which is termed the *relative wave height*. Like the wave steepness, large values of the relative wave height

indicate that the small-amplitude assumption may not be valid. A fourth dimensionless parameter often used

to assess the relevance of various wave theories is termed the *Ursell number*. The Ursell number is given by

2

(II-1-67)

'

(b) The value of the Ursell number is often used to select a wave theory to describe a wave with given

in shallow water that may necessitate the use of nonlinear wave theory, to be discussed next.

(c) The linear or small-amplitude wave theory described in the preceding sections provides a useful first

approximation to the wave motion. Ocean waves are generally not small in amplitude. In fact, from an

engineering point of view it is usually the large waves that are of interest since they result in the largest forces

and greatest sediment movement. In order to approach the complete solution of ocean waves more closely,

a perturbation solution using successive approximations may be developed to improve the linear theory

solution of the hydrodynamic equations for gravity waves. Each order wave theory in the perturbation

expansion serves as a correction and the net result is often a better agreement between theoretical and

observed waves. The extended theories can also describe phenomena such as *mass transport *where there is

a small net forward movement of the water during the passage of a wave. These higher-order or extended

solutions for gravity waves are often called *nonlinear wave theories*.

(d) Development of the nonlinear wave theories has evolved for a better description of surface gravity

waves. These include *cnoidal*, *solitary*, and *Stokes *theories. However, the development of a Fourier-series

approximation by Fenton in recent years has superseded the previous historical developments. Since earlier

theories are still frequently referenced, these will first be summarized in this section, but Fenton's theory is

recommended for regular waves in all coastal applications.

(2) Stokes finite-amplitude wave theory.

(a) Since the pioneering work of Stokes (1847, 1880) most extension studies (De 1955; Bretschneider

1960; Skjelbreia and Hendrickson 1961; Laitone 1960, 1962, 1965; Chappelear 1962; Fenton 1985) in wave

perturbation theory have assumed the wave slope *ka *is small where *k *is the wave number and *a *the amplitude

of the wave. The perturbation solution, developed as a power series in terms of *ε *= *ka*, is expected to

converge as more and more terms are considered in the expansion. Convergence does not occur for steep

waves unless a different perturbation parameter from that of Stokes is chosen (Schwartz 1974; Cokelet 1977;

Williams 1981, 1985).

(b) The fifth-order Stokes finite-amplitude wave theory is widely used in practical applications both in

deep- and shallow-water wave studies. A formulation of Stokes fifth-order theory with good convergence

properties has recently been provided (Fenton 1985). Fenton's fifth-order Stokes theory is computationally

efficient, and includes closed-form asymptotic expressions for both deep- and shallow-water limits.

Kinematics and pressure predictions obtained from this theory compare with laboratory and field

measurements better than other nonlinear theories.

(c) In general, the perturbation expansion for velocity potential *Φ *may be written as

Φ ' εΦ1 % ε2Φ2 % ...

(II-1-68)

II-1-32

Water Wave Mechanics

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