author(s):  
Dan Loewenthal ◽  
Z. Alterman
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Half Space ◽  
Line Source ◽  
Two Dimensional ◽  
Impulsive Force ◽  
Space Model ◽  
Finite Difference Technique

abstract The finite difference technique developed by Alterman and Rotenberg (1969) and Alterman and Loewenthal (1970) is applied to the problem of obtaining the motion of a two-dimensional half-space model, consisting of two homogeneous and isotropic welded quarter-spaces of different materials. Two source conditions are considered: an internal impulsive line source acting inside one of the quarter-planes, and an impulsive force acting perpendicular to the free surface.

scholarly journals No steady water waves of small amplitude are supported by a shear flow with a still free surface

2013 ◽  
Vol 717 ◽  
pp. 523-534 ◽  
Author(s):  
Vladimir Kozlov ◽  
Nikolay Kuznetsov
Keyword(s):  
Free Surface ◽  
Shear Flow ◽  
Gravity Waves ◽  
Water Waves ◽  
Small Amplitude ◽  
Finite Depth ◽  
Coordinate Frame ◽  
Two Dimensional ◽  
Horizontal Shear ◽  
Positive Coefficients

AbstractThe two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. It is proved that no small-amplitude waves are supported by a horizontal shear flow whose free surface is still, that is, it is stagnant in a coordinate frame such that the flow is time-independent in it. The class of vorticity distributions for which such flows exist includes all positive constant distributions, as well as linear and quadratic ones with arbitrary positive coefficients.

A Single-Mode Approximation for Gravity Waves in the Thermosphere

2021 ◽  
Author(s):  
Harold Knight ◽  
David Broutman ◽  
Stephen Eckermann
Keyword(s):  
Gravity Waves ◽  
Single Mode ◽  
Single Mode Approximation
Sign in / Sign up

Export Citation Format

Share Document