(1) Wave height is limited by both depth and wavelength. For a given water depth and wave period,

there is a maximum height limit above which the wave becomes unstable and breaks. This upper limit of

wave height, called *breaking wave height*, is in deep water a function of the wavelength. In shallow and

transitional water it is a function of both depth and wavelength. Wave breaking is a complex phenomenon

and it is one of the areas in wave mechanics that has been investigated extensively both experimentally and

numerically.

(2) Researchers have made some progress over the last three decades in the numerical modeling of

waves close to breaking (Longuet-Higgins and Fenton 1974; Longuet-Higgins 1974; 1976; Schwartz 1974;

Dalrymple and Dean 1975; Byatt-Smith and Longuet-Higgins 1976; Peregrine 1976; Cokelet 1977; Longuet-

Higgins and Fox 1977; Longuet-Higgins 1985; Williams 1981; 1985). These studies suggest the limiting

wave steepness to be *H/L *= 0.141 in deep water and *H/d *= 0.83 for solitary waves in shallow water with a

corresponding solitary wave celerity of *c/(gd)*1/2 = 1.29.

(3) Dalrymple and Dean (1975) investigated the maximum wave height in the presence of a steady uni-

form current using the stream function theory. Figure II-1-19 shows the influence of a uniform current on

the maximum wave height where *T*c is the wave period in a fixed reference frame and *U *is the current speed.

(4) The treatment of wave breaking in the propagation of waves is discussed in Part II-3. Information

about wave breaking in deep and shoaling water and its relation to nearshore processes is provided in

Part II-4.

(1) To ensure their proper use, the range of validity for various wave theories described in this chapter

must be established. Very high-order Stokes theories provide a reference against which the accuracy of

various theories may be tested. Nonlinear wave theories better describe mass transport, wave breaking,

shoaling, reflection, transmission, and other nonlinear characteristics. Therefore, the usage of the linear

theory has to be carefully evaluated for final design estimates in coastal practice. It is often imperative in

coastal projects to use nonlinear wave theories.

(2) Wave amplitude and period may sometimes be estimated from empirical data. When data are

lacking or inadequate, uncertainty in wave height and period estimates can give rise to a greater uncertainty

in the ultimate answer than does neglecting the effect of nonlinear processes. The additional effort necessary

for using nonlinear theories may not be justified when large uncertainties exist in the wave data used for

design. Otherwise, nonlinear wave theories usually provide safer and more accurate estimates.

(3) Dean (1968, 1974) presented an analysis by defining the regions of validity of wave theories in

terms of parameters *H/T*2 and *d/T*2 since *T*2 is proportional to the wavelength. Le Mhaut (1976) presented

a slightly different analysis (Figure II-1-20) to illustrate the approximate limits of validity for several wave

theories, including the third- and fourth-order theories of Stokes. In Figure II-1-20, the fourth-order Stokes

theory may be replaced with more popular fifth-order theory, since the latter is often used in applications.

Both Le Mhaut and Dean recommend cnoidal theory for shallow-water waves of low steepness, and Stokes'

higher order theories for steep waves in deep water. Linear theory is recommended for small steepness *H/T*2

and small *U*R values. For low steepness waves in transitional and deep water, linear theory is adequate but

other wave theories may also be used in this region. Fenton's theory is appropriate for most of the wave

parameter domain. For given values of *H*, *d*, and *T*, Figure II-1-20 should be used as a guide to select an

appropriate wave theory.

II-1-56

Water Wave Mechanics

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