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.

Long waves in a uniform channel of arbitrary cross-section

1968 ◽  
Vol 32 (2) ◽  
pp. 353-365 ◽  
Author(s):  
D. H. Peregrine
Keyword(s):  
Solitary Wave ◽  
Cross Section ◽  
Gravity Waves ◽  
Equations Of Motion ◽  
Rectangular Channel ◽  
Arbitrary Cross Section ◽  
Long Waves ◽  
Order Of Approximation ◽  
Two Dimensional ◽  
Uniform Channel

Equations of motion are derived for long gravity waves in a straight uniform channel. The cross-section of the channel may be of any shape provided that it does not have gently sloping banks and it is not very wide compared with its depth. The equations may be reduced to those for two-dimensional motion such as occurs in a rectangular channel. The order of approximation in these equations is sufficient to give the solitary wave as a solution.

The initial-value problem for long waves of finite amplitude

1964 ◽  
Vol 20 (1) ◽  
pp. 161-170 ◽  
Author(s):  
Robert R. Long
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Integration ◽  
Initial Value Problem ◽  
Finite Amplitude ◽  
Long Waves ◽  
Initial Value ◽  
Partial Differential ◽  
Long Wave ◽  
Constant Slope

Derived herein is a set of partial differential equations governing the propagation of an arbitrary, long-wave disturbance of small, but finite amplitude. The equations reduce to that of Boussinesq (1872) when the assumption is made that the disturbance is propagating in one direction only. The equations are hyperbolic with characteristic curves of constant slope. The initial-value problem can be solved very readily by numerical integration along characteristics. A few examples are included.

On an asymptotic solution of the Korteweg–de Vries equation with slowly varying coefficients

1973 ◽  
Vol 60 (4) ◽  
pp. 813-824 ◽  
Author(s):  
R. S. Johnson
Keyword(s):  
Solitary Wave ◽  
Asymptotic Solution ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Variable Depth ◽  
Exponential Form ◽  
Varying Coefficients ◽  
Numerical Integrations ◽  
De Vries ◽  
Depth Water

The variable-coefficient Korteweg–de Vries equation \[ H_X + {\textstyle\frac{3}{2}}d^{-\frac{7}{4}}HH_{\xi} + {\textstyle\frac{1}{6}}\kappa d^{\frac{1}{2}}H_{\xi\xi\xi} = 0 \] with d = d(εX) is discussed for solitary-wave initial profiles. A straightforward asymptotic solution for ε → 0 is constructed and is shown to be non-uniform both ahead of and behind the solitary wave. The behaviour ahead is rectified by matching to the appropriate exponential form and, together with the use of conservation laws for the equation, the nature of the solution behind the solitary wave is discussed. This leads to the formulation of the solution in the oscillatory ‘tail’, which is again matched directly.The results are applied to the development of the solitary wave into variable-depth water, and the predictions are compared with those obtained, for example, by Grimshaw (1970, 1971). Finally, the asymptotic behaviour of both the solitary wave and the oscillatory tail are assessed in the light of some numerical integrations of the equation.

Long waves on a beach

1967 ◽  
Vol 27 (4) ◽  
pp. 815-827 ◽  
Author(s):  
D. H. Peregrine
Keyword(s):  
Solitary Wave ◽  
Small Amplitude ◽  
Wave Equations ◽  
Equations Of Motion ◽  
Boussinesq Equations ◽  
Long Waves ◽  
Constant Depth ◽  
Reflected Wave ◽  
Long Wave ◽  
Non Linear

Equations of motion are derived for long waves in water of varying depth. The equations are for small amplitude waves, but do include non-linear terms. They correspond to the Boussinesq equations for water of constant depth. Solutions have been calculated numerically for a solitary wave on a beach of uniform slope. These solutions include a reflected wave, which is also derived analytically by using the linearized long-wave equations.

The solitary wave in water of variable depth. Part 2

1971 ◽  
Vol 46 (3) ◽  
pp. 611-622 ◽  
Author(s):  
R. Grimshaw
Keyword(s):  
Solitary Wave ◽  
Differential Equations ◽  
Bottom Topography ◽  
Experimental Results ◽  
Slow Variation ◽  
Variable Depth

This paper examines the deformation of a solitary wave due to a slow variation of the bottom topography. Differential equations which determine the slow variation of the parameters of a solitary wave are derived by a certain averaging process applied to the exact in viscid equations. The equations for the parameters are solved when the bottom topography varies only in one direction, and when the wave evolves from a region of uniform depth. The variation of amplitude with depth is determined and compared with some recent experimental results.

Water Waves in a Nonhomogeneous Incompressible Fluid

1977 ◽  
Vol 44 (4) ◽  
pp. 523-528 ◽  
Author(s):  
A. E. Green ◽  
P. M. Naghdi
Keyword(s):  
Surface Tension ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Incompressible Fluid ◽  
Water Waves ◽  
Nonlinear Differential Equations ◽  
Long Waves ◽  
Direct Approach ◽  
Hydraulic Jumps ◽  
Two Dimensional

After a brief discussion of some undesirable features of a number of different partial differential equations often employed in the existing literature on water waves, a relatively simple restricted theory is constructed by a direct approach which is particularly suited for applications to problems of fluid sheets. The rest of the paper is concerned with a derivation of a system of nonlinear differential equations (which may include the effects of gravity and surface tension) governing the two-dimensional motion of incompressible in-viscid fluids for propagation of fairly long waves in a nonhomogeneous stream of water of variable initial depth, as well as some new results pertaining to hydraulic jumps. The latter includes an additional class of possible solutions not noted previously.

Supercritical Solutions for the Buoyancy Boundary Layer

1978 ◽  
Vol 100 (4) ◽  
pp. 648-652 ◽  
Author(s):  
P. A. Iyer ◽  
R. E. Kelly
Keyword(s):  
Boundary Layer ◽  
Nusselt Number ◽  
Prandtl Number ◽  
Wall Temperature ◽  
Wave Amplitude ◽  
Reynolds Numbers ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Wave Solutions ◽  
Nonlinear Wave Solutions

Two-dimensional, finite amplitude solutions for the Prandtl buoyancy boundary layer are obtained for Reynolds numbers close to the critical values by an expansion in terms of disturbance amplitude. Supercritical nonlinear wave solutions are found to occur, rather than subcritical instabilities. The dependence of the wave amplitude and Nusselt number upon wall inclination, relative wall temperature, and Prandtl number is discussed.

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