Summary
This study is concerned with the Green function of the theory of potential flow about a body in regular (time-harmonic) water waves in deep water, that is with the linearized velocity potential of the flow due to a source of pulsating strength at a fixed position below the free surface (or a pulsating flux across the free surface) of a quiescent infinitely deep sea. An asymptotic expansion and a convergent ascending-series expansion for the Green function are obtained from two alternative complementary ‘near-field’ and ‘far-field’ single-integral representations in terms of the exponential integral. The asymptotic expansion and the ascending series allow efficient numerical evaluation of the Green function for large and small distances, respectively, from the mirror image of the singularity (submerged source or free-surface flux) with respect to the mean sea surface.
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References
J. V. Wehausen, The motion of floating bodies, Ann. Rev. Fluid Mech. 3 (1971) 237–268.
J. N. Newman, Marine hydrodynamics, M.I.T. Press, Cambridge, Mass., USA 1977.
N. E. Kochin, The theory of waves generated by oscillations of a body under the free surface of a heavy incompressible fluid, translated in Soc. Nav. Archit. Mar. Eng. Tech. Res. Bull. No. 1–10 (1952).
T. H. Havelock, The damping of the heaving and pitching motion of a ship, Philos. Mag. 33 (1942) 666–673.
T. H. Havelock, Waves due to a floating sphere making periodic heaving oscillations, Proc. R. Soc. London A 231 (1955) 1–7.
M. D. Haskind, The oscillation of a body immersed in heavy fluid, Prikl. Mat. Mekh. 8 (1944) 287–300.
M. D. Haskind, On wave motions of a heavy fluid, Prikl. Mat. Mekh. 18 (1954) 15–26.
F. John, On the motions of floating bodies II, Commun. Pure Appl. Math. 3 (1950) 45–101.
H. C. Liu, Über die Entstehung von Ringwellen an einer Flüssigkeitsorberfläche durch unter dieser gelegene, kugelige periodische Quellensysteme, Z. Angew. Math. Mech. 32 (1952) 211–226.
R. C. Thorne, Multiple expansions in the theory of surface waves, Proc. Cambridge Philos. Soc. 49 (1953) 707–716.
R. C. MacCamy, The motion of a floating sphere in surface waves, Wave Research Lab., Univ. of Calif., Berkeley, Rep. Ser. 61 Issue 4 (1954).
J. V. Wehausen and E. V. Laitone, Surface waves, Encyclopedia of Physics, Vol. 9, Springer-Verlag Berlin (1960) 446–814.
F. Ursell, The periodic heaving motion of a half-immersed sphere, Manchester Univ., Dept. of Math., Rep. to U.S. Office of Naval Research (1963).
W. D. Kim, On the harmonic oscillations of a rigid body on a free surface, J. Fluid Mech. 21 (1965) 427–451.
R. W. Yeung, A singularity-distribution method for free-surface flow problems with an oscillating body, Univ. of Calif., Berkeley, Rep. NA 73–6 (1973).
G. E. Hearn, Alternative methods of evaluating Green's function in three-dimensional ship-wave problems, J. Ship Res. 21 (1977) 89–93.
P. Guevel & J. C. Daubisse, Oscillations d'un flotteur soumis a l'action de la houle, Lecture notes E.N. S.M. Nantes, France (1978).
D. Martin, Résolution numérique du problème linéarisé de la tenue à la mer, Trans. Assoc. Tech. Maritime Aéronautique (1980).
F. Noblesse, On the theory of flow of regular water waves about a body, Massachusetts Institute of Technology, Dept. Ocean Eng., Rep. No. 80-2 (1980).
M. J. Lighthill, On waves generated in dispersive systems by travelling forcing effects, with applications to the dynamics of rotating fluids, J. Fluid Mech. 27 (1967) 725–752.
F. Noblesse, Alternative integral representations for the Green function of the theory of ship wave resistance, J. Eng. Math. 15 (1981) 241–265.
K. W. H. Eggers, Proceedings of the International Seminar on Wave Resistance, Society of Naval Architects of Japan (1976) 399.
J. J. Stoker, Water waves, Interscience Publ., New York, 1957.
M. Abramowitz & I. A. Stegun, Handbook of mathematical functions, Dover Publ., New York, 1965.
I. S. Gradshteyn & I. W. Ryzhik, Tables of integrals, series, and products, Academic Press, New York, 1965.
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This work was sponsored by the Office of Naval Research.
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Noblesse, F. The Green function in the theory of radiation and diffraction of regular water waves by a body. J Eng Math 16, 137–169 (1982). https://doi.org/10.1007/BF00042551
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DOI: https://doi.org/10.1007/BF00042551