Madsen, P. A.; Schäffer, H. A. Higher-order Boussinesq-type equations for surface gravity waves: Derivation and analysis. (English) Zbl 0930.35137 Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 356, No. 1749, 3123-3184 (1998). The authors derive Boussinesq-type equations of higher-order in dispersion (small parameter \(\mu\)) as well as the nonlinearity (small parameter \(\varepsilon\)) for waves over an uneven bottom. Formulations are given in terms of various velocity variables such as the depth-averaged velocity and particle velocity at the still water level, and at an arbitrary vertical location. The solutions are represented as power series in \(\mu\) and \(\varepsilon\). The authors study various truncations of these series (at \(O(\mu^6)\), \(O(\varepsilon^2\mu^4)\), \(O(\mu^4, \varepsilon\mu^4)\), \(O(\mu^4,\varepsilon^5 \mu^4)\) and \(O(\mu^2,\varepsilon^4 \mu^2)\)), and analyze the original and enhanced equations with emphasis on linear dispersion, shoaling and nonlinear properties at large wavenumbers. Reviewer: O.Titow (Berlin) Cited in 58 Documents MSC: 35Q35 PDEs in connection with fluid mechanics 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction Keywords:nonlinear random waves; wave-current interactions; triad interactions; nonlinear dispersive waves; power series solution; depth-averaged velocity; particle velocity; truncations; shoaling PDFBibTeX XMLCite \textit{P. A. Madsen} and \textit{H. A. Schäffer}, Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 356, No. 1749, 3123--3184 (1998; Zbl 0930.35137) Full Text: DOI