The theory of symmetrical gravity waves of finite amplitude II. The solitary wave

1952 ◽  
Vol 213 (1113) ◽  
pp. 238-249 ◽  
Keyword(s):  
Free Surface ◽  
Solitary Wave ◽  
Gravity Waves ◽  
Finite Amplitude ◽  
Linear Boundary ◽  
Dimensional Problem ◽  
Qualitative Behaviour ◽  
Linear Boundary Condition ◽  
Linear Condition ◽  
Non Linear

An exact solution is obtained to a problem which differs from the two-dimensional problem of the solitary wave only in that the non-linear condition on the free surface is replaced by a slightly modified non-linear boundary condition having the same qualitative behaviour. Expressions are obtained for the wave velocity and amplitude, and also the equation of the free surface in parametric form. Breaking conditions at the crest are also investigated.

The theory of symmetrical gravity waves of finite amplitude. I

1951 ◽  
Vol 208 (1095) ◽  
pp. 475-486 ◽  
Keyword(s):  
Boundary Condition ◽  
Gravity Waves ◽  
Finite Amplitude ◽  
Breaking Wave ◽  
Two Dimensional ◽  
Linear Boundary ◽  
Dimensional Problem ◽  
Infinite Depth ◽  
Linear Boundary Condition ◽  
Small Parameters

The two-dimensional problem of symmetric finite amplitude gravity waves in an incompressible fluid of infinite depth is treated by a method which first involves satisfying a non-linear boundary condition exactly. The higher approximations are obtained by the method of small parameters. The breaking-wave conditions are discussed and expressions are given for the free-surface equation, the kinetic and the potential energies of the fluid.

Integral properties of periodic gravity waves of finite amplitude

1975 ◽  
Vol 342 (1629) ◽  
pp. 157-174 ◽  
Keyword(s):  
Solitary Wave ◽  
Gravity Waves ◽  
Deep Water ◽  
Water Waves ◽  
Wave Theory ◽  
Finite Amplitude ◽  
Cnoidal Wave ◽  
The Mean ◽  
Exact Relations ◽  
Kinetic And Potential Energies

A number of exact relations are proved for periodic water waves of finite amplitude in water of uniform depth. Thus in deep water the mean fluxes of mass, momentum and energy are shown to be equal to 2T(4T—3F) and (3T—2V) crespectively, where T and V denote the kinetic and potential energies and c is the phase velocity. Some parametric properties of the solitary wave are here generalized, and some particularly simple relations are proved for variations of the Lagrangian The integral properties of the wave are related to the constants Q, R and S which occur in cnoidal wave theory. The speed, momentum and energy of deep-water waves are calculated numerically by a method employing a new expansion parameter. With the aid of Padé approximants, convergence is obtained for waves having amplitudes up to and including the highest. For the highest wave, the computed speed and amplitude are in agreement with independent calculations by Yamada and Schwartz. At the same time the computations suggest that the speed and energy, for waves of a given length, are greatest when the height is less than the maximum. In this respect the present results tend to confirm previous computations on solitary waves.

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