Surface wave scattering by an elastic plate submerged in water with uneven bottom

    Souvik Kundu   Affiliation
    ; Rupanwita Gayen   Affiliation


Wave interaction with a vertical elastic plate in presence of undulating bottom topography is considered, assuming linear theory and utilizing simple perturbation analysis. First order correction to the velocity potential corresponding to the problem of scattering by a vertical elastic plate submerged in a fluid with a uniform bottom is obtained by invoking the Green’s integral theorem in a suitable manner. With sinusoidal undulation at the bottom, the first-order transmission coefficient (T1) vanishes identically. Behaviour of the first order reflection coefficient (R1) depending on the plate length, ripple number, ripple amplitude and flexural rigidity of the plate is depicted graphically. Also, the resonant nature of the first order reflection is observed at a particular value of the ratio of surface wavelength to that of the bottom undulations. The net reflection coefficient due to the joint effect of the plate and the bottom undulation is also presented for different flexural rigidity of the plate. When the rigidity parameter is made sufficiently large, the results for R1 reduce to the known results for a surface piercing rigid plate in water with bottom undulation.

Keyword : bottom undulation, vertical elastic plate, perturbation analysis, first-order reflection

How to Cite
Kundu, S., & Gayen, R. (2020). Surface wave scattering by an elastic plate submerged in water with uneven bottom. Mathematical Modelling and Analysis, 25(3), 323-337.
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May 13, 2020
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S. Banerjea, P. Rakshit and P. Maiti. On the waves generated due to a line source present in an ocean with an ice cover and a small bottom undulation. Fluid Dynamics Research, 43(2):025506, 2011.

R. Chakraborty, A. Mondal and R. Gayen. Interaction of surface water waves with a vertical elastic plate: a hypersingular integral equation approach. Zeitschrift für angewandte Mathematik und Physik, 67(5):115, 2016.

P.G. Chamberlain and D. Porter. The modified mild-slope equation. Journal of Fluid Mechanics, 291:393–407, 1995.

A. Choudhary and S.C. Martha. Diffraction of surface water waves by an undulating bed topography in the presence of vertical barrier. Ocean Engineering, 122:32–43, 2016.

A.G. Davies. The reflection of wave energy by undulations on the seabed. Dynamics of Atmospheres and Oceans, 6(4):207–232, 1982.

A.G. Davies and A.D. Heathershaw. Surface-wave propagation over sinusoidally varying topography. Journal of Fluid Mechanics, 144:419–443, 1984.

G.F. Fitz-Gerald. The reflexion of plane gravity waves travelling in water of variable depth. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 284(1317):49–89, 1976.

R. Gayen and A. Mondal. Water wave interaction with two symmetric inclined permeable plates. Ocean Engineering, 124:180–191, 2016.

J.T. Kirby. A note on Bragg scattering of surface waves by sinusoidal bars. Physics of Fluids A: Fluid Dynamics, 5(2):380–386, 1993.

G. Kreisel. Surface waves. Quarterly of Applied Mathematics, 7(1):21–44, 1949.

S. Kundu, R. Gayen and R. Datta. Scattering of water waves by an inclined elastic plate in deep water. Ocean Engineering, 167:221–228, 2018.

P. Maiti and B.N. Mandal. Water wave scattering by bottom undulations in an ice-covered two-layer fluid. Applied Ocean Research, 30(4):264–272, 2008.

B.N. Mandal and S. De. Surface wave propagation over small undulations at the bottom of an ocean with surface discontinuity. Geophysical & Astrophysical Fluid Dynamics, 103(1):19–30, 2009.

B.N. Mandal and R. Gayen. Water wave scattering by bottom undulations in the presence of a thin partially immersed barrier. Applied Ocean Research, 28(2):113–119, 2006.

S.C. Martha and S.N. Bora. Oblique surface wave propagation over a small undulation on the bottom of an ocean. Geophysical & Astrophysical Fluid Dynamics, 101(2):65–80, 2007.

M.H. Meylan. A flexible vertical sheet in waves. The International Journal of Offshore and Polar Engineering, 5(02):105–110, 1994. Available from Internet:

J.W. Miles. Surface-wave scattering matrix for a shelf. Journal of Fluid Mechanics, 28(4):755–767, 1967.

S. Mohapatra. The effect of free-surface tension on scattering of water waves by small bottom undulation. ANZIAM Journal, 58(0):39–80, 2017.

S. Mohapatra and S.N. Bora. Reflection and transmission of water waves in a two-layer fluid flowing through a channel with undulating bed. Zeitschrift für Angewandte Mathematik und Mechanik, 91(1):46–56, 2011.

J.N. Newman. Propagation of water waves over an infinite step. Journal of Fluid Mechanics, 23(2):399–415, 1965.

S. Panda and S.C. Martha. Water-waves scattering by permeable bottom in twolayer fluid in the presence of surface tension. Mathematical Modelling & Analysis, 22(6):827–851, 2017.

S. Panda, S.C. Martha and A. Chakrabarti. Three-layer fluid flow over a small obstruction on the bottom of a channel. The ANZIAM Journal, 56(3):248–274, 2015.

M.A. Peter and M.H. Meylan. A general spectral approach to the timedomain evolution of linear water waves impacting on a vertical elastic plate. SIAM Journal on Applied Mathematics, 70(7):2308–2328, 2010.

P. Rakshit and S. Banerjea. Effect of bottom undulation on the waves generated due to rolling of a plate. Journal of Marine Science and Application, 10(1):7, 2011.

M. Roseau. Contribution à la théorie des ondes liquides de gravité en profondeur variable, volume 275. Service de documentation et d’information technique de l’aéronautique, 1952.

D.J. Staziker, D. Porter and D.S.G. Stirling. The scattering of surface waves by local bed elevations. Applied Ocean Research, 18(5):283–291, 1996.