Fully Nonlinear Model for Water Wave Propagation from Deep to Shallow Waters

On the linear stability of one- and two-layer Boussinesq-type equations for wave propagation over uneven beds

“…Also, in the linear case, the equations K01-1 (Kennedy et al, 2001), W95-1 and N93-1 (Nwogu, 1993) reduce to the same equations, and here we will refer to them as N93-1 when dealing with linear aspects. Similarly, S13-1 , G12-1 (Galan et al, 2012) and M98-1 coincide in the linear case, and we will refer to them as M98-1.…”

“…The equations by -and hence Nwogu (1993)-, Madsen and Schaffer (1998), Kennedy et al (2001) or Galan et al (2012) are recovered as particular cases setting some of these coefficients null (Table 1).…”

“…The proper procedure to assess the linear shoaling of BTEs originally considered the so-called shoaling gradient (Madsen and Sorensen, 1992), and, in order to obtain the error for the wave Table 1 Free coefficients for the one-layer BTEs (marked with "✓"; "-" means "does not apply"). B96-1: Beji and Nadaoka (1996); S13-1: one-layer Simarro et al (2013); G12-1: Galan et al (2012); M98-1: Madsen and Schaffer (1998); K01-1: Kennedy et al (2001); W95-1: ; N93-1: Nwogu (1993). Table 2 Free coefficients for the two-layer BTEs (marked with "✓").…”

“…The BTEs by Simarro et al (2013) depart from the sets by Madsen and Schaffer (1998) and Galan et al (2012) to further improve weakly non-linear and weakly dispersive performance, and include eight free coefficients: α, α ϵ , δ, δ ϵ , δ h , γ, γ ϵ and γ h . The equations by -and hence Nwogu (1993)-, Madsen and Schaffer (1998), Kennedy et al (2001) or Galan et al (2012) are recovered as particular cases setting some of these coefficients null (Table 1).…”

“…The error is expressed as a function of κ for the same reasons argued by Galan et al (2012). It is known that ∂ω=∂k obtained from the above dispersion equation does not satisfy A 2 c g ¼ constant, with A being the wave amplitude (i.e.…”