In the next section the wave generation
sources are derived for 1D uni- and bi-directional wave equations with arbitrary dispersive properties. The generalization for 2D wave equations, forward propagating or multi-directional propagating, is presented in Section 3. Section 4 describes the adjustment of embedded wave generation for strongly nonlinear cases. Simulation results will be shown in Section 5, and the paper finishes with conclusions. This section deals with embedded influxing in 1D dispersive equations; the next section shows that the basic ideas can be directly generalized to 2D multi-directional equations. After introducing notation and the factorization into uni-directional wave equations selleck chemicals llc based on the dispersion relation Bleomycin purchase that characterizes a second order in time dispersive wave equation,
it is shown in Section 2.2 that for uni-directional equations the generation source is not unique. This property is used in Section 2.3, together with a simple symmetry argument, to construct the influxing source for bi-directional waves for prescribed wave generation on each side. The wave elevation will be denoted by η(x,t)η(x,t). Both spatial and temporal Fourier transforms will be used repeatedly, with the following conventions. The spatial Fourier transformation η^(k) and the profile η(x)η(x) are related to each other by η(x)=∫η^(k)eikxdk,η^(k)=12π∫η(x)e−ikxdxTo the simplify formulas in the following, the notation =^ in expressions like η(x)=^η^(k) will be used to indicate the relation by Fourier transformation. For a signal s(t)s(t) and its temporal Fourier transform sˇ(ω) the relation is s(t)=∫sˇ(ω)e−iωtdω,sˇ(ω)=12π∫s(t)eiωtdt.The spatial–temporal Fourier transformation of η(x,t)η(x,t) will be denoted by an overbar: η¯(k,ω) η(x,t)=∬η¯(k,ω)ei(kx−ωt)dkdωWhen not indicated otherwise, integrals are taken over the whole real axis. A dispersive wave equation is determined by its dispersion relation, specifying the relation between the wave number k and the frequency ωω so that harmonic modes expi(kx−ωt) are physical solutions.
For a second order in time equation, the relation can be written as ω2=D(k)ω2=D(k)where D is a non-negative, even function. In modelling and simulating waves, the dispersion relation expresses the translation of the interior fluid motion to quantities at the surface, which implies a dimension reduction of one. Equations which model the waves with quantities in horizontal directions only are called Boussinesq-type of equations. The interior fluid motion in the layer below the free surface is then usually only approximately modelled. For linear waves, in the approximation of infinitesimal small wave heights, the exact dispersion relation Dex is given by Dex(k)=gktanh(kh)with g and h being the gravitational acceleration and depth of the fluid layer respectively.