EM 111021100 (Part II)
30 Apr 02
in which waves did not break when the slope m was greater than about 0.18 and that as the slope increased
the breaking position moved closer to the shoreline. This accounts for the large values of Hb/db for large
slopes; i.e., as db 6 0. For some conditions, Equations II197 and II198 are unsatisfactory for predicting
breaking depth. Further discussion of the breaking of waves with experimental results is provided in Part II4.
(6) Streamfunction wave theory. Numerical approximations to solutions of hydrodynamic equations
describing wave motion have been proposed and developed. Some common theories and associated equations
are listed in Table II12. The approach by Dean (1965, 1974), termed a symmetric, streamfunction theory,
is a nonlinear wave theory that is similar to higher order Stokes' theories. Both are constructed of sums of
sine or cosine functions that satisfy the original differential equation (Laplace equation). The theory,
however, determines the coefficient of each higher order term so that a best fit, in the least squares sense, is
obtained to the theoretically posed, dynamic, freesurface boundary condition. Assumptions made in the
theory are identical to those made in the development of the higher order Stokes' solutions. Consequently,
some of the same limitations are inherent in the streamfunction theory, and it represents an alternative
solution to the equations used to approximate the wave phenomena. However, the streamfunction
representation had successfully predicted the wave phenomena observed in some laboratory wave studies
(Dean and Dalrymple 1991), and thus it may possibly describe naturally occurring wave phenomena.
Table II12
Boundary Value Problem of Water Wave Theories (Dean 1968)
Exactly Satisfies
Theory
DE
BBC
KFSBC
DFSBC
Linear wave theory
X
X


Thirdorder Stokes
X
X


Fifthorder Stokes
X
X


Firstorder cnoidal

X


Secondorder cnoidal

X


Stream function
numerical wave
X
X
X

theory
DE = Differential equation.
BBC = Bottom boundary condition.
KFSBC = Kinematic free surface boundary condition.
DFSBC = Dynamic free surface boundary condition.
X = Exactly satisfies.
(7) Fourier approximation  Fenton's theory.
(a) Fenton's Fourier series theory, another theory developed in recent years (Fenton 1988), is somewhat
similar to Dean's stream function theory, but it appears to describe oceanic waves at all water depths better
than all previous similar theories.
up the higher order streamfunction theory of Dean had in the past limited its use to either tabular or graphical
presentations of the solutions. These tables, their use, and their range of validity may be found elsewhere
(Dean 1974).
(c) Stokes and cnoidal wave theories yield good approximations for waves over a wide range of depths
if highorder expansions are employed. Engineering practice has relied on the Stokes fifthorder theory
(Skjelbreia and Hendrickson 1961), and the stream function theory (Dean 1974). These theories are
II150
Water Wave Mechanics