Abstract
Wave interaction with rough bottom surfaces (topography), offshore drilling platforms and wave energy collectors is accompanied by the diffraction of waves. Here, an account is given of our research results in the field of wave diffraction. Specific aspects and methods used to solve the problems of the wave-diffraction theory are described in brief. The problem of wave diffraction by a submerged elliptical cylinder with elliptical lower face under an arbitrary incidence of plane waves is discussed. Wave diffraction by a submerged compound cylinder is also considered. Horizontal wave force is calculated as an example illustrating the dependence on the degree of submersion of the obstacle. Both the decrease of the submersion depth and the increase of the wave number are shown to favour the approach to the resonance range and thus worsen the reliability of the results. Scattering of magnetoacoustic waves by a cylinder is studied, the effect of magnetoelastic waves on the scattered field is demonstrated. Wave diffraction in a multiconnected domain formed by a set of vertical cylinders is analysed. The mutual influence of the cylinders is considerable for \( l/2a < 2 \), where l is the distance between the cylinders and 2a is the diameter. Interaction between diffracted fields is studied and the maximum lateral wave force that can also act outside the frontal cylinder is calculated. An exact analytical solution to the problem of wave diffraction by an asymmetrically inhomogeneous cylinder is found, the effect of asymmetry on the total scattering cross-section is analysed. An efficient numerical-analytical method is also considered, i.e. an auxiliary surface of simple shape (spherical or cylindrical) is introduced that surrounds the scatterer. In this case, the boundary-value problem can be subdivided into two problems, interior and exterior ones. The former is solved analytically in an infinite domain, the latter is solved numerically in a finite domain with a complex boundary. Furthermore, we study the wave diffraction by a prolate body of revolution consisting of a cylinder and spherical end cups. To solve the problem, a new finite-element algorithm is proposed.
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References
Belyaev, V.A., Selezov, I.T., Yakovlev, V.V.: Difraktsiya regulyarnykh voln na proizvolno raspolozhennoy sisteme vertikalnykh krugovykh tsilindrov (Diffraction of regular waves by an arbitrarily spaced system of vertical circular cylinders). Izvestiya AN SSSR. Mekhanika zhidkosti i gaza 1, 177–181 (1984)
Black, J.L., Mei, C.C., Bray, M.C.: Radiation and scattering of water waves by rigid bodies. J. Fluid Mech. 46(1), 151–164 (1971)
Born, M.: Optik. Springer, Berlin (1933)
Borovikov, V.A.: Difraktsiya na mnogougolnikakh i mnogogrannikakh (Diffraction by Polygons and Polyhedra). Nauka, Moscow (1966)
Brebbia, C.A., Walker, S.: Dynamic Analysis of Offshore Structures. Butterworth-Heinemann Ltd., London (1979)
Cai, L.-W.: Evaluation of layered multiple-scattering method for antiplane shear wave scattering from gratings. J. Acoust. Soc. Am. 120(1), 49–61 (2006)
Chakrabarti, S.K.: Wave forces on multiple vertical cylinders. J. Waterway Port Coast. Ocean Div. 104(2), 147–161 (1978)
Chen, H.S., Mei, C.C.: Wave forces on a stationary platform of elliptical shape. J. Ship Res. 17(2), 61–71 (1973)
D’Este, F., Codiglia, R., Contento, G.: On the side wall effects on the nonli-near wave loads on a TLP-like structure in a wave tank: analysis of the nonlinear near-trapping. In: Proceedings of the 12th International Offshore and Polar Engineering Conference, Kitakyushu, Japan, 26–31 May 2002, pp. 196–202
Doyle, T.E.: Iterative simulation of elastic wave scattering in arbitrary dispersions of spherical particles. J. Acoust. Soc. Am. 119(5), 2599–2610 (2006)
Fejer, J.A.: Scattering of electromagnetic waves by a plasma cylinder. Phys. Fluids 7(3), 439–445 (1964)
Ferrante, V., Vicinanza, D.: Spectral analysis of wave transformation behind submerged breakwater. In: Proceedings of the 16th International Offshore and Polar Engineering Conference, San Francisco, USA, 28 May–2 June 2006, vol. 3, pp. 727–733
Garrett, C.R.: Wave forces on a circular doc. J. Fluid Mech. 46(1), 129–139 (1971)
Haskind, M.D.: Gidrodinamicheskaya teoriya kachki korablya (The Hydrodynamic Theory of Ship Rolling). Nauka, Moscow (1973)
Hönl, H., Maue, A.W., Westpfahl, K.: Theorie der Beugung. Springer, Berlin (1961)
Isaacson, M.: Fixed and floating axisymmetric structure in waves. J. Waterway Port Coast. Ocean Eng. 108(2), 180–189 (1982)
Ivanov, E.A.: Difraktsiya elektromagnitnykh voln na dvukh telakh (Diffraction of Electromagnetic Waves by Two Bodies). Nauka i Tekhnika, Minsk (1968)
Kagemoto, H., Murai, M., Saito, M., Molin, B., Malenica, S.: Experimental and theoretical analysis of the wave decay along a long array of vertical cylinders. J. Fluid Mech. 456, 113–135 (2002)
Kamke, E.: Handbook of Ordinary Differential Equations. Chelsea Publishing (1976)
Karlsson, A., Kristensson, G.: Wave splitting in the time domain for a radially symmetric geometry. Wave Motion 12(3), 197–211 (1990)
Katsenelenbaum, B.Z.: High-Frequency Electrodynamics. Wiley-VCH, Weinheim (2006)
Kee, S.T.: Submerged horizontal and vertical membrane wave barrier. In: Proceedings of the 15th International Offshore and Polar Engineering Conference, Seoul, South Korea, 19–24 June 2005, vol. 3, pp. 587–594
Kim, J.W., Kyoung, J.H., Ertekin, R.C., Bai, K.J.: Finite-element computation of wave-structure interaction between steep Stokes waves and vertical cylinders. J. Waterway Port Coast. Ocean Eng. 5, 337–347 (2006)
Kim, M.H.: Interaction of waves with N vertical circular cylinders. J. Waterway Port Coast. Ocean Eng. 119(6), 671–689 (1993)
Kim, M.H., Natvig, B.J., Mercier, R.S., Gu, G., Wu, C.: PC-based computation for second-order wave loads on large volume multi-column structures. In: Proceedings of the 5th International Offshore and Polar Engineering Conference, Hague, Netherlands, 11–16 June 1995, pp. 561–570
Kornilov, I.E.: Primenenie metoda konechnykh elementov k raschetu obtekaniya mnogo-svyaznykh oblastei potentsialnym potokom szhimaemoi zhidkosti (Finite-element analysis of the circumfluence of multiply-connected domains by a potential flow of compressible fluid). In: Matematicheskie Metody Issledovaniya Gidrodinamicheskikh Techenii (Mathematical Methods of Investigating the Hydrodynamical Flows), pp. 38–41. Naukova Dumka, Kiev (1978)
Kratzer, A., Franz, W.: Transzendente Funktionen. Akademische Verlagsgesellschaft Geest & Portig, Leipzig (1960)
Kryvonos, Yu.G., Kornilov, I.E., Noga, Yu.V., Selezov, I.T.: Chislennoe issledovanie volnovykh polei ot proizvolnykh tel vrashcheniya i neodnorodnostei (Numerical investigation of the wave fields from arbitrary bodies of revolution and inhomogeneities). Preprint, Institute of Cybernetics AN USSR, Kiev (1983)
Kryvonos, Yu.G., Selezov, I.T.: Upravlenie diagrammoi napravlennosti pri rasseyanii tsilindricheskoi akusticheskoi volny na tsilindre v poluogranichennoi oblasti (Controlling the directional diagram during the scattering of a cylindrical acoustic wave by a cylinder in a semibounded domain). In: Kibernetika i vychislitelnaya tekhnika. Slozhnye sistemy upravleniya (Cybernetics and Computational Techniques. Complex Control Systems), vol. 8, pp. 102–109. Naukova Dumka, Kiev (1971)
Linton, C.M., Evans, D.V.: The interaction of waves with arrays of vertical circular cylinders. J. Fluid Mech. 215, 549–569 (1990)
Malenica, S., Taylor, R.E., Huang, J.B.: Second-order water wave diffraction by an array of vertical cylinders. J. Fluid Mech. 390, 349–373 (1999)
Massel, S.B.: Interaction of water waves with cylinder barrier. J. Waterways Harbors Coast Eng. Div. 102(2), 165–187 (1976)
Mavrakos, S.A., Koumoutsakos, P.: Hydrodynamic interaction among vertical axisymmetric bodies restrained in waves. Appl. Ocean Res. 9(3), 128–140 (1987)
McLachlan, N.W.: Theory and Application of Mathieu Functions. Dover, New York (1964)
Miles, J.W., Gilbert, F.: Scattering of gravity waves by a circular dock. J. Fluid Mech. 34(4), 783–793 (1968)
Morse, P.M., Feshbach, H.: Methods of Theoretical Physics. Parts 1 & 2. McGraw-Hill, New York (1953)
Nozaka, M., Takaku, K.: Scattering of electromagnetic waves by a cylindrically inhomogeneous plasma. J. Phys. Soc. Jpn. 24(1), 172–184 (1968)
Parkinson, R.G., Kharadly, M.M.Z.: Experimental study of scattering of electromagnetic waves by radially inhomogeneous cylinders. Can. J. Phys. 49(23), 2989–2996 (1971)
Peter, M.A., Meylan, M.N.: Water-wave scattering by a semi-infinite periodic array of arbitrary bodies. J. Fluid Mech. 575, 473–494 (2007)
Selezov, I.T.: Rasprostranenie magnitouprugikh voln napryazheniya ot tsilindricheskoi polosti v provodyashchei srede (Propagation of magnetoelastic stress waves from a cylindrical cavity in the conducting medium). Zhurnal prikladnoi mekhaniki i tekhnicheskoi fiziki 2, 15–20 (1969)
Selezov, I.T.: Dyfraktsiia khvyl na tsylindri v pruzhnomu pivprostori (Wave diffraction by a cylinder in an elastic half-space). Dopovidi AN URSR. Ser. A 3, 259–264 (1970)
Selezov, I.T.: Diffraction of waves by convex bodies in semibounded regions. Soviet Appl. Mech. 6(3), 258–263 (1970)
Selezov, I.T.: Nekotorye zadachi difraktsii i rasseyaniya voln prostranstvennymi neodnorodnostyami v okeane i atmosphere (Some problems of wave diffraction and scattering by spatial inhomogeneities in the ocean and atmosphere). In: Teoreticheskie i eksperimentalnye issledovaniya poverkhnostnykh i vnutrennikh voln (Theoretical and Experimental Studies of Surface and Internal Waves), pp. 7–18. Institute of Marine Hydrophysics, Sevastopol (1980)
Selezov, I.T.: Numerical solution of the boundary value problems of the wave diffraction by arbitrary revolution surfaces. In: Numerical Methods. Colloquia Mathematica Societatis János Bolyai, vol. 50, pp. 473–480. North-Holland, Amsterdam (1987)
Selezov, I.T.: Modelirovanie volnovykh i difraktsionnykh protsessov v sploshnykh sredakh (Modeling the Wave and Diffraction Processes in Continuous Media). Naukova Dumka, Kiev (1989)
Selezov, I.T.: Rasprostranenie i transformatsiya poverkhnostnykh gravitatsionnykh voln v zhidkosti konechnoi glubiny (Propagation and transformation of surface gravity waves in a finite-depth fluid). In: Itogi nauki i tekhniki. Ser. Mekhanika zhidkosti i gaza (Science and Technology Review. Ser. Mechanics of Fluids and Gases), vol. 24, pp. 3–76. VINITI, Moscow (1990)
Selezov, I.T.: Diffraction of waves by radially inhomogeneous inclusions. Phys. Express 1(2), 101–115 (1993)
Selezov, I.T.: Propagation and diffraction of waves in locally inhomogeneous and bound structures. Int. J. Fluid Mech. Research 23(1–2), 81–95 (1996)
Selezov, I.T., Korsunskii, S.V.: Nestatsionarnye i nelineynye volny v elektroprovodyashchikh sredakh (Nonstationary and Nonlinear Waves in Electrically Conducting Media). Naukova Dumka, Kiev (1991)
Selezov, I.T., Kryvonos, Yu.G.: Rasseyanie akusticheskikh voln na tsilindre v poluogra-nichennoi oblasti (Scattering of acoustic waves by a cylinder in a semibounded domain). Gidromekhanika 15, 99–106 (1969)
Selezov, I.T., Kryvonos, Yu.G.: Rozsiiannia akustychnoi khvyli na sferi v napivobmezhenii oblasti (Scattering of an acoustic wave by a sphere in a semibounded domain). Dopovidi AN URSR. Ser. A 7, 632–636 (1969)
Selezov, I.T., Kryvonos, Yu.G.: Rozsiiannia elektromahnitnykh khvyl na tsylindri v napivobmezhenii oblasti (Scattering of electromagnetic waves by a cylinder in a semibounded domain). Dopovidi AN URSR. Ser. A. 4, 367–372 (1970)
Selezov, I.T., Kryvonos, Yu.G.: Statsionarna zadacha rozsiiannia tsylindrychnoi mahnito-akustychnoi khvyli na idealnoprovidnomu tsylindri (A stationary problem of scattering of a cylindrical magnetoacoustic wave by an ideally conducting cylinder). Dopovidi AN URSR. Ser. A 2, 169–173 (1971)
Selezov, I.T., Kryvonos, Yu.G., Yakovlev, V.V.: Rasseyanie voln lokalnymi neodnorodnostyami v sploshnykh sredakh (Wave Scattering by Local Inhomogeneities in Continuous Media). Naukova Dumka, Kiev (1985)
Selezov, I.T., Lazarenko, M.A.: Rozsiiannia ta dyfraktsiia pruzhnykh khvyl na sferi, rozmischenii u pivprostori (Scattering and diffraction of elastic waves by a sphere located in a half-space). Dopovidi AN URSR. Ser. A 2, 179–182 (1966)
Selezov, I.T., Selezova, L.V.: The flow of an ionized gas around an oscillating boundary in the presence of a magnetic field. Magnetohydrodynamics 3(1), 5–10 (1967)
Selezov, I.T., Sidorchuk, V.N., Yakovlev, V.V.: Transformatsiya voln v pribrezhnoi zone shelfa (Wave Transformation in the Coastal Shelf Zone). Naukova Dumka, Kiev (1983)
Selezov, I.T., Svechnikova, K.M.: Difraktsiya poverkhnostnykh gravitatsionnykh voln na zatoplennom tsilindre, raspolozhennom na krugovom osnovanii (Diffraction of surface gravity waves by a submerged cylinder located on a circular base). Dinamicheskie sistemy 24, 33–37 (2008)
Selezov, I.T., Tkachenko, V.A., Yakovlev, V.V.: Difraktsiya voln tsunami na podvodnom beregovom sklone krugovogo ostrova (Diffraction of tsunami waves by an underwater coastal slope of a circular island). In: Generatsiya tsunami i vykhod voln na bereg (Tsunami Generation and Wave Run-Up on the Coast), pp. 41–46. Radio i svyaz, Moscow (1984)
Selezov, I.T., Vanina, E.G.: Difraktsiya gidrodinamicheskikh voln na polupogruzhennom vertikalnom ellipticheskom tsilindre s neploskim tortsom (Diffraction of hydrodynamic waves by a partially submerged elliptical cylinder with nonflat base). Dopovidi AN URSR. Ser. A 5, 33–37 (1986)
Selezov, I.T., Yakovlev, V.V.: Problems of plane-wave diffraction by a variable-density cylinder. Soviet Physics—Acoustics 23(5), 460–462 (1977)
Selezov, I.T., Yakovlev, V.V.: Difraktsiya voln na simmetrichnykh neodnorodnostyakh (Wave Diffraction by Symmetric Inhomogeneities). Naukova Dumka, Kiev (1978)
Sheikh, R., Swan, C.: Wave slamming on vertical surface-piercing cylinders: the role of nonlinear wave scattering. In: Proceedings of the 15th International Offshore and Polar Engineering Conference, Seoul, South Korea, 19–24 June 2005, vol. 3, pp. 652–659
Shenderov, E.L.: Difraktsiya tsilindricheskoi zvukovoi volny na tsilindre (Diffraction of a cylindrical acoustic wave by a cylinder). Akusticheckii zhurnal 7(3), 370–374 (1961)
Shenderov, E.L.: Volnovye zadachi gidroakustiki (Wave Problems of Hydroacoustics). Sudostroyenie, Leningrad (1972)
Sleeman, B.D.: Acoustic scattering by inhomogeneous media. Q. J. Mech. Appl. Math. 33(4), 373–383 (1980)
Spring, B.H., Monkmeyer, P.L.: Interaction of plane waves with vertical cylinders. In: Proceedings of the 14th International Conference on Coastal Engineering, Copenhagen, Denmark, 24–28 June 1974, vol. 3, pp. 1828–1847. American Society of Civil Engineers, New York (1975)
Sun, L., Li, Y., Teng, B., Zhang, N.: Wave action on structures of combined cylinders. China Ocean Eng. 19(3), 375–384 (2005)
Swan, C., Sheikh, R.: The interaction between steep waves and a surface-piercing column. Philos. Trans. R. Soc. Lond. A 373(2033), 20140114 (2015)
Takayama, T., Goda, Y.: Wave forces on a semi-submerged vertical cylinder of elliptical shape. Coast. Eng. Jpn. 16, 147–164 (1973)
Tikhonov, A.N., Samarskii, A.A.: Equations of Mathematical Physics. Dover Publication, New York (2011)
Tsubogo, T.: Near-resonance of diffraction waves by an array of plates on shallow water in oblique waves. In: Proceedings of the 10th International Offshore and Polar Engineering Conference, Seattle, USA, 27 May–2 June 2000, vol. 3, pp. 154–164
Twersky, V.: Multiple scattering of radiation by an arbitrary configuration of parallel cylinders. J. Acoust. Soc. Am. 24(1), 42–46 (1952)
Vanina, E.G.: Rasseyanie poverkhnostnykh gravitatsionnykh voln ellipticheskimi neodnorodnostyami (Scattering of surface gravity waves by elliptical inhomogeneities). Dopovidi AN URSR. Ser. A 4, 29–32 (1984)
Vasilenko, V.A.: Splain-funktsii: teoriya, algoritmy, programmy (Spline Functions: Theory, Algorithms, Programs). Nauka, Novosibirsk (1983)
Vissers, F.P.L.M., Kamp, L.P.J., Sluijter, F.W.: Some generalizations of radial Bremmer series. Wave Motion 13(1), 43–51 (1991)
Walker, S.: Boundary elements in fluid-structure interaction problems of rotational shells. Appl. Math. Model. 4, 345–350 (1980)
Williams, A.N.: Wave diffraction by elliptical breakwaters in shallow water. Ocean Eng. 12(1), 25–43 (1985)
Williams, A.N., Li, W.: Water wave interaction with an array of bottom-mounted surface-piercing porous cylinders. Ocean Eng. 27, 841–866 (2000)
Yilmaz, O.: Hydrodynamic interactions of waves with group of truncated vertical cylinders. J. Waterway Port Coast. Ocean Eng. 124, 272–279 (1998)
Young, J.W., Bertrand, J.C.: Multiple scattering by two cylinders. J. Acoust. Soc. Am. 58(6), 1190–1195 (1975)
Zavyalov, Yu.S., Kvasov, B.I., Miroshnichenko, V.D.: Metody splain-funktsii (Methods of Spline Functions). Nauka, Moscow (1980)
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Selezov, I.T., Kryvonos, Y.G., Gandzha, I.S. (2018). Analytical and Numerical Solutions to the Wave Diffraction Problems. In: Wave Propagation and Diffraction. Foundations of Engineering Mechanics. Springer, Singapore. https://doi.org/10.1007/978-981-10-4923-1_4
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