Sunil Chandra Barman

The first solution for the flexural-gravity wave propagation at the blocking and saddle points for a two-layer fluid is presented here. The compression on the plate-covered surface leads to wave blocking, where the group velocity vanishes. The blocking and saddle points are actually associated with a double and triple real root of the dispersion relation since the group velocity is the derivative of the dispersion relation. The solution in terms of eigenfunctions is restricted in the case of real and complex roots with multiplicity one of the dispersion relation. Now, we are able to overcome this restriction, and we newly obtain the complete solution for the velocity potential by establishing the orthogonal mode-coupling relation of the corresponding eigenfunctions for the flexural-gravity wave propagation at the blocking and saddle points using the Fourier transform. Further, the convergency of the newly derived solution at the blocking point is analyzed using the spectral representation of the associated eigenfunction. Next, we solve a simple model problem of flexural-gravity wave scattering by a crack in a floating ice sheet using the newly derived analytical solution at the blocking point. We show, by deriving the energy balance relation using Green’s theorem at the blocking point, that the validity of the energy identity is satisfied for the model problem. The ice sheet deflection and interface elevation are analyzed in the frequency domain at the blocking point, which is simulated in the time domain within and outside the blocking point frequencies.

Flexural-gravity wave blocking occurs in the presence of lateral compressive force. The existing expansion formulae for the velocity potential were limited in the case of distinct real roots of the dispersion relation with the multiplicity one. This limitation is overcome, and we obtain the complete solution for the flexural-gravity wave-maker problem at blocking and saddle points by deriving the expansion formula for velocity potential using the Fourier transform corresponding to the roots of the dispersion relation with multiplicity two and three, respectively. Further, we check the convergence of the newly derived expansion formulae by using the spectral representation of the associated eigenfunction. It has been proved that the eigenfunctions related to the expansion formulae are linearly dependent and satisfy the orthogonal mode-coupling relation. Next, we solve a model problem of flexural-gravity wave scattering by a crack using the derived expansion formulae and establish the energy balance relation by applying Green’s integral theorem at blocking points for infinite water depth. In addition, we have presented the validation of energy balance relation involving the generalized scattering coefficients.

Flexural-gravity wave scattering by an array of vertical porous barriers of various configurations is investigated in finite water depth from the viewpoint of blocking dynamics. A scattering matrix is introduced for the velocity potentials using the canonical eigenfunction expansion method developed for a single propagating wave mode to account for the multiple propagating wave modes. Subsequently, appropriate matching conditions are applied at the interface boundaries and edges to solve the physical problem. Apart from multiple barriers of equal length, the efficiency of four different barrier configurations of unequal lengths is investigated. This study shows that out of these four barrier configurations, the convex and increasing order of the barrier arrangements are more effective as wave-dissipating systems than the concave and decreasing order of the barriers. Bragg reflection occurs in the case of two or more barriers for a specific value of porosity and suitable barrier configuration, whose amplitude decreases with an increase in the number of barriers due to the dissipation of wave energy. The presence of three propagating wave modes in the blocking paradigm leads to mode conversion within a certain range of the frequency space. Both the scattering and dissipation coefficients are influenced by the wave energy transfer rates and the amplitudes of incident, reflected, and transmitted wave modes. This investigation exhibits the presence of discontinuities in the scattering coefficients at frequencies where blocking and mode conversion occur. The frequency domain results are used to simulate the plate displacement in the time domain by applying the Fourier transform.

Flexural-gravity wave scattering due to an array of polynyas is investigated from the perspective of the blocking dynamics. The canonical eigenfunction expansion method is generalized to account for multiple propagating wave modes within blocking frequencies. Bragg scattering occurs due to the presence of multiple gaps in the floating ice sheet, and the number of sub-harmonic peaks in wave reflection becomes one/two less than the number of gaps as the reflection coefficient varies with a change in gap/ice-sheet length. In addition, the amplitudes of harmonic peaks in wave reflection increase with an increase in the number of gaps. The variation of wave reflection with an increase in wavenumber/length of the ice sheet depicts that common zero minima occur for an even number of gaps, while common sub-harmonic maxima occur for an odd number of gaps. The scattering coefficients vary between zero and unity within the blocking frequencies, despite the individual amplitudes of the scattered waves becoming more than unity for certain frequencies. Noticeably, higher amplitudes of the scattered waves are associated with lower energy transfer rates and vice versa. Extrema in wave reflection occur for higher values of frequency within the primary and secondary blocking points. In addition, removable discontinuities are found in the scattering coefficient at the blocking frequencies, whereas a jump discontinuity is observed for certain frequencies within the blocking limits due to the incident wave mode conversion. Moreover, irregularities in the ice sheet's deflection are observed for any frequency within the blocking limit due to the superposition of three propagating wave modes.

The influence of wave blocking on the scattering of a flexural-gravity wave by a linear crack in a thin ice cover resting over a two-layer ocean having a rigid flat seafloor is investigated. The wave dispersion curve reveals the existence of multiple propagating wave modes within the blocking frequencies, either in the surface or interface mode. The Sommerfeld radiation condition depends on multiple propagating wave modes within the blocking frequency and contributes to wave energy propagation. The solution process involves the appropriate transition of wave modes within the blocking frequencies, which is obtained with the help of the dispersion curve. The reflection and transmission coefficients are generalized in the case of multiple propagating wave modes, and the associated energy balance relation is derived using Green's integral theorem. The scattering matrix is generated to describe all the possible transmitted and incident wave modes. The role of lateral compressive force and the density ratio on the scattering process, ice deflection, and interface elevation are shown graphically. This study reveals the occurrence of removable and jump discontinuities in the reflection and transmission coefficients at the saddle point as well as blocking frequencies and at the frequency for which incident wave mode changes. Irregular plate deflection and interface elevation patterns are found due to the superposition of multiple propagating wave modes within the primary and secondary blocking frequencies.

Bragg scattering of long flexural gravity waves due to an array of submerged trenches is studied under the shallow water approximation and small amplitude structural response in the presence of lateral compressive force. The group velocity vanishes at two different points in the frequency space for specific values of the compressive force, which are referred to as primary and secondary blocking points. Between these two blocking points, three propagating modes exist for each frequency, of which two are associated with the positive group velocity and one with the negative group velocity. Of the three propagating wave modes, the contribution to the energy relation by the lowest wavenumber is predominant near the secondary blocking frequency. In contrast, the higher wavenumber is dominant in the proximity of the primary blocking frequency. The study reveals the occurrence of Bragg scattering of flexural gravity waves in the presence of compressive force for more than two submerged trenches, which is analogous to that of surface gravity waves. However, within the blocking limits of the compressive force, the superposition of multiple propagating wave modes and the change in the incident wave mode contribute to certain irregularities and an increase in wave amplitude in the Bragg reflection pattern. The response amplitude of the structure and the pulse rate increase with an increase in the number of trenches.

The scattering of flexural-gravity waves in a thin floating plate is investigated in the presence of compression. In this case, wave blocking occurs, which is associated with both a zero in the group velocity and coalition of two or more roots of the related dispersion relation. There exists a region in the frequency space in which there are three real roots of the dispersion equation and hence three propagating modes. This multiplicity leads to mode conversion when scattering occurs. In one of these modes, the energy propagation direction is opposite to the wavenumber, making enforcement of the Sommerfeld radiation condition challenging. The focus here is on a canonical problem in flexural-gravity wave scattering, the scattering of waves by a crack. Formulae are developed that apply uniformly at all frequencies, including through the blocking frequencies. This solution is developed by tracking the movement of the dispersion relation roots carefully in the complex plane. The mode conversion is verified by the scattering matrix of the process and through an energy identity. This energy identity for the case of more than one progressive modes is established using Green's theorem and later applied in the scattering matrix to identify the incident and transmitted waves in the scattering process and derive the radiation condition. Appropriate scaling of the reflection and transmission coefficients are provided with the energy identity. The solution method is illustrated with numerical examples.