Finite-Amplitude, Long Waves on Liquid Films Flowing Down a Plane

1970 ◽  
Vol 9 (1) ◽  
pp. 107-113 ◽  
Author(s):  
W. B. Krantz ◽  
S. L. Goren
Keyword(s):  
Liquid Films ◽  
Finite Amplitude ◽  
Long Waves

Propagation of long waves of finite amplitude at the interface of two viscous fluids

1985 ◽  
Vol 11 (4) ◽  
pp. 481-502 ◽  
Author(s):  
G. Ooms ◽  
A. Segal ◽  
S.Y. Cheung ◽  
R.V.A. Oliemans
Keyword(s):  
Finite Amplitude ◽  
Viscous Fluids ◽  
Long Waves

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.

Model equations for long waves in nonlinear dispersive systems

1972 ◽  
Vol 272 (1220) ◽  
pp. 47-78 ◽  
Keyword(s):  
Initial Value Problem ◽  
Finite Amplitude ◽  
Long Waves ◽  
Existence Theory ◽  
Classical Solutions ◽  
Initial Value ◽  
Main Result Theorem ◽  
General Terms ◽  
Model Equations ◽  
The Right

Several topics are studied concerning mathematical models for the unidirectional propagation of long waves in systems that manifest nonlinear and dispersive effects of a particular but common kind. Most of the new material presented relates to the initial-value problem for the equation u t + u x + u u x − u x x t = 0 , ( a ) , whose solution u ( x,t ) is considered in a class of real nonperiodic functions defined for ࢤ∞ < x < ∞, t ≥0. As an approximation derived for moderately long waves of small but finite amplitude in particular physical systems, this equation has the same formal justification as the Korteweg-de Vries equation u t + u x + u u x − u x x x = 0 , ( b ) with which ( a ) is to be compared in various ways. It is contended that ( a ) is in important respects the preferable model, obviating certain problematical aspects of ( b ) and generally having more expedient mathematical properties. The paper divides into two parts where respectively the emphasis is on descriptive and on rigorous mathematics In §2 the origins and immediate properties of equations ( a ) and ( b ) are discussed in general terms, and the comparative shortcomings of ( b ) are reviewed. In the remainder of the paper (§§ 3,4) - which can be read independently Preceding discussion _ an exact theory of ( a ) is developed. In § 3 the existence of classical solutions is proved: and following our main result, theorem 1, several extensions and sidelights are presented. In § 4 solutions are shown to be unique, to depend continuously on their initial values, and also to depend continuously on forcing functions added to the right-hand side of ( a ). Thus the initial-value problem is confirmed to be classically well set in the Hadamard sense. In appendix 1 a generalization of ( a ) is considered, in which dispersive effects within a wide class are represented by an abstract pseudo-differential operator. The physical origins of such an equation are explained in the style of § 2, two examples are given deriving from definite physical problems, and an existence theory is outlined. In appendix 2 a technical fact used in § 3 is established.

scholarly journals The modulation of short gravity waves by long waves or currents

1988 ◽  
Vol 29 (4) ◽  
pp. 410-429 ◽  
Author(s):  
R. Grimshaw
Keyword(s):  
Gravity Waves ◽  
Finite Amplitude ◽  
Basic Flow ◽  
Long Waves ◽  
Water Current ◽  
Kinematic Equation ◽  
Local Dispersion ◽  
Short Waves ◽  
First Case ◽  
A Current

AbstractThe modulation of short gravity waves by long waves or currents is described for the situation when the flow is irrotational and when the short waves are described by linearised equations. Two cases are distinguished depending on whether the basic flow can be characterised as a deep-water current, or a shallow-water current. In both cases the basic flow has a current which has finite amplitude, while in the first case the free surface slope of the basic flow can be finite, but in the second case is small. The modulation equations are the local dispersion relation of the short waves, the kinematic equation for conservation of wave crests and the wave action equation. The results incorporate and extend the earlier work of Longuet-Higgins and Stewart [10, 11].

The evolution of time-periodic long waves of finite amplitude

1970 ◽  
Vol 44 (1) ◽  
pp. 195-208 ◽  
Author(s):  
O. S. Madsen ◽  
C. C. Mei ◽  
R. P. Savage
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Finite Amplitude ◽  
Long Waves ◽  
Shallow Water Waves ◽  
Wave Maker ◽  
Steep Waves ◽  
Time Periodic ◽  
Moderate Magnitude ◽  
Approximate Equations

The breakdown of shallow water waves into forms exhibiting several secondary crests is analyzed by numerical computations based on approximate equations accounting for the effects of non-linearity and dispersion. From detailed results of two cases it is shown that when long waves are such that the parameter σ = ν*L*2/h*3 is of moderate magnitude, either due to initially steep waves generated at a wave-maker or due to forced amplification by decreasing depth, waves periodic in time do not remain simply periodic in space. Numerical results are compared with experiments for waves propagating past a slope and onto a shelf.

scholarly journals Approximate symmetries and similarity solutions for wave equations on liquid films

2020 ◽  
pp. 18-18
Author(s):  
Sameerah Jamal ◽  
Andronikos Paliathanasis
Keyword(s):  
Conservation Laws ◽  
Small Amplitude ◽  
Wave Equations ◽  
Liquid Films ◽  
Similarity Solutions ◽  
Lie Symmetries ◽  
Long Waves ◽  
Approximate Symmetries ◽  
Approximate Similarity ◽  
Perturbation Parameters

We study the exact and approximate Lie symmetries for two equations which describe long waves with small amplitude on liquid films. Specifically, we study the 1+2 Benney-Luke and the 1+1 Benney-Lin equations, both from an exact and approximate perspective. To induce approximate symmetries, we show that terms involving derivatives higher than two are necessarily selected as the perturbation parameters. We construct conservation laws for both equations, and illustrate how the approximate point symmetries can be used to determine approximate similarity solutions.

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