The effect of free-surface tension on scattering of water waves by small bottom undulation

We examine the influence of surface tension at the free-surface on the problem of scattering of normal incident waves by small bottom undulation in a two-layer fluid within the framework of linearised wave theory. In such a two-layer fluid, there exist two different modes of time-harmonic waves for the given frequency, one of which corresponds to a free-surface disturbance and the other to an interfacial wave motion. Using perturbation technique, the problem is reduced up to first-order to a coupled boundary value problem which is solved by a method based on Green's integral theorem with the introduction of appropriate Green's functions. The first-order approximations to the reflection and transmission coefficients evaluated in terms of computable integrals for both the modes. For a sinusoidal bottom topography, these coefficients are determined and the numerical results depicted graphically. When the wave is incident due to the free-surface disturbance or the interfacial wave motion in a two-layer fluid, we always find energy transfer from the incident wave mode to the other wave mode. We conclude that realistic changes in surface tension on the free-surface have a significant effect on the problem of water wave scattering by bottom undulation in a two-layer fluid. References P. G. Chamberlain and D. Porter, ``Wave scattering in a two-layer fluid of varying depth'', J. Fluid Mech. 524 (2005) 207–228; doi:10.1017/S0022112004002356 G. J. D. Chapman and R. Porter, ``Second-order sloshing over an arbitrary bed'', J. Fluid Mech. 524 (2005) 331–355; doi:10.1017/S0022112004002502 A. G. Davies, ``The reflection of wave energy by undulations of the sea bed'', Dyn. Atmos. Oceans 6 (1982) 207–232; doi:10.1016/0377-0265(82)90029-X D. V. Evans, ``The influence of surface tension on the reflection of water waves by a plane vertical barrier'', Proc. Camb. Phil. Soc. 64 (1968) 795–810; doi:10.1017/S0305004100043504 D. V. Evans, ``The effect of surface tension on the waves produced by a heaving circular cylinder'', Proc. Camb. Phil. Soc. 64 (1968) 833–847; doi:10.1017/S030500410004353X R. Harter, I. D. Abrahams and M. J. Simon, ``The effect of surface tension on trapped modes in water wave problems'' Proc. R. Soc. Lond. A 463 , (2007) 147–180; doi:10.1098/rspa.2007. 0063 R. Harter, M. J. Simon and I. D. Abrahams, ``The effect of surface tension on localized free-surface oscillations about surface-piercing bodies'', Proc. R. Soc. A 464 (2008) 3039–3054; doi:10.1098/rspa.2008. 0059 H. Lamb, Hydrodynamics , (6th ed. Cambridge University Press, Cambridge, 1932). C. M. Linton and M. McIver, ``The interaction of waves with horizontal cylinders in two-layer fluids'', J. Fluid Mech. 304 (1995) 213–229; doi:10.1017/S002211209500440X C. M. Linton and J. R. Cadby, ``Scattering of oblique waves in a two-layer fluid'', J. Fluid Mech. 461 (2002) 343–364; doi:10.1017/S002211200200842X P. Maiti and B. N. Mandal, ``Scattering of oblique waves by bottom undulations in a two-layer fluid'', J. Appl. Math. \(and\) Computing 22 (2006) 21–39; doi:10.1007/BF02832035 B. N. Mandal and S. De, Water wave scattering (CRC Press, Florida, 2015). C. C. Mei, ``Resonant reflection of surface water waves by periodic sandbars'', J. Fluid Mech. 152 (1985) 315–335; doi:10.1017/S0022112085000714 J. W. Miles, ``Oblique surface wave diffraction by a cylindrical obstacle'', Dyn. Atmos. Oceans 6 (1981) 121–123; doi:10.1016/0377-0265(81)90019-1 S. Mohapatra and S. N. Bora, ``Water wave interaction with a sphere in a two-layer fluid flowing through a channel of finite depth'', Arch. Appl. Mech. 79 (2009) 725–740; doi:10.1007/s00419-008-0248-z Mohapatra, S. and Bora, S. N. Scattering of internal waves in a two-layer fluid flowing through a channel with small undulations. Ocean Dynamics , 59 (2009) 615–625; doi:10.1007/s10236-009-0214-5 S. Mohapatra and S. N. Bora, ``Reflection and transmission of water waves in a two-layer fluid flowing through a channel with undulating bed'', Z. Angew. Math. Mech. [ZAMM] 91(1) (2011) 46–56; doi:10.1002/zamm.200800216 S. Mohapatra and S. N. Bora, ``Propagation of oblique waves over small bottom undulation in an ice-covered two-layer fluid'', Geophysical and Astrophysical Fluid Dynamics 103 (2009) 347–374; doi:10.1080/03091920903071077 R. Porter and D. Porter, ``Scattered and free waves over periodic beds'', J. Fluid Mech. 483 (2003) 129–163; doi:10.1017/S0022112003004208 P. F. Rhodes-Robinson, ``On the forced surface waves due to a vertical wave-maker in the presence of surface tension'' Math. Proc. Camb. Phil. Soc. 70 (1971) 323–337; doi:10.1017/S0305004100049926 P. F. Rhodes-Robinson, ``Note on the reflexion of water waves at a wall in the presence of surface tension'', Math. Proc. Camb. Phil. Soc. 92 (1982) 369–374; doi:10.1017/S0305004100060011