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Boundary integrals for oscillating bodies in stratified fluids

Published online by Cambridge University Press:  21 September 2021

Bruno Voisin Open the ORCID record for Bruno Voisin [Opens in a new window]
Bruno Voisin*
Affiliation:
Laboratoire des Écoulements Géophysiques et Industriels, Université Grenoble Alpes, CNRS, Grenoble INP, 38000Grenoble, France
*
†Email address for correspondence: [email protected]
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Abstract

The theoretical foundations of the boundary integral method are considered for inviscid monochromatic internal waves, and an analytical approach is presented for the solution of the boundary integral equation for oscillating bodies of simple shape: an elliptic cylinder in two dimensions, and a spheroid in three dimensions. The method combines the coordinate stretching introduced by Bryan and Hurley in the frequency range of evanescent waves, with analytic continuation to the range of propagating waves by Lighthill's radiation condition. Not only are the waves obtained for arbitrary oscillations of the body, with application to radial pulsations and rigid vibrations, but also the distribution of singularities equivalent to the body, allowing later inclusion of viscosity in the theory. Both a direct representation of the body as a Kirchhoff–Helmholtz integral involving single and double layers together, and an indirect representation involving a single layer alone, are considered. The indirect representation is seen to require a certain degree of symmetry of the body with respect to the horizontal and the vertical. As the surface of the body is approached the single- and double-layer potentials exhibit the same discontinuities as in classical potential theory.

JFM classification

Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Stratified flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 927 , 25 November 2021 , A3
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© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Abramowitz, M. & Stegun, I.A. 1972 Handbook of Mathematical Functions, 10th edn. Dover.Google Scholar
Allakhverdiev, K.B. & Pletner, Y.D. 1993 A fundamental solution of the two-dimensional operator of gravity-gyroscopic waves and some initial-boundary problems. Phys. Dokl. 38, 234–236.Google Scholar
Appleby, J.C. & Crighton, D.G. 1986 Non-Boussinesq effects in the diffraction of internal waves from an oscillating cylinder. Q. J. Mech. Appl. Maths 39, 209–231.CrossRefGoogle Scholar
Appleby, J.C. & Crighton, D.G. 1987 Internal gravity waves generated by oscillations of a sphere. J. Fluid Mech. 183, 439–450.CrossRefGoogle Scholar
Backus, G. & Rieutord, M. 2017 Completeness of inertial modes of an incompressible inviscid fluid in a corotating ellipsoid. Phys. Rev. E 95, 053116.CrossRefGoogle Scholar
Balmforth, N.J. & Peacock, T. 2009 Tidal conversion by supercritical topography. J. Phys. Oceanogr. 39, 1965–1974.CrossRefGoogle Scholar
Barcilon, V. & Bleistein, N. 1969 Scattering of inertial waves in a rotating fluid. Stud. Appl. Maths 48, 91–104.CrossRefGoogle Scholar
Bardakov, R.N., Vasil'ev, A.Y. & Chashechkin, Y.D. 2007 Calculation and measurement of conical beams of three-dimensional periodic internal waves excited by a vertically oscillating piston. Fluid Dyn. 42, 612–626.CrossRefGoogle Scholar
Bleistein, N. 1984 Mathematical Methods for Wave Phenomena. Academic.Google Scholar
Brebbia, C.A., Telles, J.C.F. & Wrobel, L.C. 1984 Boundary Element Techniques. Springer.CrossRefGoogle Scholar
Brouzet, C., Ermanyuk, E.V., Moulin, M., Pillet, G. & Dauxois, T. 2017 Added mass: a complex facet of tidal conversion at finite depth. J. Fluid Mech. 831, 101–127.CrossRefGoogle Scholar
Bryan, G.H. 1889 The waves on a rotating liquid spheroid of finite ellipticity. Phil. Trans. R. Soc. Lond. A 180, 187–219.Google Scholar
Cayley, A. 1870 On the geodesic lines on an oblate spheroid. Phil. Mag. 40, 329–340.CrossRefGoogle Scholar
Cheng, A.H.-D. & Cheng, D.T. 2005 Heritage and early history of the boundary element method. Engng Anal. Bound. Elem. 29, 268–302.CrossRefGoogle Scholar
Copley, L.G. 1968 Fundamental results concerning integral representations in acoustic radiation. J. Acoust. Soc. Am. 44, 28–32.CrossRefGoogle Scholar
Courant, R. & Hilbert, D. 1962 Methods of Mathematical Physics, vol. II. Wiley.Google Scholar
Crighton, D.G., Dowling, A.P., Ffowcs Williams, J.E., Heckl, M. & Leppington, F.G. 1992 Modern Methods in Analytical Acoustics. Springer.CrossRefGoogle Scholar
Davis, A.M.J. 2012 Generation of internal waves from rest: extended use of complex coordinates, for a sphere but not a disk. J. Fluid Mech. 703, 374–390.CrossRefGoogle Scholar
Davis, A.M.J. & Llewellyn Smith, S.G. 2010 Tangential oscillations of a circular disk in a viscous stratified fluid. J. Fluid Mech. 656, 342–359.CrossRefGoogle Scholar
Davydova, M.A. 2004 Asymptotic behavior of a boundary layer solution to one hydrodynamic problem. Comp. Maths Math. Phys. 44, 1038–1050.Google Scholar
Davydova, M.A. 2006 a On the dynamic potential theory for the equation of a weakly stratified fluid. Differ. Equ. 42, 537–547.CrossRefGoogle Scholar
Davydova, M.A. 2006 b Asymptotic solution of the problem on nonstationary vibrations of a continuously stratified medium with small dissipation. Differ. Equ. 42, 1623–1632.CrossRefGoogle Scholar
Davydova, M.A. & Chashechkin, Y.D. 2004 The structure of three-dimensional periodic boundary layers in a continuously stratified fluid. J. Appl. Maths Mech. 68, 391–397.CrossRefGoogle Scholar
Echeverri, P. & Peacock, T. 2010 Internal tide generation by arbitrary two-dimensional topography. J. Fluid Mech. 659, 247–266.CrossRefGoogle Scholar
Echeverri, P., Yokossi, T., Balmforth, N.J. & Peacock, T. 2011 Tidally generated internal-wave attractors between double ridges. J. Fluid Mech. 669, 354–374.CrossRefGoogle Scholar
Ermanyuk, E.V. 2002 The rule of affine similitude for the force coefficients of a body oscillating in a uniformly stratified fluid. Exp. Fluids 32, 242–251.CrossRefGoogle Scholar
Ermanyuk, E.V. & Gavrilov, N.V. 2002 Oscillations of cylinders in a linearly stratified fluid. J. Appl. Mech. Tech. Phys. 43, 503–511.CrossRefGoogle Scholar
Ermanyuk, E.V. & Gavrilov, N.V. 2003 Force on a body in a continuously stratified fluid. Part 2. Sphere. J. Fluid Mech. 494, 33–50.CrossRefGoogle Scholar
Flynn, M.R., Onu, K. & Sutherland, B.R. 2003 Internal wave excitation by a vertically oscillating sphere. J. Fluid Mech. 494, 65–93.CrossRefGoogle Scholar
Gabov, S.A. 1984 a Explicit solution and existence of a limiting amplitude for a problem in the dynamics of a stratified fluid. Sov. Maths Dokl. 30, 190–194.Google Scholar
Gabov, S.A. 1984 b The angular potential for an equation of S.L. Sobolev and its applications. Sov. Maths Dokl. 30, 405–409.Google Scholar
Gabov, S.A. 1985 The solution of a problem of stratified fluid dynamics and its stabilization as $t \to \infty$. USSR Comp. Maths Math. Phys. 25 (3), 47–55.CrossRefGoogle Scholar
Gabov, S.A. & Krutitskii, P.A. 1987 On the non-stationary Larsen problem. USSR Comp. Maths Math. Phys. 27 (4), 148–154.CrossRefGoogle Scholar
Gabov, S.A. & Krutitskii, P.A. 1989 On the small vibrations of a section placed at the boundary of separation between two stratified liquids. USSR Comp. Maths Math. Phys. 29 (2), 154–162.CrossRefGoogle Scholar
Gabov, S.A., Malysheva, G.Y. & Sveshnikov, A.G. 1983 A composite equation related to oscillations of a stratified compressible fluid. Differ. Equ. 19, 866–873.Google Scholar
Gabov, S.A., Malysheva, G.Y., Sveshnikov, A.G. & Shatov, A.K. 1984 On some equations arising in the dynamics of a rotating stratified and compressible fluid. USSR Comp. Maths Math. Phys. 24 (6), 162–170.CrossRefGoogle Scholar
Gabov, S.A. & Mamedov, K.S. 1983 The potential function and problems on oscillations of an exponentially stratified fluid. Sov. Maths Dokl. 27, 309–312.Google Scholar
Gabov, S.A. & Orazov, B.B. 1986 The equation $(\partial ^2/\partial t^2)[{u_{xx}-u}]+u_{xx} = 0$ and several problems associated with it. USSR Comp. Maths Math. Phys. 26 (1), 58–64.CrossRefGoogle Scholar
Gabov, S.A. & Pletner, Y.D. 1985 An initial-boundary value problem for the gravitational-gyroscopic wave equation. USSR Comp. Maths Math. Phys. 25 (6), 64–68.CrossRefGoogle Scholar
Gabov, S.A. & Pletner, Y.D. 1987 a The gravitational-gyroscopic wave equation: the angular potential and its applications. USSR Comp. Maths Math. Phys. 27 (1), 66–73.CrossRefGoogle Scholar
Gabov, S.A. & Pletner, Y.D. 1987 b Solvability of an exterior initial-boundary value problem for the gravitational gyroscopic wave equation. USSR Comp. Maths Math. Phys. 27 (3), 44–49.CrossRefGoogle Scholar
Gabov, S.A. & Pletner, Y.D. 1987 c On the Dirichlet problem for the equation of gravitational-gyroscopic waves. Sov. Maths Dokl. 36, 43–46.Google Scholar
Gabov, S.A. & Pletner, Y.D. 1988 The problem of the oscillations of a flat disc in a stratified liquid. USSR Comp. Maths Math. Phys. 28 (1), 41–47.CrossRefGoogle Scholar
Gabov, S.A. & Shevtsov, P.V. 1983 Basic boundary value problems for the equation of oscillations of a stratified fluid. Sov. Maths Dokl. 27, 238–241.Google Scholar
Gabov, S.A. & Shevtsov, P.V. 1984 On a differential equation of the type of Sobolev's equation. Sov. Maths Dokl. 29, 411–414.Google Scholar
Gabov, S.A. & Simakov, S.T. 1989 The theory of internal and surface waves in a stratified liquid. USSR Comp. Maths Math. Phys. 29 (1), 155–164.CrossRefGoogle Scholar
Garrett, C. & Kunze, E. 2007 Internal tide generation in the deep ocean. Annu. Rev. Fluid Mech. 39, 57–87.CrossRefGoogle Scholar
Gaul, L., Kögl, M. & Wagner, M. 2003 Boundary Element Methods for Engineers and Scientists. Springer.CrossRefGoogle Scholar
Ghaemsaidi, S.J. & Peacock, T. 2013 3D Stereoscopic PIV visualization of the axisymmetric conical internal wave field generated by an oscillating sphere. Exp. Fluids 54, 1454.CrossRefGoogle Scholar
Gibson, W.C. 2014 The Method of Moments in Electromagnetics, 2nd edn. Chapman and Hall/CRC.CrossRefGoogle Scholar
Gorodtsov, V. 2013 Small wave–vortex disturbances in stratified fluid. Procedia IUTAM 8, 111–118.CrossRefGoogle Scholar
Gorodtsov, V.A. & Teodorovich, E.V. 1980 On the generation of internal waves in the presence of uniform straight-line motion of local and nonlocal sources. Izv. Atmos. Ocean. Phys. 16, 699–704.Google Scholar
Gorodtsov, V.A. & Teodorovich, E.V. 1982 Study of internal waves in the case of rapid horizontal motion of cylinders and spheres. Fluid Dyn. 17, 893–898.CrossRefGoogle Scholar
Gray, E.P., Hart, R.W. & Farrell, R.A. 1983 The structure of the internal wave Mach front generated by a point source moving in a stratified fluid. Phys. Fluids 26, 2919–2931.CrossRefGoogle Scholar
Happel, J. & Brenner, H. 1983 Low Reynolds Number Hydrodynamics, 2nd edn. Springer.Google Scholar
Hart, R.W. 1981 Generalized scalar potentials for linearized three-dimensional flows with vorticity. Phys. Fluids 24, 1418–1420.CrossRefGoogle Scholar
Hendershott, M.C. 1969 Impulsively started oscillations in a rotating stratified fluid. J. Fluid Mech. 36, 513–527.CrossRefGoogle Scholar
Hinch, E.J. 2020 Think Before You Compute. Cambridge University Press.CrossRefGoogle Scholar
Hobson, E.W. 1931 The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press.Google Scholar
Hurley, D.G. 1970 Internal waves in a wedge-shaped region. J. Fluid Mech. 43, 97–120.CrossRefGoogle Scholar
Hurley, D.G. 1972 A general method for solving steady-state internal gravity wave problems. J. Fluid Mech. 56, 721–740.CrossRefGoogle Scholar
Hurley, D.G. 1997 The generation of internal waves by vibrating elliptic cylinders. Part 1. Inviscid solution. J. Fluid Mech. 351, 105–118.CrossRefGoogle Scholar
Hurley, D.G. & Hood, M.J. 2001 The generation of internal waves by vibrating elliptic cylinders. Part 3. Angular oscillations and comparison of theory with recent experimental observations. J. Fluid Mech. 433, 61–75.CrossRefGoogle Scholar
Hurley, D.G. & Keady, G. 1997 The generation of internal waves by vibrating elliptic cylinders. Part 2. Approximate viscous solution. J. Fluid Mech. 351, 119–138.CrossRefGoogle Scholar
Ivers, D.J., Jackson, A. & Winch, D. 2015 Enumeration, orthogonality and completeness of the incompressible Coriolis modes in a sphere. J. Fluid Mech. 766, 468–498.CrossRefGoogle Scholar
Jackson, J.D. 1999 Classical Electrodynamics, 3rd edn. Wiley.Google Scholar
Kapitonov, B.V. 1980 Potential theory for the equation of small oscillations of a rotating fluid. Maths USSR Sb. 37, 559–579.CrossRefGoogle Scholar
Kerswell, R.R. 1995 On the internal shear layers spawned by the critical regions in oscillatory Ekman boundary layers. J. Fluid Mech. 298, 311–325.CrossRefGoogle Scholar
Kharik, V.M. 1993 Generation of internal waves in a rotating, stratified fluid: some symmetry properties. J. Math. Phys. 34, 206–213.CrossRefGoogle Scholar
Kharik, V.M. & Pletner, Y.D. 1990 a The problem of gravity-gyroscopic waves, which are excited by the oscillations of a curve. J. Math. Phys. 31, 1280–1283.CrossRefGoogle Scholar
Kharik, V.M. & Pletner, Y.D. 1990 b The problem of gravity-gyroscopic waves, which are excited by the oscillations of a curve. J. Math. Phys. 31, 1422–1425.CrossRefGoogle Scholar
King, B., Zhang, H.P. & Swinney, H.L. 2009 Tidal flow over three-dimensional topography in a stratified fluid. Phys. Fluids 21, 116601.CrossRefGoogle Scholar
Kistovich, Y.V. & Chashechkin, Y.D. 2001 Some exactly solvable problems of the radiation of three-dimensional periodic internal waves. J. Appl. Mech. Tech. Phys. 42, 228–236.CrossRefGoogle Scholar
Korobkin, A.A. 1990 Motion of a body in anisotropic fluid. Arch. Mech. 42, 627–638.Google Scholar
Korpusov, M.O., Pletner, Y.D. & Sveshnikov, A.G. 1997 a On the solvability of an initial-boundary value problem for the internal-wave equation. Comp. Maths Math. Phys. 37, 602–605.Google Scholar
Korpusov, M.O., Pletner, Y.D. & Sveshnikov, A.G. 1997 b Oscillation of a two-sided line segment in a stratified fluid. Comp. Maths Math. Phys. 37, 936–942.Google Scholar
Korpusov, M.O., Pletner, Y.D. & Sveshnikov, A.G. 1997 c Unsteady waves in a stratified fluid excited by the variation of the normal velocity component on a line segment. Comp. Maths Math. Phys. 37, 1076–1085.Google Scholar
Korpusov, M.O., Pletner, Y.D. & Sveshnikov, A.G. 1998 Oscillation of a set of curvilinear segments in a stratified fluid. Comp. Maths Math. Phys. 38, 1519–1526.Google Scholar
Krishna, D.V. & Sarma, L.V. 1969 Motion of an axisymmetric body in a rotating stratified fluid confined between two parallel planes. J. Fluid Mech. 38, 833–842.CrossRefGoogle Scholar
Krutitskii, P.A. 1988 Small non-stationary vibrations of vertical plates in a channel with a stratified liquid. USSR Comp. Maths Math. Phys. 28 (6), 166–176.CrossRefGoogle Scholar
Krutitskii, P.A. 1992 a An explicit solution of the Dirichlet problem for an equation of composite type in a multiply connected domain. Dokl. Maths 46, 63–69.Google Scholar
Krutitskii, P.A. 1992 b Solution of a hyperbolic boundary value problem as a result of applying the principle of limiting amplitude to an initial-boundary value problem for an equation of composite type. Dokl. Maths 46, 118–125.Google Scholar
Krutitskii, P.A. 1994 The limit amplitude of the initial and boundary-value problem for an equation of composite type in a multiply connected domain. Comp. Maths Math. Phys. 34, 951–957.Google Scholar
Krutitskii, P.A. 1995 An explicit solution of the pseudo-hyperbolic initial boundary value problem in a multiply connected region. Math. Meth. Appl. Sci. 18, 897–925; Erratum. Math. Meth. Appl. Sci. 19, 253 (1996).CrossRefGoogle Scholar
Krutitskii, P.A. 1996 a The second initial-boundary-value problem for the gravitational-gyroscopic wave equation. Comp. Maths Math. Phys. 36, 113–123.Google Scholar
Krutitskii, P.A. 1996 b Asymptotics of the gradient of the solution and the inception of singularities in the initial-boundary problem for an equation of a composite type. Dokl. Maths 54, 912–917.Google Scholar
Krutitskii, P.A. 1996 c The second initial boundary value problem for the gravitation-gyroscopic wave equation in exterior domains. Math. Notes 60, 29–41.CrossRefGoogle Scholar
Krutitskii, P.A. 1996 d Reduction of the second initial-boundary value problem for the equation of gravitational-gyroscopic waves to a uniquely solvable integral equation. Differ. Equ. 32, 1383–1392.Google Scholar
Krutitskii, P.A. 1996 e An initial-boundary value problem for the evolutionary PDE of composite type with the mixed boundary condition. Appl. Anal. 61, 209–217.CrossRefGoogle Scholar
Krutitskii, P.A. 1997 a The first initial-boundary value problem for the gravity-inertia wave equation in a multiply connected domain. Comp. Maths Math. Phys. 37, 113–123.Google Scholar
Krutitskii, P.A. 1997 b The pseudohyperbolic problem arising in the dynamics of stratified rotating fluids. Appl. Maths Lett. 10 (2), 117–122.CrossRefGoogle Scholar
Krutitskii, P.A. 1997 c An initial-boundary value problem for the pseudo-hyperbolic equation of gravity-gyroscopic waves. J. Maths Kyoto Univ. 37, 343–365.Google Scholar
Krutitskii, P.A. 1998 Exterior initial-boundary value problem for a certain evolution equation in a multiply connected domain. Differ. Equ. 34, 1629–1638.Google Scholar
Krutitskii, P.A. 2000 The first initial boundary value problem for a compound type equation in a three-dimensional multiply connected region. Math. Notes 68, 217–231.CrossRefGoogle Scholar
Krutitskii, P.A. 2001 Evolutionary equation of inertial waves in 3-D multiply connected domain with Dirichlet boundary condition. Intl J. Maths Math. Sci. 25, 587–602.CrossRefGoogle Scholar
Krutitskii, P.A. 2003 a The jump problem for the equation of internal waves in a stratified rotating fluid. In Progress in Analysis (ed. H.G.W. Begehr, R.P. Gilbert & M.W. Wong), vol. 2, pp. 1185–1195. World Scientific.CrossRefGoogle Scholar
Krutitskii, P.A. 2003 b Initial-boundary value problem for an equation of internal gravity waves in a 3-D multiply connected domain with Dirichlet boundary condition. Adv. Maths 177, 208–226.CrossRefGoogle Scholar
Lai, R.Y.S. & Lee, C.-M. 1981 Added mass of a spheroid oscillating in a linearly stratified fluid. Intl J. Engng Sci. 19, 1411–1420.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.Google Scholar
Landau, L.D. & Lifshitz, E.M. 1984 Electrodynamics of Continuous Media, 2nd edn. Pergamon.Google Scholar
Larsen, L.H. 1969 Oscillations of a neutrally buoyant sphere in a stratified fluid. Deep-Sea Res. 16, 587–603.Google Scholar
Le Dizès, S. & Le Bars, M. 2017 Internal shear layers from librating objects. J. Fluid Mech. 826, 653–675.CrossRefGoogle Scholar
Legendre, A.M. 1806 Analyse des triangles tracés sur la surface d'un sphéroïde. Mém. Cl. Sci. Math. Phys. Inst. Fr. 1er semestre, 130–161.Google Scholar
Lighthill, J. 1978 Waves in Fluids. Cambridge University Press.Google Scholar
Lighthill, J. 1986 An Informal Introduction to Theoretical Fluid Mechanics. Oxford University Press.Google Scholar
Llewellyn Smith, S.G. & Young, W.R. 2003 Tidal conversion at a very steep ridge. J. Fluid Mech. 495, 175–191.CrossRefGoogle Scholar
Machicoane, N., Cortet, P.-P., Voisin, B. & Moisy, F. 2015 Influence of the multipole order of the source on the decay of an inertial wave beam in a rotating fluid. Phys. Fluids 27, 066602.CrossRefGoogle Scholar
Makarov, S.A., Neklyudov, V.I. & Chashechkin, Y.D. 1990 Spatial structure of two-dimensional monochromatic internal-wave beams in an exponentially stratified liquid. Izv. Atmos. Ocean. Phys. 26, 548–554.Google Scholar
Martin, P.A. 2006 Multiple Scattering. Cambridge University Press.CrossRefGoogle Scholar
Martin, P.A. & Llewellyn Smith, S.G. 2011 Generation of internal gravity waves by an oscillating horizontal disc. Proc. R. Soc. Lond. A 467, 3406–3423.Google Scholar
Martin, P.A. & Llewellyn Smith, S.G. 2012 Internal gravity waves, boundary integral equations and radiation conditions. Wave Motion 49, 427–444.CrossRefGoogle Scholar
Mathur, M., Carter, G.S. & Peacock, T. 2014 Topographic scattering of the low-mode internal tide in the deep ocean. J. Geophys. Res. Oceans 119, 2165–2182.CrossRefGoogle Scholar
McLaren, T.I., Pierce, A.D., Fohl, T. & Murphy, B.L. 1973 An investigation of internal gravity waves generated by a buoyantly rising fluid in a stratified medium. J. Fluid Mech. 57, 229–240.CrossRefGoogle Scholar
Milne-Thomson, L.M. 1968 Theoretical Hydrodynamics, 5th edn. Dover.CrossRefGoogle Scholar
Miropol'skii, Y.Z. 1978 Self-similar solutions of the Cauchy problem for internal waves in an unbounded fluid. Izv. Atmos. Ocean. Phys. 14, 673–679.Google Scholar
Morse, P.M. & Feshbach, H. 1953 Methods of Theoretical Physics. Part II. Feshbach Publishing.Google Scholar
Mowbray, D.E. & Rarity, B.S.H. 1967 A theoretical and experimental investigation of the phase configuration of internal waves of small amplitude in a density stratified liquid. J. Fluid Mech. 28, 1–16.CrossRefGoogle Scholar
Musgrave, R.C., Pinkel, R., MacKinnon, J.A., Mazloff, M.R. & Young, W.R. 2016 Stratified tidal flow over a tall ridge above and below the turning latitude. J. Fluid Mech. 793, 933–957.CrossRefGoogle Scholar
Nycander, J. 2006 Tidal generation of internal waves from a periodic array of steep ridges. J. Fluid Mech. 567, 415–432.CrossRefGoogle Scholar
Olver, F.W.J., Lozier, D.W., Boisvert, R.F. & Clark, C.W. 2010 NIST Handbook of Mathematical Functions. NIST/Cambridge University Press.Google Scholar
Peacock, T., Echeverri, P. & Balmforth, N.J. 2008 An experimental investigation of internal tide generation by two-dimensional topography. J. Phys. Oceanogr. 38, 235–242.CrossRefGoogle Scholar
Pétrélis, F., Llewellyn Smith, S. & Young, W.R. 2006 Tidal conversion at a submarine ridge. J. Phys. Oceanogr. 36, 1053–1071.CrossRefGoogle Scholar
Pierce, A.D. 1963 Propagation of acoustic-gravity waves from a small source above the ground in an isothermal atmosphere. J. Acoust. Soc. Am. 35, 1798–1807.CrossRefGoogle Scholar
Pierce, A.D. 2019 Acoustics, 3rd edn. Springer.CrossRefGoogle Scholar
Pletner, Y.D. 1988 Gyroscopic gravitational waves caused by the oscillations of a straight section. USSR Comp. Maths Math. Phys. 28 (2), 139–145.CrossRefGoogle Scholar
Pletner, Y.D. 1990 a On the vibrations of a two-sided disc in a stratified fluid. USSR Comp. Maths Math. Phys. 30 (1), 203–212.CrossRefGoogle Scholar
Pletner, Y.D. 1990 b Vibrations of a flat disk at the interface between two stratified liquids. USSR Comp. Maths Math. Phys. 30 (3), 70–80.CrossRefGoogle Scholar
Pletner, Y.D. 1991 The fundamental solution of the equation of internal waves and some initial-boundary value problems. Comp. Maths Math. Phys. 31 (4), 79–88.Google Scholar
Pletner, Y.D. 1992 Fundamental solutions of Sobolev-type operators and some initial boundary-value problems. Comp. Maths Math. Phys. 32, 1715–1728.Google Scholar
Pletner, Y.D. & Tverskaya, L.V. 1989 The problem of the vibrations of a rotating stratified liquid excited by a planar disc. USSR Comp. Maths Math. Phys. 29 (2), 94–99.CrossRefGoogle Scholar
Pletner, Y.D. & Tverskaya, L.V. 1991 On the problem of the oscillations of a rotating stratified fluid excited by a plane two-sided disc. Comp. Maths Math. Phys. 31 (5), 76–83.Google Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.CrossRefGoogle Scholar
Pozrikidis, C. 2002 A Practical Guide to Boundary Elements Methods with the Software Library BEMLIB. Chapman and Hall/CRC.CrossRefGoogle Scholar
Rieutord, M., Georgeot, B. & Valdetarro, L. 2001 Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum. J. Fluid Mech. 435, 103–144.CrossRefGoogle Scholar
Rieutord, M. & Noui, K. 1999 On the analogy between gravity modes and inertial modes in spherical geometry. Eur. Phys. J. B 9, 731–738.CrossRefGoogle Scholar
Robinson, R.M. 1969 The effects of a vertical barrier on internal waves. Deep-Sea Res. 16, 421–429.Google Scholar
Robinson, R.M. 1970 The effects of a corner on a propagating internal gravity wave. J. Fluid Mech. 42, 257–267.CrossRefGoogle Scholar
Sarma, L.V.K.V. & Krishna, D.V. 1972 Oscillation of axisymmetric bodies in a stratified fluid. Zastosow. Matem. 13, 109–121.Google Scholar
Schwartz, L. 1966 Mathematics for the Physical Sciences. Hermann.Google Scholar
Simakov, S.T. 1989 On the small vibrations of a stratified capillary liquid. J. Appl. Maths Mech. 53, 50–56.CrossRefGoogle Scholar
Skazka, V.V. 1981 Asymptotic estimates for $t \to \infty$ of mixed problems for an equation of mathematical physics. Siber. Math. J. 22, 95–106.CrossRefGoogle Scholar
Sobolev, S.L. 1954 On a new problem of mathematical physics. Izv. Akad. Nauk SSSR Ser. Mat. 18, 3–50 [in Russian]; English translation in Selected Works of S.L. Sobolev (ed. G.V. Demidenko & V.L. Vaskevich), vol. 1, pp. 279–332. Springer (2006).Google Scholar
Sommerville, D.M.Y. 1933 Analytical Conics, 3rd edn. Bell and Sons.Google Scholar
Stewartson, K. 1952 On the slow motion of a sphere along the axis of a rotating fluid. Proc. Camb. Phil. Soc. 48, 168–177.CrossRefGoogle Scholar
Sturova, I.V. 2001 Oscillations of a circular cylinder in a linearly stratified fluid. Fluid Dyn. 36, 478–488.CrossRefGoogle Scholar
Sturova, I.V. 2006 Oscillations of a cylinder piercing a linearly stratified fluid layer. Fluid Dyn. 41, 619–628.CrossRefGoogle Scholar
Sturova, I.V. 2011 Hydrodynamic loads acting on an oscillating cylinder submerged in a stratified fluid with ice cover. J. Appl. Mech. Tech. Phys. 52, 415–426.CrossRefGoogle Scholar
Sundukova, A.V. 1991 Fundamental solution of the gravitational-gyroscopic wave equation and the solvability of the internal and external Dirichlet problems. Comp. Maths Math. Phys. 31 (10), 87–93.Google Scholar
Sutherland, B.R., Dalziel, S.B., Hughes, G.O. & Linden, P.F. 1999 Visualization and measurement of internal waves by ‘synthetic schlieren’. Part 1. Vertically oscillating cylinder. J. Fluid Mech. 390, 93–126.CrossRefGoogle Scholar
Sutherland, B.R. & Linden, P.F. 2002 Internal wave excitation by a vertically oscillating elliptical cylinder. Phys. Fluids 14, 721–731.CrossRefGoogle Scholar
Sveshnikov, A.G., Shishmarev, I.A. & Pletner, Y.D. 1989 Sergei Aleksandrovich Gabov (24.05.1948–19.04.1989). Vestn. Moskov. Univ. Ser. 3 Fiz. Astron. (5), 92–93 [in Russian].Google Scholar
Voisin, B. 1991 Internal wave generation in uniformly stratified fluids. Part 1. Green's function and point sources. J. Fluid Mech. 231, 439–480.CrossRefGoogle Scholar
Voisin, B. 2003 Limit states of internal wave beams. J. Fluid Mech. 496, 243–293.CrossRefGoogle Scholar
Voisin, B. 2009 Added mass in density-stratified fluids. In Actes du 19ème Congrès Français de Mécanique (ed. C. Rey, P. Bontoux & A. Chrisochoos). Available at http://hdl.handle.net/2042/37312.Google Scholar
Voisin, B. 2020 Near-field internal wave beams in two dimensions. J. Fluid Mech. 900, A3.CrossRefGoogle Scholar
Voisin, B., Ermanyuk, E.V. & Flór, J.-B. 2011 Internal wave generation by oscillation of a sphere, with application to internal tides. J. Fluid Mech. 666, 308–357.CrossRefGoogle Scholar
Watson, G.N. 1944 A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press.Google Scholar
Zhang, H.P., King, B. & Swinney, H.L. 2007 Experimental study of internal gravity waves generated by supercritical topography. Phys. Fluids 19, 096602.CrossRefGoogle Scholar