Boussinesq-type equations are capable of providing accurate description of wave evolution in coastal regions. The onset of recent developments in the field of Boussinesq models was triggered by two events. The first was the increasing availability of the computer resources needed to run the models. The second was the development of variants of the theory which could be optimized to obtain better dispersion properties at larger values, thus allowing the model to treat a larger range of water depths.
The earliest depth-averaged model that included both weakly dispersive and nonlinear effects was derived by Boussinesq in 1871. The equations were derived for horizontal bottoms only. Later, Mei and LeMehaute (1966) and Peregrine (1967) derived Boussinesq equations for variable depth. Mei and LeMehaute (1966) used the velocity at the bottom as the dependent variable, whereas Peregrine (1967) used the depth-averaged velocity.
The past decade saw the advent and widespread applications of Boussinesq-type equation models for studying wave propagation. Due to the wide popularity in the coastal engineering community, the equations derived by Peregrine (1967) are often referred to as the standard Boussinesq equations, which are able to describe the nonlinear transformation of irregular and multidirectional waves in shallow water. The vertical velocity is assumed to vary linearly over the depth.
The Boussinesq equations include the lowest-order effects of frequency dispersion and nonlinearity. They can thus account for the transfer of energy between different frequency components, changes in the shape of the individual waves and the evolution of wave groups, in the shoaling irregular wave train (e.g., Freilich and Guza 1984).
However, the standard Boussinesq equations (Peregrine, 1967) had two major limitations that they are only applicable to relatively shallow water depths:
(1) The depth-averaged model poorly described the frequency dispersion of wave propagation in intermediate depths.
(2) the weakly nonlinear assumption limited the largest wave height that could accurately be modeled.
To keep errors in the phase velocity less than 5%, the water depth has to be less than about one-fifth of the equivalent deep-water wavelength (McCowan 1987).
Recently, a number of attempts have been made to extend the range of applicability of the equations to deeper water by improving the dispersion characteristic of the equations. Witting (1984) used a different form of the exact, fully nonlinear, depth-integrated momentum equation for one horizontal dimension, expressed in terms of the velocity at free surface. A Taylor-series-type expansion was used to relate the different velocity variables in the governing equations, with the coefficients of the expansion determined to yield the best linear dispersion characteristics. By retaining terms up to the fourth order in dispersion, Witting obtained relatively accurate result for both deep and shallow water waves. However, the expansions presented by Witting are only valid in water of constant depth.
Murray (1989) and Madsen et al. (1991) examined the dispersion properties of various forms of the Boussinesq equations as well as Witting’s (1984) Pade approximation of the linear dispersion relation for Airy waves. Based on the excellent characteristic of Pade approximant, the writers have introduced an additional third-order term to the momentum equation to improve the dispersion properties of the Boussinesq equations. The third-order is derived from the long wave equations and reduces to zero in shallow water, resulting in the standard form of the equations for shallow water. The equations assume a constant water depth and, thus, are not applicable to shoaling waves.
On the other hand, by defining the dependent variable as the velocity at an arbitrary depth, Nwogu (1993) achieved a rational polynomial approximation to the exact linear dispersion relationship without the need to add higher order terms to the equations. Although the arbitrary location could be chosen to give a Pade approximation to the linear dispersion relationship, Nwogu (1993) chose an alternative value which minimized the error in the linear phase speed over certain depth range.
Both Madsen et al’s and Nwogu’s expressions give more accurate dispersion relation in intermediate water depths than the standard Boussinesq equations do. They have shown by examples that the extended equations are able to simulate wave propagation from much deeper water to shallow water. Of these two models, the Boussinesq equations derived by Nwogu (1993) are obtained through a consistent derivation from the continuity and Euler equations of motion.
Since Nwogu’s extended Boussinesq equations were proposed, it quickly became one of the most cited works in the field of water wave modeling by using Boussinesq equations. The popularity of Nwogu’s equations among coastal modelers is due to the relatively simple form of the equations as well as the balanced ability of representing wave nonlinearity and dispersion.
Liu (1994) and Wei et al. (1995) presented a set of highly nonlinear Boussinesq-type equations that not only can be applied to intermediate water depth but also are capable of simulating wave propagation with strong nonlinear interaction. Wei et al. (1995) have also developed a high-order numerical scheme to solve these equations.
Although the higher-order Boussinesq equations for the improvement of the nonlinear and dispersive properties in water waves have been attempted and successful in certain aspects, most of them involved numerous additional derivatives and hence made the accurate numerical solution increasingly difficult to obtain. For this reason, Nwogu’s extended Boussinesq equations remain to be a good choice for coastal wave modeling, especially for engineering applications.
References can be found here.