(e) The incompressible assumption (a) above implies that there is another mathematical function termed

the *stream function *Ψ. Some wave theories are formulated in terms of the stream function Ψ, which is

orthogonal to the potential function Φ. Lines of constant values of the potential function (equipotential lines)

and lines of constant values of the stream function are mutually perpendicular or orthogonal. Consequently,

if Φ is known, Ψ can be found, or vice versa, using the equations

MΦ

MΨ

(II-1-3)

'

M*x*

M*z*

MΦ

MΨ

(II-1-4)

'&

M*z*

M*x*

termed the *Cauchy-Riemann conditions *(Whitham 1974; Milne-Thompson 1976). Both Φ and Ψ satisfy the

the assumptions outlined above, the Laplace equation governs the flow beneath waves. The Laplace equation

in two dimensions with x = horizontal, and z = vertical axes in terms of velocity potential Φ is given by

M2Φ

M2Φ

'0

(II-1-5)

%

2

2

M*x*

M*z*

(f) In terms of the stream function, Ψ, Laplace's equation becomes

M2Ψ

M2Ψ

(II-1-6)

'0

%

2

2

M*x*

M*z*

(g) The linear theory formulation is usually developed in terms of the potential function, Φ.

In applying the seventh assumption to waves in water of varying depth (encountered when waves approach

a beach), the local depth is usually used. This can be justified, but not without difficulty, for most practical

cases in which the bottom slope is flatter than about 1 on 10. A progressive wave moving into shallow water

will change its shape significantly. Effects due to the wave transformations are addressed in Parts II-3 and

II-4.

(h) The most fundamental description of a simple sinusoidal oscillatory wave is by its length *L *(the

horizontal distance between corresponding points on two successive waves), height *H *(the vertical distance

to its crest from the preceding trough), period *T *(the time for two successive crests to pass a given point), and

depth *d *(the distance from the bed to SWL).

(i) Figure II-1-1 shows a two-dimensional, simple progressive wave propagating in the positive x-

direction, using the symbols presented above. The symbol *η *denotes the displacement of the water surface

relative to the SWL and is a function of *x *and time *t*. At the wave crest, *η *is equal to the amplitude of the

wave *a*, or one-half the wave height *H/2*.

(2) Wave celerity, length, and period.

(a) The speed at which a wave form propagates is termed the *phase velocity *or *wave celerity C*. Since

the distance traveled by a wave during one wave period is equal to one wavelength, wave celerity can be

related to the wave period and length by

II-1-6

Water Wave Mechanics

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