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Erschienen in:
2011 | OriginalPaper | Buchkapitel
1. Theory of Water Waves
verfasst von : A. J. Hermans
Erschienen in: Water Waves and Ship Hydrodynamics
Verlag: Springer Netherlands
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Abstract
This chapter contains the formulation of boundary and initial boundary value problems in water waves. The basic equations here are the Euler equations and the equation of continuity for a non-viscous incompressible fluid moving under gravity. Throughout the book, in most considerations the motion is assumed to be irrotational and hence the existence of a velocity potential function is ensured in simply connected regions. In this case the equation of continuity for the velocity of the fluid is then reduced to the familiar Laplace equation for the velocity potential function.
Water waves are created normally by the presence of a free surface along which the pressure is constant. For the irrotational motion, on the free surface one than obtains the non-linear Bernoulli equation for the velocity potential function from the Euler equation. Based on small amplitude waves, linearised problems for the velocity potential function and for the free surface elevation are formulated.
At first we follow the derivation as can be found in J.J. Stoker (Water Waves, Interscience, 1957) and J.V. Wehausen and E.V. Laitone in (Encyclopedia of Physics, vol. 9, Springer, pp. 446–814, 1960) to obtain equations for the wave potential in still water and as a superposition on a constant parallel flow potential. The coefficients in the free surface equations are constant. Then we derive linear equation for the superposition of small amplitude waves on a flow disturbed by some three dimensional object. If we consider the magnitude of the steady velocity vector to be small, we obtain for the time-dependent wave potential function a linear equation with non-constant coefficients.