Part C | 27.1
602 Part C Coastal Design
angle parabolic propagation [27.24] and interaction
effects among triads of longshore wavenumbers [27.22,
25]. Several models also contain a dissipation mechanism representing spectral wave breaking, consisting
of a lumped parameter breaking description [27.7] coupled with an adjustable distribution of dissipation over
the frequency range. The distribution is an adjustable
balance between a frequency-independent weighting
of dissipation and one that is weighted as frequency
squared. The latter distribution was shown by Kaihatu
et al. [27.26] to be an inner surf zone asymptote of the
dissipation dependence on frequency. Smoothed versions of these phase-resolving models that treat triad
interactions in terms of averaged bispectral quantities [27.22, 25, 27] have also been developed; these
models are closer akin to the phase-averaged spectral
models discussed previously.
27.1.3 Depth-Integrated
and Boussinesq-Type Approaches
While frequency domain models are advantageous due
to their relatively small computational demand, a significant effort in the nearshore wave model community towards developing phase-resolving, time domain
Boussinesq models has occurred in the past decade. Assuming that both nonlinearity and frequency dispersion
are weak and are of the same order of magnitude, Peregrine [27.28] derived the standard Boussinesq equations for variable depth in terms of depth-averaged
velocity and free surface displacement. Numerical results based on the standard Boussinesq equations or the
equivalent formulations have been shown to give predictions that compared quite well with field data [27.29]
and laboratory data [27.30, 31].
As mentioned above, the standard Boussinesq equations are derived based on an assumption between the
nonlinearity of the wave, " D a=h, and the frequency
dispersion of the wave, D kh. The precise relation of
these parameters comes from a nondimensionalization
of the full potential flow equations using a shallow water scaling and is given by
O ../ D O
2
1 ;
which is the true Boussinesq assumption as related to
nonlinear long surface waves. With this assumption,
nonlinear two-dimensional-vertical potential flow can
be reduced to the one-dimensional 1-D-horizontal equation set for a wave over constant water depth as
continuity: Á t C .Áu/ x C hu x D 0 ;
momentum: u t C gÁ x C uu x
1
3
hu xxt D 0 ;
where the subscripts represent partial derivatives and
u is the depth-averaged horizontal velocity. The nondispersive, shallow-water wave equations are easily
extracted from this equation set by neglecting the last
term (
1
3
hu xxt / on the left-hand side of the momentum
equation; it is this single term that adds dispersion to
the shallow water base.
The mathematical effect of the additional dispersive
term can be quantified by examining the linear form of
the standard Boussinesq equations given above and substituting in the linear wave solution form
Á D Á o e
iÂ
; u D u o e
iÂ
; Â D kx !t :
After some algebra, the dispersion relation of this approximate equation set can be expressed as
!
2
D
ghk
2
1 C
1
3
.kh/
2
;
which is also the Œ0; 2 Pade approximation of the full
linear dispersion relation given in the Linear Wave Approaches above (Sect. 27.1.1). In the practical sense,
this single additional term in the momentum equation
allows for accurate linear propagation of waves up to
kh 0:5, which is approximately a fivefold increase in
applicability compared to the shallow water model, using the same accuracy thresholds in phase and group
velocity [27.32].
As it is required that both frequency dispersion
and nonlinear effects are weak and of the same order,
the standard Boussinesq equations are not applicable
to very shallow water depth, where the nonlinearity
becomes more important than the frequency dispersion, nor to the deep water depth, where the frequency
dispersion is of order 1. The standard Boussinesq equations break down when the depth is greater than
1=10 of the wavelength. For many engineering applications, where the incident wave energy spectrum
consists of many frequency components, a lesser depth
restriction is desirable. To extend the applications to
shorter waves (or deeper water depth) many modified
forms of Boussinesq-type equations have been introduced [27.22, 33, 34]. Although the methods of derivation are different, the resulting dispersion relations of
the linear components of these modified Boussinesq
equations are similar, and may be viewed as a slight
modification of the .2; 2/ Pade approximation of the
full dispersion relation for linear water waves [27.35].
It has been demonstrated that the modified Boussinesq
equations are able to simulate wave propagation from
intermediate water depth (the water depth to wavelength ratio is about 0.5) to shallow water including the
wave–current interaction [27.36].
602 Part C Coastal Design
angle parabolic propagation [27.24] and interaction
effects among triads of longshore wavenumbers [27.22,
25]. Several models also contain a dissipation mechanism representing spectral wave breaking, consisting
of a lumped parameter breaking description [27.7] coupled with an adjustable distribution of dissipation over
the frequency range. The distribution is an adjustable
balance between a frequency-independent weighting
of dissipation and one that is weighted as frequency
squared. The latter distribution was shown by Kaihatu
et al. [27.26] to be an inner surf zone asymptote of the
dissipation dependence on frequency. Smoothed versions of these phase-resolving models that treat triad
interactions in terms of averaged bispectral quantities [27.22, 25, 27] have also been developed; these
models are closer akin to the phase-averaged spectral
models discussed previously.
27.1.3 Depth-Integrated
and Boussinesq-Type Approaches
While frequency domain models are advantageous due
to their relatively small computational demand, a significant effort in the nearshore wave model community towards developing phase-resolving, time domain
Boussinesq models has occurred in the past decade. Assuming that both nonlinearity and frequency dispersion
are weak and are of the same order of magnitude, Peregrine [27.28] derived the standard Boussinesq equations for variable depth in terms of depth-averaged
velocity and free surface displacement. Numerical results based on the standard Boussinesq equations or the
equivalent formulations have been shown to give predictions that compared quite well with field data [27.29]
and laboratory data [27.30, 31].
As mentioned above, the standard Boussinesq equations are derived based on an assumption between the
nonlinearity of the wave, " D a=h, and the frequency
dispersion of the wave, D kh. The precise relation of
these parameters comes from a nondimensionalization
of the full potential flow equations using a shallow water scaling and is given by
O ../ D O
2
1 ;
which is the true Boussinesq assumption as related to
nonlinear long surface waves. With this assumption,
nonlinear two-dimensional-vertical potential flow can
be reduced to the one-dimensional 1-D-horizontal equation set for a wave over constant water depth as
continuity: Á t C .Áu/ x C hu x D 0 ;
momentum: u t C gÁ x C uu x
1
3
hu xxt D 0 ;
where the subscripts represent partial derivatives and
u is the depth-averaged horizontal velocity. The nondispersive, shallow-water wave equations are easily
extracted from this equation set by neglecting the last
term (
1
3
hu xxt / on the left-hand side of the momentum
equation; it is this single term that adds dispersion to
the shallow water base.
The mathematical effect of the additional dispersive
term can be quantified by examining the linear form of
the standard Boussinesq equations given above and substituting in the linear wave solution form
Á D Á o e
iÂ
; u D u o e
iÂ
; Â D kx !t :
After some algebra, the dispersion relation of this approximate equation set can be expressed as
!
2
D
ghk
2
1 C
1
3
.kh/
2
;
which is also the Œ0; 2 Pade approximation of the full
linear dispersion relation given in the Linear Wave Approaches above (Sect. 27.1.1). In the practical sense,
this single additional term in the momentum equation
allows for accurate linear propagation of waves up to
kh 0:5, which is approximately a fivefold increase in
applicability compared to the shallow water model, using the same accuracy thresholds in phase and group
velocity [27.32].
As it is required that both frequency dispersion
and nonlinear effects are weak and of the same order,
the standard Boussinesq equations are not applicable
to very shallow water depth, where the nonlinearity
becomes more important than the frequency dispersion, nor to the deep water depth, where the frequency
dispersion is of order 1. The standard Boussinesq equations break down when the depth is greater than
1=10 of the wavelength. For many engineering applications, where the incident wave energy spectrum
consists of many frequency components, a lesser depth
restriction is desirable. To extend the applications to
shorter waves (or deeper water depth) many modified
forms of Boussinesq-type equations have been introduced [27.22, 33, 34]. Although the methods of derivation are different, the resulting dispersion relations of
the linear components of these modified Boussinesq
equations are similar, and may be viewed as a slight
modification of the .2; 2/ Pade approximation of the
full dispersion relation for linear water waves [27.35].
It has been demonstrated that the modified Boussinesq
equations are able to simulate wave propagation from
intermediate water depth (the water depth to wavelength ratio is about 0.5) to shallow water including the
wave–current interaction [27.36].