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By J. D Ramboz

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Example text

The pressure variable p introduced here, therefore measures the deviation from the hydrostatic pressure distribution; we shall find that p ^ 0 during the passage of a wave. 57) D and S + | + S =o - (i 58) - These equations are written exclusively in terms of nondimensional variables, where 8 = ho/k is the long wavelength or shallowness parameter; we shall have much to write about 8 later. 56), but with the transformations on JC and y replaced by r^Xr. 60) D _ d d_ v d and 1 3 . , 1 3t; 3w all expressed in nondimensional variables; 8 is defined exactly as above: 8 = ho/X.

29), at least in the suitably approximate forms that we usually encounter. Now we turn to the extension of this dynamic condition (for an inviscid fluid) which accommodates the effects of surface tension (which supports a pressure difference across a curved surface). 30) The boundary conditions for water waves 17 where \/R is the mean curvature 1-1 A R /q K2' and K\9 K2 are the principal radii of curvature. 3. 30), is usually called Laplace's formula and, in general, T varies with temperature; here we shall treat F as a constant.

The form of this expansion is governed by the way in which the parameter appears in the equation and, perhaps, also how it appears in the boundary/initial conditions. Usually, a rather simple iterative construction will suggest how this expansion proceeds. In order to explain and describe how these ideas are relevant in theories of wave propagation (and, therefore, to our study of water waves), we consider the partial differential equation Utt ~ uxx = e(u2 + uxx)xx. 94) The small parameter, £, in this equation (which here represents the characteristics of both small amplitude and long waves) suggests that we seek a solution in the form oo u(x, f; e) ~ ] T *""«(*.

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