Abstract
The reflection and transmission coefficients arising from the scattering of linear water waves by a one-dimensional topography are known to possess certain symmetry properties. In this paper it is shown that the same relations hold in the mild-slope approximation to the full linear theory. These relations are used in the development of a decomposition method where solutions for relatively simple depth profiles may be combined to give solutions for more complicated ones. The use of the decomposition method provides explicit error bounds in cases where they were previously unavailable. Examples are given, including the case where the depth profile consists of a sequence of an arbitrary number of identical, equally spaced humps. An example of how the decomposition may be applied to Kirby's extended mild-slope equation (for which the symmetry relations are also valid) is presented in the case of a ripple bed.
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References
J. C. W. Berkhoff, Computation of combined refraction-diffraction. Proc. 13th Int. Conf. on Coastal Eng., July 1972, Vancouver, Canada. ASCE, New York, N. Y. (1973) 471–490.
J. C. W. Berkhoff, Mathematical models for simple harmonic linear water waves. Delft Hydr. Rep. W 154-IV,.
C. Lozano and R. E. Meyer, Leakage and response of waves trapped by round islands. The Physics of Fluids 19 (1976) 1075–1088.
R. Smith and T. Sprinks, Scattering of surface waves by a conical island. Journal of Fluid Mechanics 72 (1975) 373–384.
J. Miles, Variational approximations for gravity waves in water of variable depth. Journal of Fluid Mechanics 232 (1991) 681–688.
C. Eckart, Surface waves on water of variable depth. Wave Report No. 100. Marine Physical Laboratory of the Scripps Institution of Oceanography, University of California (1951).
C. Eckart, The propagation of gravity waves from deep to shallow water. In: Gravity Waves: Proc. NBS Semicentennial Symp. on Gravity Waves, NBS, June 15–18, 1951, Washington: National Bureau of Standards 156–175.
P. G. Chamberlain, Wave scattering over uneven depth using the mild-slope equation. Wave Motion 17 (1993) 267–285.
D. Porter and D. S. G. Stirling, Integral Equations. Cambridge University Press (1990).
D. J. Staziker, Private Communication (1993).
J. N. Newman, Propagation of water waves past long two-dimensional obstacles. Journal of Fluid Mechanics 23 (1965) 23–29.
J. N. Newman, The interaction of stationary vessels with regular waves. Proc. 11th Symp. on Naval Hydrodynamics, London 1976, I. Mech E. (1977) 491–507.
D. V. Evans, The application of a new source potential tothe problem of the transmission of water waves over a shelf of arbitrary profile. Proc. Camb. Phil. Soc. 71 (1972) 391–410.
A. G. Davies and A. D. Heathershaw, Surface-wave propagation over sinusoidally varying topography. Journal of Fluid Mechanics 144 (1984) 419–443.
N. Booij, A note on the accuracy of the Mild-Slope equation. Coastal Engineering 7 (1983) 191–203.
V. Faber, D. L. Seth and G. M. Wing, Invariant imbedding in two dimensions. Proc. in Honor of G. M. Wing's 65th Birthday, Marcel Dekker, Inc. publishers (1989).
G. M. Phillips and P. J. Taylor, Theory and Applications of numerical Analysis. Academic Press (1973) 243 et seq.
J. T. Kirby, A general wave equation for waves over rippled beds. Journal of Fluid Mechanics 162 (1986) 171–186.
T. J. O'Hare and A. G. Davies, A comparison of two models for surface-wave propagation over rapidly varying topography. Applied Ocean Research 15 (1993) 1–11.
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Chamberlain, P.G. Symmetry relations and decomposition for the mild-slope equation. J Eng Math 29, 121–140 (1995). https://doi.org/10.1007/BF00051739
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DOI: https://doi.org/10.1007/BF00051739