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In the present manuscript, the time dependent capillary gravity wave motion in the presence of current and undulated permeable bottom is analyzed. Spectral method is used to simulate the time dependent surface elevation. Also Laplace-Fourier transform method is used to obtain the integral form of surface elevation and asymptotic form of the associa...
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... Fig. 16, it is observed that, for large time, there is a small change in the amplitude of the surface elevation due to the change in the values of porosity parameter. Similar conclusion in the case of x component of particle velocity and for plane capillary gravity wave motion In Fig. 18, the effect of porosity parameter on surface elevation ...
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... Fig. 16, it is observed that, for large time, there is a small change in the amplitude of the surface elevation due to the change in the values of porosity parameter. Similar conclusion in the case of x component of particle velocity and for plane capillary gravity wave motion In Fig. 18, the effect of porosity parameter on surface elevation for large time and space is studied. It is observed that, there is a small increase in the amplitude of surface elevation due to the increase in the values of the porosity parameter for both large time and ...
Citations
... Bhatacharjee and Sahoo [38] studied the flexural-wave movement in a sea in the presence of currents and presented the effect of uniform currents on the phase and group velocities, wavelength and wave particle motion in the sea. Mohanty [39] discussed the timevariant capillary gravity wave movement in the presence of currents in a fluid that sealed by an irregular permeable base surface. Das et al. [40] employed the spectral techniques to highlight the wave blocking phenomenon in the case of flexural gravity wave proliferation in the presence of compression and currents. ...
... Mohanty et al. (2014) discussed the time-dependent flexural gravity wave motion in the presence of current using Green's function technique. Mohanty (2021) discussed the capillary gravity wave motion in the presence of current. Sahoo et al. (2001) derived an orthogonal relation and studied the scattering of surface waves by a floating elastic plate. ...
The role of wave blocking on flexural gravity wave motion with oblique incidence and current is
discussed. The oblique wave incidence due to a finite floating elastic plate is investigated using
the eigenfunction expansion method. The role of different wave propagation modes on reflection
coefficient is discussed. The reflection and transmission coefficients, the surface elevation, and plate
deflection in the presence of current are derived. The effects of flexural rigidity and the uniform
current speed on reflection and transmission coefficients are analyzed. The resonating nature of
the reflection coefficient for some fixed values of plate length is observed. Significant effects of different
combinations of wave propagation modes of plate-covered regions and open-water regions on the
reflection and transmission coefficients are observed. The wave energy propagation from reflection
coefficient decreases whenever the Froude number increases. It has been found that the amplitude
of the plate deflection increases as the value of the Froude number increases. The amplitude and
wavelength of the reflection coefficient are higher for the combination of the first propagating mode
of the plate-covered region and the second propagating mode of the open water region. Moreover, it is
found that zero reflection and full transmission occurs periodically. Co-propagating waves are found to
transmit more energy, while counter-propagating waves reflect more energy. Higher values of flexural
rigidity result in reduced wave energy transmission.
... All the above works are discussed in the absence of ocean current. The Bragg resonance analysis for small bottom undulation in the presence of current and surface tension for wave scattering problems is studied in [23]. It can be noted that along with the undulated bottom, a few researchers also considered the permeability in it and studied the Bragg resonance. ...
... The permeable bed is also having its own importance, as can be observed from the research work of Gayen & Islam [24]. Numerous fascinating phenomena such as wave-energy dissipation and damping, are triggered by its existence [20,[23][24][25]. To explore the wave damping brought on by the porous nature of the seabed, Corvaro et al. [26] conducted laboratory experiments. ...
... In this section, Bragg scattering in the presence of current and undulated permeable bottom is discussed. The associated bottom boundary condition (2.6) is given by Martha et al. [20] and Mohanty [23] ∂φ ∂z − Gφ = 0, on z = h + B(x), (3.25) where ( 1) ...
Three different types of resonances, i.e., blocking, trapping and Bragg resonance associated with flexural gravity wave motion in the presence of current and permeable bottom are discussed. Also, using the different modes of wave propagation, the interconnection among all these resonances is analyzed. An uneven pattern in the plate deflection is seen inside the blocking resonance zone. The uniform convergence of the infinite series associated with the trapped mode is proved. We have found that the trapped mode does not exist if the magnitude of the opposing current is higher or whenever we move the cylinder away from the plate covered surface. It is also observed that two propagating modes within the blocking frequencies create the environment for the occurrence of trapped waves. At the same time, all three modes have a remarkable contribution to Bragg resonance. The Bragg reflection increases in the blocking zone and decreases outside of it. We have seen that the amplitude of the Bragg reflection increases with the increase in current speed and porosity parameter. Also, the discontinuities in Bragg reflection coefficients are seen at blocking frequencies. The model is also carefully validated numerically for trapped and Bragg resonances against the results available in the existing literature.
... There are numerous important impacts of surface gravity waves and current interaction [20]. It impacts the design and placement of marine facilities [8] as well as the direction of littoral movement [38] in the coastal area. ...
... The fluid in each layer is assumed to be incompressible, inviscid and immiscible having constant but different densities and the flow is assumed to be irrotational. The fluid densities are denoted by s i (s 1 < s 2 ), i = 1, 2, where the former subscript refers to the 3 In the presence of uniform current speed U, Ψ i is of the form as given by Ψ i (x, y, t) = U x + ψ i (x, y, t) [20], where ψ i satisfies the Laplace equation obtained from the equation of continuity as: ...
In the present work, the scattering of water waves by undulating bottom in a two-layer fluid system is investigated by the inclusion of current, surface tension and interfacial tension to better understand the phenomenon of wave blocking. The perturbation technique followed by the Fourier transform method is applied to solve the coupled boundary value problem. The associated velocity potentials, Bragg reflection coefficients, and Bragg transmission coefficients are obtained in integral forms. A particular case of the undulating bottom, namely sinusoidal bottom undulation, has been taken into consideration for showing the effects of current speed and surface tension. A shift in the Bragg resonant frequency is observed with a change in the current speed. Further, the combined effects of Bragg resonance and wave blocking are investigated. For some values of opposing current, the group velocity vanishes at two distinct points in the frequency space; the maxima is known as the primary blocking point and the minima is called as the secondary blocking point. For each frequency, there exist three propagating modes between these two blocking points. Certain abnormalities and a sharp increase in the Bragg reflection and transmission coefficients are caused by the superposition of various propagating wave modes and triad interaction within the blocking points as well as a change in the incident wave mode.
Mathematics Subject Classification: 76B15, 42A38, 35Q35.
... The bottom boundary condition is given by (Martha et al. 20 , Mohanty 30 ) ...
... Mohanty and Sidharth [22] studied the transient flexural gravity wave motion in an ocean with a permeable bed in the presence of current. Later, the time-dependent capillary gravity wave motion in the presence of undulated permeable ocean bottom has been discussed by Mohanty [23]. ...
Wave generation problem generated by a moving oscillatory pressure disturbance (time harmonic) at the free surface of a finite depth ocean in the presence of surface tension is analyzed for permeable ocean bed. The governing initial boundary value problem is solved by using Laplace-Fourier transform technique. The method of stationary phase is used to evaluate the asymptotic solution of surface elevation. The asymptotic form of free surface is represented graphically in a number of figures. The influence of current speed, porosity parameter and surface tension on the behavior of surface elevation in the far field has been studied. Dispersion relation is also taken into account. Further, the phase and group velocities of wave motion for variation of parameters have been demonstrated through a large number of figures, and appropriate conclusions are drawn.
... Mohanty and Sidharth [24] studied the transient flexural gravity wave motion in an ocean with a permeable bed in the presence of current. Later, Mohanty [26] discussed the time-dependent capillary gravity wave motion in the presence of undulated permeable ocean bottom. ...
... Here we follow the procedure developed by Debnath and Rosenblat [7] to evaluate the integrals (26) and (27) separately for large values of |x| and t. For convenience of calculation, we first calculate J and then I. ...
Generation of magneto-hydrodynamic surface wave by a moving oscillatory disturbance in a finite depth ocean bounded below by a horizontal porous bottom is presented. The governing initial-boundary value problem is solved using Laplace-Fourier transform and an integral form of the surface elevation is obtained. The asymptotic solutions for the free surface elevation are obtained using stationary phase method. The results show that both the steady-state and transient components exist where the latter decays asymptotically and steady-state is reached. It is seen that the bottom permeability has a significant impact on surface elevation. The dispersion relation is also taken into account and the phase and group velocities for deep water and shallow water are analyzed for different values of the parameters in number of figures. The results for limiting case in which the current speed and the porosity vanishes are compared with the calculations investigated earlier.
The present work discusses the wave resonances in the presence of permeable bottom in the stratified fluid and in three dimensions. The blocking and Bragg resonances are analyzed and the role of primary and secondary blocking points on Bragg reflection and transmission in surface and interface mode are discussed. The effect of Bragg reflection and transmission due to obliquely incident wave modes are studied, and the interconnection between blocking and Bragg resonance is demonstrated. The nature of wave energy propagation obtained in the case of the frequency domain is simulated in the time domain using an asymptotic solution and spectral method. Two different types of initial disturbances, namely impulsive and Gaussian disturbance, are considered to analyze the time-dependent problem. In the case of initial impulsive disturbance, the asymptotic form of the plate deflection and interface elevations are obtained using the method of stationary phase. Also, for the initial Gaussian response, the spectral method is used to simulate the plate deflection and interface elevation. We observe that the flexural rigidity, compressive force, position of interface, and porosity parameter have significant impacts on the plate deflection and interface elevation. The dead water effect or the increase in the internal wave amplitude is observed as the densities of the two fluids approach each other.
A mathematical model is developed to analyze the non-linear wave interactions (diffraction and radiation) with submerged body of arbitrary geometry in time domain using combination of high-order spectral method (HOSM) and boundary element method (BEM). In this method, HOSM used for the representation of free surface potential under kinematic and dynamic boundary conditions, and for body representation of the BEM is utilized. The current numerical model is validated with the previous study. Further, the present model is utilized to study the effect of various incident waves on free surface and also to study the vertical and horizontal drift forces on the submerged circular cylindrical-shaped body. The current numerical model can be implemented on practical applications to compute the forces and analyze the effect of it.KeywordsHigh-order spectral methodBoundary element methodFinite difference method
The finite amplitude water wave theory in the presence of undulated porous bed and surface current is discussed. The nonlinear case is handled using Homotopy Analysis Method (HAM), whereas the linearized problem is solved using perturbation method and Fourier transform method. The eigenfunction expansion method is used to derive the solution of the associated linearized problem. These solutions are used in the zeroth-order approximation of the nonlinear case. For the nonlinear case, a system of decoupled differential equation is derived from where the surface elevation and velocity potentials are obtained. The effect of current and porosity parameter on group velocities is discussed. The effect of current on reflection coefficients is analyzed in the case of undulated bottom topography which is sinusoidal in nature. It is observed that if the wave number of the rippled bed is twice that of the wave number of the gravity wave, then the Brag resonance occur.KeywordsCurrentGravity waveUndulated bottomHomotopy analysis method
