in which *ε *= *ka *is the *perturbation expansion parameter*. Each term in the series is smaller than the preceding

term by a factor of order *ka*. In this expansion, *Φ*1 is the first-order theory (linear theory), *Φ*2 is the second-

order theory, and so on.

(d) Substituting Equation II-1-68 and similar expressions for other wave variables (i.e., surface elevation

η, velocities *u *and *w*, pressure *p*, etc.) into the appropriate governing equations and boundary conditions

describing the wave motion yields a series of higher-order solutions for ocean waves. Equating the

coefficients of equal powers of *ka *gives recurrence relations for each order solution. A characteristic of the

perturbation expansion is that each order theory is expressed in terms of the preceding lower order theories

(Phillips 1977; Dean and Dalrymple 1991; Mei 1991). The first-order Stokes theory is the linear (Airy)

theory.

(e) The Stokes expansion method is formally valid under the conditions that *H/d (kd)*2 for *kd < 1 *and

This condition restricts the wave heights in shallow water and the Stokes theory is not generally applicable

to shallow water. For example, the maximum wave height in shallow water allowed by the second-order

Stokes theory is about one-half of the water depth (Fenton 1985). The mathematics of higher-order Stokes

theories is cumbersome and is not presented here. See Ippen (1966) for a detailed derivation of the Stokes

second-order theory.

(f) In the higher-order Stokes solutions, superharmonic components (i.e., higher frequency components

at two, three, four, etc. times the fundamental frequency) arise. These are superposed on the fundamental

component predicted by linear theory. Hence, wave crests are steeper and troughs are flatter than the

sinusoidal profile (Figure II-1-10). The fifth-order Stokes expansion shows a secondary crest in the wave

trough for high-amplitude waves (Peregrine 1972; Fenton 1985). In addition, particle paths for Stokes waves

are no longer closed orbits and there is a *drift *or *mass transport *in the direction of wave propagation.

(g) The linear dispersion relation is still valid to second order, and both wavelength and celerity are

independent of wave height to this order. At third and higher orders, wave celerity and wavelength depend

on wave height, and therefore, for a given wave period, celerity and length are greater for higher waves.

Some limitations are imposed on the finite-amplitude Stokes theory in shallow water both by the water depth

and amplitude nonlinearities. For steeper waves in shallow water, higher-order terms in Stokes expansion

may increase in magnitude to become comparable or larger than the fundamental frequency component

(Fenton 1985; Chakrabarti 1987). When this occurs, the Stokes perturbation becomes invalid.

(h) Higher-order Stokes theories include aperiodic (i.e., not periodic) terms in the expressions for water

particle displacements. These terms arise from the product of time and a constant depending on the wave

period and depth, and give rise to a continuously increasing net particle displacement in the direction of wave

propagation. The distance a particle is displaced during one wave period when divided by the wave period

gives a mean drift velocity (z), called the *mass transport velocity*. To second-order, the mass transport

velocity is

2

π*H*

(II-1-69)

2

sinh2 (2π*d*/*L*)

indicating that there is a net transport of fluid by waves in the direction of wave propagation. If the mass

transport leads to an accumulation of mass in any region, the free surface must rise, thus generating a pressure

gradient. A current, formed in response to this pressure gradient, will reestablish the distribution of mass.

Water Wave Mechanics

II-1-33

Integrated Publishing, Inc. |