Vellore Institute of Technology (VIT) is a private university located in&nbsp;Tamil Nadu, India. Founded in 1984, as Vellore Engineering College, the institution offers 20 undergraduate, 34 postgraduate, four integrated and four research programs. It has campuses in Vellore, Amravati, Bhopal and Chennai.

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VIT University

In the present manuscript, the time-dependent capillary gravity wave motion in the presence of a current and an undulated permeable bottom is analyzed. The spectral method is used to simulate the time-dependent surface elevation. Also, the Laplace–Fourier transform method is used to obtain the integral form of the surface elevation, and the asymptotic form of the associated highly oscillatory integral is derived using the method of stationary phase. The reflection and transmission coefficients due to the small bottom undulation are obtained using the perturbation method and the Fourier transform method and also alternatively using Green’s function technique and Green’s identity. The nature of wave energy propagation obtained from plane capillary gravity wave motion is verified through time domain simulation and using the spectral method. It is found that, in the case of co-propagating waves, the wave energy propagates faster and also the surface profiles in terms of wave packets move faster for larger values of the Froude number. Also, the maximum value of the reflection and transmission coefficients decreases due to increasing values of the Froude number. For the sinusoidal bottom topography, the Bragg resonance occurs if the ratio of the wave numbers of the wave and the rippled bed is one by two. © 2020, Springer-Verlag GmbH Austria, part of Springer Nature.

Department of Mathematics

School of Advanced Sciences

Vellore Campus

Acta Mechanica

Time-dependent wave motion with undulated bottom

The present study deals with the transient flexural gravity wave motion associated with floating elastic plate in the presence of permeable bottom. Integral form of the floating plate deflection is obtained analytically using Laplace–Fourier transform method and the asymptotic solution is derived for large time using stationary phase method. Also, using fast Fourier transform method the time domain simulation of floating plate deflection is obtained. The effect of current, compressive force and porosity parameter on phase and group velocities and floating plate deflection are analyzed. It is observed that in the absence of current, due to the presence of porosity parameter there exist two points where there is a situation of no wave propagation. It is also observed that the current and compressive force have significant effect on transient flexural gravity wave motion whereas the porosity parameter has small impact on the gravity wave motion associated with floating elastic plate. The properties of the roots of the dispersion relation associated with the flexural gravity wave motion in the presence of permeable bed are discussed analytically and numerically. © 2020, Springer Nature B.V.

Meccanica

Time dependent wave motion in a permeable bed

The present manuscript deals with the generation of flexural gravity waves due to transient axi-symmetric disturbances in two-layer fluid. Roles of structural rigidity and compressive force on the transient flexural gravity wave motion are analyzed under the assumption of small amplitude water wave theory and structural responses in the case of finite water depth. Using Laplace and Hankel transforms, integral forms of the velocity potentials, structural deflection, surface and interface elevations are derived for the flexural gravity wave motion. Asymptotic solutions for surface and interface elevations are derived in special cases using method of stationary phase for large time and space. © 2019, Springer Nature India Private Limited.

International Journal of Applied and Computational Mathematics

Transient Axi-Symmetric Disturbances in Two-Layer Fluid

In the present manuscript, flexural gravity wave generation due to an initial disturbance in the case of two-layer fluid of finite depth in the presence of current is analyzed in three dimensions. From the plane progressive wave solution, effect of wave and current direction and the combined effect of current and compressive force on flexural gravity wave motion are discussed. Using Laplace and Fourier transform, the solution of the transient wave structure interaction problem is obtained and integral forms of velocity potentials, structural deflection and interface elevations are derived. Assuming the initial disturbances is on the plate covered surface asymptotic solutions for structural deflection and interface elevations are obtained using method of stationary phase for large time and space. The study reveals that the effect of opposing current and critical compressive force in the presence of current has significant impact on the plate deflection. © 2019, Springer Nature B.V.

Transient flexural gravity waves in two-layer fluid

Combined effects of current and surface tension on time-dependent transient boundary value problems (BVPs) associated with capillary-gravity wave motion in a two-layer fluid are analyzed under the assumption of small amplitude water wave theory in three dimensions. Plane wave solutions are discussed by analyzing the roots of the dispersion relation and phase velocity. Using the Laplace transform technique and Green’s function method, the time-dependent initial boundary value problem (IBVP) for the capillary-gravity wave motion are analyzed in both the cases of finite and infinite water depths. Using the method of stationary phase, the time-dependent Green’s function and surface and interface profiles are computed and analyzed for various values of physical parameters in two different cases. The time harmonic solutions are obtained as one of the special cases of the time-dependent IBVP in both the cases of finite and infinite water depths. It is observed that the direction of current and wave changes the phase velocity of the capillary-gravity wave motion along with the free surface and interface elevations. The position of the source significantly changes the amplitude of the source potentials in surface and internal modes. Further, the location of the interface has a significant influence on the pattern of surface and interface elevations. © 2014, Springer-Verlag Wien.

World University root node

Time-dependent capillary-gravity waves in the presence of current

In the present study, a combined effect of current and compressive force on time dependent flexural gravity wave motion in both the cases of single and two-layer fluids is analyzed in finite and infinite water depths in two dimensions. The roots of the dispersion relation associated with the plane flexural gravity waves are analyzed via contour plots and by plotting various terms of the dispersion relation separately. The characteristic of plane flexural gravity waves is studied by analyzing the phase and group velocities along with the law of conservation of energy flux to understand the combined effect of current and compressive force on the wave motion. The integral form of the time dependent Green's function in the presence of current is obtained using the Laplace transform method and used in Green's identity to derive the time dependent velocity potential for the flexural gravity wavemaker problem. The time harmonic Green's function and velocity potentials are obtained as a special case from the time dependent problems. Numerical results are computed and analyzed in particular cases using the method of stationary phase to obtain the asymptotic results for Green's function and the deflection of ice sheet. The integral form of Green's function derived here will be suitable to deal with physical problem when the roots of the dispersion relation for the flexural gravity wave problem coalesces which were otherwise not possible in the eigenfunction expansion method used for time harmonic problems. © 2013 Elsevier Ltd.

Journal of Fluids and Structures

Time dependent flexural gravity waves in the presence of current

In the present paper, expansion formulae for the velocity potentials and associated orthogonal modecoupling relations for 3D Laplace equation satisfying higher order boundary conditions arising in fluid structure interaction problems are derived. Characteristics of the eigensystems are derived in terms of several identities and the convergence of the developed expansion formulae is demonstrated. As an application of the expansion formulae, flexural gravity wave scattering due to a crack in a floating ice sheet is investigated. The present method of solution can be easily applied to a large class of problems in the area of wave structure interaction arising in ocean engineering and various branches of applied mathematics, mathematical physics and engineering. © 2011 The Authors 2011. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.