Liu, Philip L.-F.; Yoon, Sung B.; Kirby, James T. Nonlinear refraction-diffraction of waves in shallow water. (English) Zbl 0593.76028 J. Fluid Mech. 153, 185-201 (1985). Summary: The parabolic approximation is developed to study the combined refraction/diffraction of weakly nonlinear shallow-water waves. Two methods of approach are used. In the first method Boussinesq equations are used to derive evolution equations for spectral-wave components in a slowly varying two-dimensional domain. The second method modifies the K-P equation [B. B. Kadomtsev and V. I. Petviashvili, Dokl. Akad. Nauk SSSR 192, 753-756 (1970; Zbl 0217.250)] to include varying depth in two dimensions. Comparisons are made between present numerical results, experimental data and previous numerical calculations. Cited in 10 Documents MSC: 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 35Q99 Partial differential equations of mathematical physics and other areas of application 76M99 Basic methods in fluid mechanics Keywords:parabolic approximation; combined refraction/diffraction; weakly nonlinear shallow-water waves; Boussinesq equations; evolution equations for spectral-wave components; slowly varying two-dimensional domain Citations:Zbl 0217.250 PDFBibTeX XMLCite \textit{P. L. F. Liu} et al., J. Fluid Mech. 153, 185--201 (1985; Zbl 0593.76028) Full Text: DOI References: [1] DOI: 10.1017/S0022112080000481 · Zbl 0437.76017 · doi:10.1017/S0022112080000481 [2] Whalin, Res. Rep. 87 pp 7932– (1971) [3] Liu, J. Geophys. Res. 88 pp 4421– (1983) [4] Kirby, J. Fluid Mech. 136 pp 453– (1983) [5] Kirby, J. Geophys. Res. 89 pp 745– (1984) [6] Kadomtsev, Sov. Phys. Dokl. 15 pp 539– (1970) [7] DOI: 10.1017/S0022112072000540 · Zbl 0242.76008 · doi:10.1017/S0022112072000540 [8] Freilich, Phil. Trans. R. Soc. Lond. A311 pp 1– (1984) [9] Dalrymple, J. Waterway, Port, Coastal and Ocean Engineering, ASCE 110 pp 67– (1984) [10] Bryant, J. Fluid Mech. 115 pp 525– (1982) [11] Tsay, J. Geophys. Res. 87 pp 7932– (1982) [12] DOI: 10.1016/0378-3839(77)90006-0 · doi:10.1016/0378-3839(77)90006-0 [13] Skovgaard, J. Waterways, Port, Harbors and Coastal Engng Div., ASCE 101 pp 15– (1975) [14] DOI: 10.1017/S0022112078002037 · doi:10.1017/S0022112078002037 [15] DOI: 10.1017/S0022112079001397 · Zbl 0415.76012 · doi:10.1017/S0022112079001397 [16] DOI: 10.1016/0378-3839(84)90023-1 · doi:10.1016/0378-3839(84)90023-1 [17] DOI: 10.1017/S0022112080001887 · Zbl 0463.76022 · doi:10.1017/S0022112080001887 [18] Liu, J. Fluid Mech. 141 pp 265– (1984) [19] Liu, J. Waterway, Port, Coastal and Ocean Engrg. Div. ASCE 131 pp 59– (1984) [20] Liu, J. Fluid Mech. 131 pp 59– (1983) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.