The second order equation corresponding

to the dispersion relation ω2=D(k)ω2=D(k) can be written as ∂t2η=−Dηhere DD is a pseudo-differential operator when D   is not a polynomial, but in all cases it is uniquely Docetaxel cost defined by multiplication in Fourier space as Dη(x)=^D(k)η^(k)From the nonnegativity and evenness of D  , the operator DD will be a symmetric, positive definite operator. Defining the positive root of D  , as the function ΩΩ Ω(k)=D(k)introduce the odd function Ω1(k)=sign(k)Ω(k)Ω1(k)=sign(k)Ω(k)Then the wave expi(kx−Ω1(k)t)=expik(x−C(k)t) is for all values of k   to the right travelling with positive phase speed C(k)=Ω1(k)/kC(k)=Ω1(k)/k; similarly expi(kx+Ω1(k)t) is to the left travelling with speed −C(k)−C(k). By defining the corresponding skew symmetric operator A1=^iΩ1(k)the operator DD can be factorized as D=A1⁎A1=−(A1)2The second order in time equation is then factorized as (∂t2+D)η=(∂t−A1)(∂t+A1)ηThe first order in time operators describe

to the right and left travelling waves, selleck chemicals llc which are precisely the solutions of the uni-directional equations (∂t+A1)ηr=0,(∂t−A1)ηℓ=0for which the dispersion relations are ω=Ω1(k)ω=Ω1(k) and ω=−Ω1(k)ω=−Ω1(k) respectively. For construction of the embedded sources of the bi-directional equation, this factorization will be used. In the following we will need the property that the function D   is monotonically increasing for k>0k>0, so that Ω1(k)Ω1(k) has a unique inverse for all real k   which we will denote by K  1: ω=Ω1(k)⇔k=K1(ω).ω=Ω1(k)⇔k=K1(ω).For later reference, recall that the group velocity is the even function given by Vg(k)=dΩ1(k)dkThe

exact dispersion given above corresponds to a monotone concave function Ω1Ω1, so that the phase velocity decreases for shorter waves; this will also be a reasonable assumption for approximations that are not only meant Urease to be valid for long waves, such as the shallow water equations. Note furthermore that the scaling property of the exact dispersion relation and group velocity with depth is given by Ω1(k,h)=ghM(kh)Vg(k,h)=c0m(kh)withc0=ghrespectively, where m is the derivative of M. For reliable wave models with approximate dispersion, the same scaling properties will be satisfied, at least in a restricted interval of wave numbers. In models that are used for analytic or numerical investigations, the approximation of the exact dispersion relation will satisfy in the relevant intervals the same scaling properties. As one example we mention the Variational Boussinesq Model (VBM) described in Adytia and van Groesen (2012) and Klopman et al. (2010). In that model, the dependence of the fluid potential in the vertical direction z   is prescribed by an a priori chosen function F(z)F(z).