Havelock wavemakers, Westergaard dams and the Rayleigh hypothesis
- P. A. Martin
- … show all 1 hide
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
Water of constant finite depth fills a semi-infinite channel, with a wavemaker, W, at one end. The generation of small-amplitude gravity waves by harmonic oscillations of W leads to a linear boundary-value problem for a velocity potential, ϕ. For vertical, plane wavemakers, there is a theory due to Havelock in which ϕ is represented as a convergent series of eigenfunctions, with coefficients determined by the boundary condition on W. We show that the same representation (with different coefficients) can also be used for some wavemakers with other shapes; the allowable geometries and forcings are determined. This is a hydrodynamic analogue of the so-called Rayleigh hypothesis in the theory of gratings. Similar results obtain for the hydrodynamic loading of dams due to short-duration earthquakes.
- J. Avilés and F.J. Sánchez-Sesma. Hydrodynamic pressures on dams with nonvertical upstream face. J. Engng. Mech. 112 (1986) 1054–1061.
- A.T. Chwang, Hydrodynamic pressures on sloping dams during earthquakes. Part 2. Exact theory. J. Fluid Mech. 87 (1978) 343–348.
- R.G. Dean and R.A. Dalrymple, Water Wave Mechanics for Engineers and Scientists. Englewood Cliffs: Prentice-Hall (1984).
- J.A. DeSanto, Scattering from a perfectly reflecting arbitrary periodic surface: an exact theory. Radio Sci. 16 (1981) 1315–1326.
- M. Greenhow and M. Simon, A note on the equivalent wavemaker method. Appl. Ocean Res. 7 (1985) 107–112.
- T.H. Havelock, Forced surface-waves on water. Phil. Mag. Series 7, 8 (1929) 569–576.
- N.R. Hill and V. Celli, Limits of convergence of the Rayleigh method for surface scattering. Phys. Rev. B 17 (1978) 2478–2481.
- H. Ikuno and K. Yasuura, Improved point-matching method with application to scattering from a periodic surface. IEEE Trans. AP 21 (1973) 657–662.
- F. John, On the motion of floating bodies II. Comm. Pure Appl. Math. 3 (1950) 45–101.
- P.J. Kachoyan and W.D. McKee, Wave forces on steeply-sloping sea walls. J. Engng. Math. 19 (1985) 351–362.
- B.A. Lippmann, Note on the theory of gratings. J. Opt. Soc. Amer. 43 (1953) 408.
- P.A. Martin, On the null-field equations for water-wave scattering problems. IMA J. Appl. Math. 33 (1984) 55–69.
- P.A. Martin, On the T-matrix for water-wave scattering problems. Wave Motion 7 (1985) 177–193.
- D. Maystre and M. Cadilhac, Singularities of the continuation of fields and validity of Rayleigh's hypothesis. J. Math. Phys. 26 (1985) 2201–2204.
- W.D. McKee, Wave forces on steeply-sloping sea walls: oblique incidence. J. Engng. Math. 22 (1987) 87–99.
- R.F. Millar, The Rayleigh hypothesis and a related least-squares solution to scattering problems for periodic surfaces and other scatterers. Radio Sci. 8 (1973) 785–796.
- R.F. Millar, Singularities and the Rayleigh hypothesis for solutions to the Helmholtz equation. IMA J. Appl. Math. 37 (1986) 155–171.
- N.M. Newmark and E. Rosenblueth, Fundamentals of Earthquake Engineering. Englewood Cliffs: Prentice-Hall (1971).
- Y. Okuno and K. Yasuura, Numerical algorithm based on the mode-matching method with a singular-smoothing procedure for analysing edge-type scattering problems. IEEE Trans. AP 30 (1982) 580–587.
- R. Petit (ed.), Electromagnetic Theory of Gratings, Topics in Current Physics 22. Berlin: Springer-Verlag (1980).
- D.J. Pizer, A Generalised Wavemaker Problem, M.Sc. Dissertation, University of Manchester (1985).
- F. Raichlen and J.J. Lee, An inclined-plate wave generator, Proc. 16th Coastal Engineering Conf., Hamburg (1978) 388–399.
- Lord Rayleigh (J.W. Strutt), The Theory of Sound, 2nd edn, Vol. 2. London: Macmillan & Co. (1896).
- Lord Rayleigh, On the dynamical theory of gratings. Proc. Roy. Soc. A 79 (1907) 399–416.
- F.J. Sánchez-Sesma and J. Avilés, Authors' closure to discussion on . J. Engng. Mech. 114 (1988) 1103–1106.
- W.A. Schlup, On the convergence of the Rayleigh ansatz for hard-wall scattering on arbitrary periodic surface profiles. J. Phys. A: Math. Gen. 17 (1984) 2607–2619.
- M.J. Simon and F. Ursell, Uniqueness in linearized two-dimensional water-wave problems. J. Fluid. Mech. 148 (1984) 137–154.
- F. Ursell, On the heaving motion of a cylinder on the surface of a fluid. Quart. J. Mech. Appl. Math. 2 (1949) 218–231.
- F. Ursell, R.G. Dean and Y.S. Yu, Forced small-amplitude water waves: a comparison of theory and experiment. J. Fluid Mech. 7 (1959) 33–52.
- P.M.van den Berg and J.T. Fokkema, The Rayleigh hypothesis in the theory of reflection by a grating. J. Opt. Soc. Amer. 69 (1979) 27–31.
- H.M. Westergaard, Water pressures on dams during earthquakes. Trans. ASCE 98 (1933) 418–433.
- Y.C. Wu, Waves generated by an inclined-plate wave generator. Int. J. Numer. Meth. in Fluids 8 (1988) 803–811.
- Y.C. Wu, Plunger-type wavemaker theory. J. Hydraulic Res. 26 (1988) 483–491.
- Y.C. Wu and D.J. Yu, Trefftz method for hydrodynamic pressure on rigid dams with non-vertical upstream face. Int. J. Numer. Meth. in Fluids 9 (1989) 1–7.
- Havelock wavemakers, Westergaard dams and the Rayleigh hypothesis
Journal of Engineering Mathematics
Volume 26, Issue 2 , pp 267-280
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers
- Additional Links
- Industry Sectors
- P. A. Martin (1)
- Author Affiliations
- 1. Department of Mathematics, University of Manchester, M13 9PL, Manchester, England