Nonlinear, Dispersive Long Waves in Water of Variable Depth.

1996 ◽  
Author(s):  
James T. Kirby
Keyword(s):  
Long Waves ◽  
Variable Depth

Long waves induced by short-wave groups over an uneven bottom

1984 ◽  
Vol 139 ◽  
pp. 219-235 ◽  
Author(s):  
Chiang C. Mei ◽  
Chakib Benmoussa
Keyword(s):  
Group Velocity ◽  
Short Wave ◽  
Long Waves ◽  
Finite Region ◽  
Variable Depth ◽  
Wave Groups ◽  
One Dimensional ◽  
Uneven Bottom ◽  
Short Waves ◽  
Horizontal Bottom

Unidirectional and periodically modulated short waves on a horizontal or very nearly horizontal bottom are known to be accompanied by long waves which propagate together with the envelope of the short waves at their group velocity. However, for variable depth with a horizontal lengthscale which is not too great compared with the group length, long waves of another kind are further induced. If the variation of depth is only one-dimensional and localized in a finite region, then the additional long waves can radiate away from this region, in directions which differ from those of the short waves and their envelopes. There are also critical depths which define caustics for these new long waves but not for the short waves. Thus, while obliquely incident short waves can pass over a topography, these second-order long waves may be trapped on a ridge or away from a canyon.

scholarly journals The tidal oscillation in an elliptic basin of variable depth-II

1936 ◽  
Vol 155 (884) ◽  
pp. 12-32 ◽  
Author(s):  
George Ridsdale Goldsbrough
Keyword(s):  
Infinite Series ◽  
Elliptic Cylinder ◽  
Long Waves ◽  
Analogous Problem ◽  
Algebraic Polynomials ◽  
Variable Depth ◽  
Finite Sets ◽  
Tidal Oscillation ◽  
The Subject ◽  
Arithmetical Calculation

The problem of the long waves in an elliptic lake with a paraboloidal law of depth was solved in a previous paper.* It appeared that the solutions could be expressed in terms of certain algebraic polynomials, from whose general properties the character of the motions could be readily derived. The subject of the present paper is the more important problem of the same basin subjected to rotation. The analogous problem of a rotating elliptic lake of uniform depth has been solved by Goldstein who used infinite series of elliptic cylinder functions. The law of depth used in the present paper, however, enables the solutions to be expressed in terms of finite sets of polynomials. The earlier modes can be completely determined without recourse to long arithmetical calculation and the interpretation of the analysis is easier. In the course of the work many properties of the polynomial are investigated.

scholarly journals The tidal oscillations in an elliptic basin of variable depth

1930 ◽  
Vol 130 (812) ◽  
pp. 157-167 ◽  
Keyword(s):  
Bessel Function ◽  
Simple Formula ◽  
Elliptic Cylinder ◽  
Long Waves ◽  
The Other ◽  
Transcendental Function ◽  
Algebraic Polynomials ◽  
Variable Depth ◽  
Numerical Approximations

The problem of the "long" waves in a circular basin of uniform depth involves in its solution a transcendental function–the Bessel function, and the determination of the free periods requires a knowledge of the zeros of this function or an allied function. On the other hand, when the basin, still circular, has a certain variable depth, it was shown by Lamb that the solution is expressed in terms of simple algebraic polynomials and the free periods of oscillation are expressed by an extremely simple formula. In similar fashion, the solution of the problem of the "long" waves in an elliptic basin of uniform depth involves the use of elliptic cylinder functions, and the free periods are only obtained as the result of lengthy numerical approximations.

The solitary wave in water of variable depth

1970 ◽  
Vol 42 (3) ◽  
pp. 639-656 ◽  
Author(s):  
R. Grimshaw
Keyword(s):  
Solitary Wave ◽  
Differential Equations ◽  
Asymptotic Solution ◽  
Wave Amplitude ◽  
Finite Amplitude ◽  
Long Waves ◽  
Two Dimensional ◽  
Variable Depth ◽  
Constant Depth

Equations are derived for two-dimensional long waves of small, but finite, amplitude in water of variable depth, analogous to those derived by Boussinesq for water of constant depth. When the depth is slowly varying compared to the length of the wave, an asymptotic solution of these equations is obtained which describes a slowly varying solitary wave; also differential equations for the slow variations of the parameters describing the solitary wave are derived, and solved in the case when the solitary wave evolves from a region of uniform depth. For small amplitudes it is found that the wave amplitude varies inversely as the depth.

A new form of Boussinesq equations for long waves in water of non-uniform depth

2012 ◽  
Vol 60 (3) ◽  
pp. 631-643
Author(s):  
J.K. Szmidt ◽  
B. Hedzielski
Keyword(s):  
Differential Equations ◽  
Order Approximation ◽  
Fluid Velocity ◽  
Power Series Expansion ◽  
Periodic Problem ◽  
Long Waves ◽  
Variable Depth ◽  
Governing Equations ◽  
Non Linear ◽  
New Form

Abstract The paper describes the non-linear transformation of long waves in shallow water of variable depth. Governing equations of the problem are derived under the assumption that the non-viscous fluid is incompressible and the fluid flow is a rotation free. A new form of Boussinesq-type equations is derived employing a power series expansion of the fluid velocity components with respect to the water depth. These non-linear partial differential equations correspond to the conservation of mass and momentum. In order to find the dispersion characteristic of the description, a linear approximation of these equations is derived. A second order approximation of the governing equations is applied to study a time dependent transformation of waves in a rectangular basin of water of variable depth. Such a case corresponds to a spatially periodic problem of sea waves approaching a near-shore zone. In order to overcome difficulties in integrating these equations, the finite difference method is applied to transform them into a set of non-linear ordinary differential equations with respect to the time variable. This final set of these equations is integrated numerically by employing the fourth order Runge - Kutta method.

Sign in / Sign up

Export Citation Format

Share Document