where zα is now a function of time and is given by zα+h = 0.465(h+η). The
volume flux density uf is given by:
(h + η)
[∇(uα ⋅ ∇h) + (∇ ⋅ uα )∇h]
= (h + η) uα + ( zα + h) 
uf
2
(9)
( zα + h )2 (h + η)2
+

∇(∇ ⋅ uα )
2
6
The fully nonlinear equations are able to implicitly model the effects of wave
by explicitly specifying a current field, U.
Linear Dispersion Properties
The linear dispersion relation of the Boussinesq model that relates the
wavelength, L, to the wave period, T, is given by Nwogu (1993) as:
L2 1  (α + 1/ 3)(kh)2
C = 2 =
2
(10)
1  α(kh)
2
T
where C is the phase speed, k = 2π/L is the wave number, and α =
[(zα+h)2/h2  1]/2. Depending on the elevation of the velocity variable or the value
of α, different dispersion relations are obtained. If the velocity at the seabed
(zα = h) is used, α = 1/2. Alternatively, if the velocity at the stillwater level
(zα = 0) is used, α = 0. The dispersion relation of the classical form of the
Boussinesq equations which uses the depthaveraged velocity as the velocity
variable corresponds to α = 1/3. Witting (1984) obtained the value α = 2/5 from
the Pad (2,2) approximant of tanh kh.
The phase speeds for different values of α, normalized with respect to the
linear theory phase speed are plotted as a function of relative depth in Figure 1.
The relative depth is defined as the ratio of the water depth, h, to the equivalent
2
deepwater wavelength Lo = gT /2π. The deepwater depth limit corresponds to
h/L = 0.5. The different dispersion equations are all equivalent in relatively
shallow water (h/L < 0.02), but gradually diverge from the exact solution with
increasing depth. An optimal depth zα = 0.535h gives errors of less than 2 per
cent in the phase velocity from shallowwater depths up to the deepwater limit.
Nonlinear Properties
In an irregular sea state, different frequency components interact to generate
forced waves at the sum and difference frequencies of the primary waves because
of the nonlinear nature of the boundary condition at the free surface. Consider a
6
Chapter 2 Theoretical Background