A practical alternative to the “mild-slope” wave equation

Abstract The “mild-slope” equation which describes wave propagation in shoaling water is normally expressed in an elliptic form. The resulting computational effort involved in the solution of the boundary value problem renders the method suitable only for small sea areas. The parabolic approximation to this equation considerably reduces the computation involved but must omit the reflected wave. Hence this method is not suited to the modelling of harbour systems or areas near to sea walls where reflections are considerable. This paper expresses the “mild-slope” equation in the form of a pair of first-order equations, which constitute a hyperbolic system, without the loss of the reflected wave. A finite-difference numerical scheme is described for the efficient solution of the equations which includes boundaries of arbitrary reflecting power.

[1]  Yoshiyuki Ito,et al.  A METHOD OF NUMERICAL ANALYSIS OF WAVE PROPAGATION APPLICATION TO WAVE DIFFRACTION AND REFRACTION , 1972 .

[2]  N. Booij,et al.  A note on the accuracy of the mild-slope equation , 1983 .

[3]  J. N. Newman.  Propagation of water waves past long two-dimensional obstacles , 1965, Journal of Fluid Mechanics.

[4]  P. Liu,et al.  On weak reflection of water waves , 1983, Journal of Fluid Mechanics.

[5]  Joseph B. Keller,et al.  Surface waves on water of non-uniform depth , 1958, Journal of Fluid Mechanics.

[6]  A. C. Radder,et al.  Verification of numerical wave propagation models for simple harmonic linear water waves , 1982 .

[7]  J. N. Newman.  Propagation of water waves over an infinite step , 1965, Journal of Fluid Mechanics.

[8]  A. C. Radder,et al.  On the parabolic equation method for water-wave propagation , 1978, Journal of Fluid Mechanics.

[9]  C. K. Thornhill,et al.  Part I. The diffraction theory of sea waves and the shelter afforded by breakwaters , 1952, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[10]  S. C. Lowell.  The propagation of waves in shallow water , 1949 .

[11]  N. Booij,et al.  Gravity waves on water with non-uniform depth and current , 1981 .

[12]  P. Liu,et al.  Numerical solution of water‐wave refraction and diffraction problems in the parabolic approximation , 1982 .

[13]  J. Keller,et al.  Geometrical theory of diffraction. , 1962, Journal of the Optical Society of America.

[14]  Robert A. Dalrymple,et al.  Propagation of obliquely incident water waves over a trench , 1983, Journal of Fluid Mechanics.

[15]  K. Friedrichs.  Waves on a shallow sloping beach , 1948 .

[16]  Robert M. Ayer,et al.  Wave propagation over a rectangular trench , 1981, Journal of Fluid Mechanics.

[17]  Philip Holmes,et al.  WAVE REFRACTION AND DIFFRACTION IN A CAUSTIC REGION: A NUMERICAL SOLUTION AND EXPERIMENTAL VALIDATION. , 1980 .

[18]  R. E. Meyer,et al.  Leakage and response of waves trapped by round islands , 1976 .

[19]  H. Barlow.  Surface Waves , 1958, Proceedings of the IRE.

[20]  A. Majda,et al.  Absorbing boundary conditions for the numerical simulation of waves , 1977 .

[21]  A. Sommerfeld.  Mathematische Theorie der Diffraction , 1896 .

[22]  O. C. Zienkiewicz,et al.  Diffraction and refraction of surface waves using finite and infinite elements , 1977 .

[23]  James R. Houston,et al.  Combined refraction and diffraction of short waves using the finite element method , 1981 .

[24]  Philip L.-F. Liu,et al.  A finite element model for wave refraction and diffraction , 1983 .

[25]  R. E. Meyer.  Theory of Water-Wave Refraction , 1979 .