Water wave propagation across a shearing current. (English) Zbl 0611.76014
An analogue of the mild-slope equation is obtained for small-amplitude water waves propagating across a shearing current. This is shown to conserve wave action flux. The results of numerical solutions of this equation are presented for both jet and shear-layer currents. These support two conjectures which have appeared in the literature.
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:
mild-slope equation; small-amplitude water waves; shearing current; wave action flux; numerical solutions; shear-layer currentsReferences:
| [1] | Mei, C.-C., The Applied Dynamics of Ocean Surface Waves (1983), Wiley: Wiley New York · Zbl 0562.76019 |
| [2] | Peregrine, D. H., Interaction of water waves and currents, Adv. Appl. Mech., 16, 9-129 (1976) · Zbl 0471.76018 |
| [3] | Radder, A. C., On the parabolic equation method for water-wave propagation, J. Fluid Mech., 95, 159-176 (1979) · Zbl 0415.76012 |
| [4] | Kirby, J. T., A general wave equation for waves over rippled beds, J. Fluid Mech., 162, 171-186 (1986) · Zbl 0596.76017 |
| [5] | Liu, P. L.-F., Wave-current interactions on slowly varying topography, J. Geophys. Res., 88, 4421-4426 (1983) |
| [6] | Evans, D. V., The transmission of deep-water waves across a vortex sheet, J. Fluid Mech., 68, 389-401 (1975) · Zbl 0324.76013 |
| [7] | Smith, J., On surface gravity waves crossing weak current jets, J. Fluid Mech., 134, 277-299 (1983) · Zbl 0541.76018 |
| [8] | McKee, W. D., Waves on a shearing current: a uniformly valid asymptotic solution, (Proc. Cambridge Phil. Soc., 75 (1974)), 295-301 · Zbl 0277.35009 |
| [9] | Meyer, R. E., Theory of water-wave refraction, Adv. Appl. Mech., 19, 53-141 (1979) · Zbl 0472.76023 |
| [10] | Kirby, J. T., Comments on ‘The effects of a jetlike current on gravity waves in shallow water’, J. Phys. Oceanography, 16, 395-397 (1986) |
| [11] | McKee, W. D., A two turning-point problem in fluid mechanics, (Math. Proc. Cambridge Phil. Soc., 77 (1975)), 581-590 · Zbl 0314.76013 |
| [12] | Morse, P. M.; Feshbach, H., Methods of Theoretical Physics (1953), McGraw-Hill: McGraw-Hill New York · Zbl 0051.40603 |
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