CHAPTER 3
WAVE MODEL
3.1 Governing Wave Equation
Severed methods have been used to derive the governing equation for the propagation
of waves over varying topography and varying currents. Among the methods used are per
turbation techniques, multiple scales, a Lagrangian and the use of Greens second identity.
With the assumption of a mild slope and weak current conditions each of these methods
should yield the same equation. For illustrative purposes the method of using Greens sec
ond identity will be used here to derive a linear equation. A Lagrangian derivation of the
same equation is presented in Appendix C. The linear equation is used since second order
waves exhibit singularities as the water depth goes to zero.
A velocity potential t is assumed such that the water particle velocities are given by
V, where V is the three-dimensional gradient operator
d d d -
(3.1)
and *,j, and k are the unit vectors in the x,y, and z directions, respectively. The velocities
are a superposition of steady currents and wave induced motion. The velocity potential is
given by
t = to + eft (3.2)
where to is defined such that its gradient yields U,V, and W which are the steady current
terms, and ft is the wave component of the velocity potential. An undefined scaling e
is used so as to separate the wave and steady current part of the velocity potential. The
wave part of the velocity potential is composed of two parts: /, which contains the depth
dependency and which is a function of the horizontal coordinates and time. The depth
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