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A model of oblique wave diffraction by multiple arbitrary porous barriers in infinitely deep water is proposed to study the role of the porous breakwater in mitigating wave effects and dissipating wave energy. The Havelock's expansion of water wave potential along with suitable matching conditions and the single term Galerkin approximation method are used to handle the mathematical boundary value problem. The role of the arbitrary porous barrier is studied by analyzing the reflection coefficient, dissipated energy, and wave forces on the barriers. Significant changes are found in wave reflection and forces due to the consideration of porous barriers as compared to rigid barriers in the fluid region. As the separation length between the vertical barriers increases, the reflection coefficient becomes oscillatory as a function of the wavenumber, which is due to multiple reflections by the barriers. It is seen that the wave load reduces significantly with an increase in the number of barriers. It is also found that more energy is dissipated if the permeability of the barriers is increased. The correctness of the present method is confirmed by comparing the results available in the literature. [ABSTRACT FROM AUTHOR]

The Moore–Gibson–Thompson (MGT) generalized thermoelasticity theory together with the Eringen's nonlocal elasticity model is used to study the propagation of harmonic plane waves in an infinite elastic medium that conducts heat. Two sets of coupled longitudinal thermoelastic waves have been initiated. These waves are dispersive in nature and experience attenuation. In conjunction with the coupled waves, there also exists one uncoupled vertically shear–type (SV-type) wave. The SV-type wave is dispersive in nature but without any attenuation. The elastic nonlocality parameter influences all these waves. Furthermore, the SV-type wave faces critical frequency, while the coupled longitudinal waves may face critical frequencies conditionally. We also explore the problem of reflection of thermoelastic waves at a stress-free insulated and isothermal plane boundary of the medium considered here. The reflection coefficients and their respective energy ratios of the reflected waves to that of incident wave are determined analytically and numerically by considering an appropriate material. The numerical results are presented graphically to highlight the effects of various parameters of interest. Some important observations about the prediction of the MGT model in comparison to the other existing models (Lord–Shulman (LS) and the Green–Nagdhi theory of type III (GN-III) models) are highlighted. [ABSTRACT FROM AUTHOR]

On the basis of the linearized theory of water waves, the present study has demonstrated a semi-analytical method to assess the hydrodynamic performance of a pair of partially immersed barriers just above a thick bottom-standing barrier. By means of the eigenfunction expansion method, a system of the first kind Fredholm-type integral equation involving a horizontal component of velocity as unknown functions is developed for the interaction of water waves with both types of barriers. The multiterm Galerkin approximation is adopted to determine these unknown functions having square root singularities at the submerged edge of the thin barriers and one-third singularities at the corners of the thick barrier. In order to overcome such types of singularities, Chebychev polynomials for half-singularities and ultra-spherical Gegenbauer polynomials for one-third singularities with suitable weight functions have been taken into consideration. The numerical examples of both reflection and transmission coefficients are presented to examine the hydrodynamic performance of breakwater. Some fascinating results like resonant frequencies are obtained for practical engineering. At the same time, reflection coefficients for the present breakwater agree reasonably for the limiting cases with the previously available results. [ABSTRACT FROM AUTHOR]

A breakwater model is developed to analyse the wave propagation over an asymmetric rectangular trench in presence of two thin vertical partially immersed barriers. With the help of eigenfunction expansion, the problem is reduced to four weakly singular integral equations involving horizontal component of velocity across the gaps below the two barriers and above the two corners of the trench. The integral equations are solved by employing Galerkin technique involving expansion in terms of Chebyshev polynomials for the two unknown velocities across the gaps below the barrier multiplied by an weight function having square root singularity at an end and Gegenbauer polynomials for the other two unknown velocities above the two corners of the trench multiplied by an weight function having one‐third singularity at an end. Numerical results for the reflection and transmission coefficients are presented graphically against wavenumber for different configurations of the barriers and trench. The accuracy of the numerical results is validated by comparing the reflection coefficients and the known results associated with barriers and trench available in the literature. [ABSTRACT FROM AUTHOR]

The Lord–Shulman (LS) model of generalized thermoelasticity and the Eringen's model of nonlocal elasticity are used to formulate the new constitutive relations and the field equations. The propagation of plane thermoelastic waves are then probed in a homogeneous isotropic nonlocal thermoelastic solid by employing this new model. We found two sets of the coupled longitudinal waves and one independent shear‐type wave. All these waves are found to be dispersive and attenuating in nature in the presence of nonlocality in the medium. We found that the shear‐type wave faces a critical frequency, while the coupled longitudinal waves may face critical frequency conditionally. A reflection of thermoelastic waves at a stress‐free thermally insulated and isothermal boundary of a thermoelastic half‐space are also considered in case of incident coupled longitudinal thermoelastic wave. The amplitude ratios of the reflected waves to that incident wave are determined analytically. For a particular model, various graphs are plotted to analyze the behavior of the phase speeds, attenuation coefficients and reflection coefficients. To investigate the effect of elastic nonlocal parameter on the variations of phase speeds, attenuation coefficients and amplitude ratios of the reflected waves are presented graphically. Some interesting results are noticed: All the waves are detected to be influenced by the nonlocality of the medium. The longitudinal waves are influenced by the thermal parameters but the shear‐type wave is independent of the thermal effect. Moreover the presence of elastic nonlocality reduces the classical shear‐type wave speed. [ABSTRACT FROM AUTHOR]

Propagation of water waves over an infinite trench is examined here assuming linear theory where waves are incident from the direction of either negative infinity or positive infinity. For each case, the problem is reduced to solving coupled weakly singular integral equations of first kind involving horizontal component of velocity above the two edges of the trench. The integral equations are solved employing Galerkin approximation in terms of simple polynomials multiplied by appropriate weight functions whose forms are dictated by the edge conditions at the corners of the trench. The numerical estimates for reflection and transmission coefficients are depicted graphically against the wave number for different distance between the trenches and also for different heights of the asymmetric trench. [ABSTRACT FROM AUTHOR]

The problem of oblique water wave diffraction by three asymmetrically arranged vertical porous barriers with different widths in infinitely deep water is investigated. Reflection and transmission coefficients are evaluated, applying single-term Galerkin approximation. The amount of energy loss is derived employing Green's integral theorem. Wave force is determined using the linear Bernoulli equation of dynamic pressure jump, on the porous barriers. Forces and energy of the wave are studied to understand the effect of the dissipation of wave energy due to the presence porous barriers. Numerical results for the reflection coefficient, wave force, and wave energy dissipation are analysed graphically. The comparison of the numerical results are carried out for the validity of the derived solution. It is observed that porosity affects significantly on reflection coefficient and amount of energy dissipation. It is also found that more energy is dissipated if the permeability of the barriers is increased. [ABSTRACT FROM AUTHOR]

Scattering of surface waves by two permeable vertical walls having apertures at different depths in uniform finite depth water has been studied assuming linear theory. The problem is reduced to a set of coupled integral equations involving the difference of velocity potentials across each wall as unknown function. A multi-term Galerkin's technique is employed to approximate these unknown functions and to obtain approximate solution of the integral equations. Numerical estimates for the reflection and transmission coefficients are then obtained by solving a linear system. It is observed from graphical representations of the reflection coefficient that in comparison with non-identical walls the minima of the reflection coefficient become very close to zero at discrete frequencies in case of identical walls. Very good agreement between some earlier known results and our present results is established. [ABSTRACT FROM AUTHOR]

In this paper, we investigated the reflection of thermoelastic plane waves from the isothermal stress-free boundary of a homogeneous, isotropic and thermally conducting solid half-space in the context of the new linear theory of generalized thermoelasticity under heat transfer with memory-dependent derivative. It has been found that three types of basic waves consisting of two sets of coupled longitudinal waves and one independent vertically shear type wave may travel with distinct phase speeds. The formulae for various reflection coefficients are determined in case of an incident coupled dilatational elastic wave at an isothermal stress-free boundary of the medium. For an appropriate material, the reflection coefficients are computed numerically and presented graphically for various values of the angle of incidence and discussed the effect of various parameters of interest. At the end, the phase speeds and the attenuations coefficients of the coupled longitudinal waves are shown graphically to compare our results with the existing results. [ABSTRACT FROM AUTHOR]

In this paper, the propagation of time-harmonic plane waves is investigated in an infinite elastic solid material by employing the modified Green–Lindsay (MGL) model of generalized thermoelasticity. It is found that three basic waves consisting of two sets of coupled longitudinal waves and one independent vertically shear-type (SV-type) wave may travel with distinct speeds. The sets of coupled waves are found to be dispersive, attenuating and influenced by the thermoelastic coupling effect. In contrast to the Green–Lindsay (GL) as well as the Lord–Shulman (LS) models, the SV-type wave is not only dispersive in nature but also experiences attenuation. Reflection phenomenon of an incident coupled longitudinal wave from stress-free and thermally insulated boundary surface of a thermoelastic solid half-space is addressed. Using these boundary conditions, the formulae for various reflection coefficients and their respective energy ratios are presented. For a particular model, various graphs are plotted to analyze the behavior of the phase speeds, reflection coefficients and their respective energy ratios. The characteristics of employing the MGL model are discussed by comparing the numerical results obtained for the present model with those obtained in the case of the GL model. [ABSTRACT FROM AUTHOR]

In this paper, the propagation of time‐harmonic thermoelastic plane waves is studied in an infinite nonlocal elastic continuum. The type III Green‐Naghdi model (with energy dissipation) of generalized thermoelasticity and the Eringen's nonlocal elasticity model are adopted to address this problem. We found two sets of the coupled longitudinal waves which are dispersive in nature and experience attenuation. In addition to the coupled waves, there also exists one independent vertically shear‐type wave which is dispersive but experiences no attenuation. All these waves are found to be influenced by the elastic nonlocality parameter. Furthermore, the shear‐type wave is found to face a critical frequency, while the coupled longitudinal waves may face critical frequencies conditionally. Reflection phenomenon of an incident coupled longitudinal waves from a rigid and thermally insulated boundary surface of a homogeneous and isotropic nonlocal thermoelastic half‐space is investigated. Using these boundary conditions, the formulae for various reflection coefficients and their respective energy ratios are presented. For a particular model, various graphs are plotted to analyze the behavior of the phase speeds, reflection coefficients and their respective energy ratios. The amplitude ratios of the reflected waves and their respective energy ratios are determined analytically. For a particular model, the effect of elastic nonlocality parameter on the variations of phase speeds, attenuation coefficients, amplitude ratios and corresponding energy ratios of the reflected waves are presented graphically. Finally, analysis of the various results have been interpreted. [ABSTRACT FROM AUTHOR]

The generalized thermoelasticity theory based upon the Green and Naghdi model III of thermoelasticity as well as the Eringen's nonlocal elasticity model is used to study the propagation of harmonic plane waves in a nonlocal thermoelastic medium. We found two sets of coupled longitudinal waves, which are dispersive in nature and experience attenuation. In addition to the coupled waves, there also exists one independent vertically shear‐type wave, which is dispersive but experiences no attenuation. All these waves are found to be influenced by the elastic nonlocality parameter. Furthermore, the shear‐type wave is found to face a critical frequency, while the coupled longitudinal waves may face critical frequencies conditionally. The problem of reflection of the thermoelastic waves at the stress‐free insulated and isothermal boundary of a homogeneous, isotropic nonlocal thermoelastic half‐space has also been investigated. The formulae for various reflection coefficients and their respective energy ratios are determined in various cases. For a particular material, the effects of the angular frequency and the elastic nonlocal parameter have been shown on phase speeds and the attenuation coefficients of the propagating waves. The effect of the elastic nonlocality on the reflection coefficients and the energy ratios has been observed and depicted graphically. Finally, analysis of the various results has been interpreted. [ABSTRACT FROM AUTHOR]

Abstract A study of obliquely incident water wave scattering by two, three and four unequal partially immersed vertical barriers in water of uniform finite depth has been carried out in this paper employing Havelock's expansion of water wave potential. A formulation involving integral equations in terms of either horizontal component of velocities across the gap below each barrier or difference of potentials across each barrier are obtained using the Havelock's inversion formulae. A multi-term Galerkin approximation technique with Chebychev's polynomials (multiplied by appropriate weights) as basis functions is adapted to solve these integral equations and to compute the reflection and transmission coefficients numerically. The numerical results are depicted graphically against the wavenumber in several figures for various arrangements of the vertical barriers. From these figures zeros of reflection coefficient are observed when the vertical barriers are immersed upto equal depths below the mean free surface. However, this observation is not true always for non-identical vertical barriers. For two non-identical partially immersed barriers reflection coefficient never vanishes whereas for three non-identical partially immersed barriers reflection coefficient vanishes at discrete frequencies if the two outer barriers have equal lengths of submergence. For four partially immersed barriers arranged symmetrically about a vertical line, zeros of reflection coefficient are always observed. The known results of a single barrier are recovered as special cases so as to establish the correctness of the present method. [ABSTRACT FROM AUTHOR]

This paper demonstrates the reflection phenomenon of the magneto-thermoelastic plane waves from a stress-free surface of a homogeneous, isotropic, thermally and electrically conducting solid half-space. We employ the generalized thermoelasticity model with memory-dependent derivative (MDD) for this study. We find that three basic plane waves consisting of two sets of coupled longitudinal waves and one independent shear type wave may travel with distinct speeds in the medium. The speeds of the coupled longitudinal waves are plotted graphically to predict a comparison for the MDD and Lord–Shulman models. At last, for an appropriate material, the reflection coefficients are computed numerically and presented graphically with the angle of incidence of the incident (vertically polarized) shear type wave and the obtained results are explained. [ABSTRACT FROM AUTHOR]

A solution to the problem of wave interaction with multiple flexible porous barriers is presented using the Galerkin approximation technique in deep water. The study has analyzed the reflection coefficient, dissipated energy, and wave forces on the barriers and presented results for different inclination angles, separation lengths, porosity, and flexural rigidity of the barriers. The comparison of the results with those available in the literature adds credibility to the study. The finding that the flexural rigidity of the porous barriers has a significant effect on wave reflection, dissipated energy, and wave forces is an important contribution to the field. The observation that an increase in the number of barriers reduces the wave load significantly is also noteworthy. Overall, this study provides valuable insights into the behavior of wave interaction with multiple flexible porous barriers, which could have practical implications for coastal protection and wave energy conversion systems. [ABSTRACT FROM AUTHOR]

A coupled model of wave interaction with a rectangular bar in the presence of two trenches are proposed to study in the context of linear potential theory. The physical problem is converted to a set of integral equations using the eigenfunction expansion method by matching pressure and velocity at the interface boundaries of the structure. After solving the integral equations, the analytical solutions utilize the multi-term Galerkin method involving the Gegenbauer polynomials of order 1 / 6 with suitable weights as basis functions. The convergence and correctness of the method are confirmed by comparing the results with the one available in the literature. Energy balance relations for the present problem are derived and used to check the accuracy of the computed results. The effects of various parameters are analyzed through different graphs for a rectangular bar in the presence of submerged rectangular trenches. It is found that the reflection coefficient is increasing with increasing the rectangular bar length, thickness, and trench thickness. It can be concluded that the physical parameters of the structure have a significant effect on the present study. Moreover, it is observed that the value of the reflection coefficient reduces for the presence of a single trench compared to the presence of multiple trenches. The present study may be useful for understanding the performance of submerged rectangular trenches to protect the rectangular bar from wave action. [ABSTRACT FROM AUTHOR]

This paper considers the reflection and transmission of flexural gravity waves by a pair of bottom-mounted rectangular barriers of arbitrary height and width. Depth of the fluid considered here is finite but not necessarily uniform. The fluid region is bounded by an infinite long thin ice sheet of homogeneous properties. Using linearized water wave theory the velocity potential associated with the motion is formulated. Eigenfunction matching technique is used to convert the boundary value problem to a set of integral equations which are solved by using multi-term Galerkin method. The reflection and transmission coefficients are calculated and represented graphically. An expression for the energy balance relation is given and checked carefully. A model of the problem in which the fluid is bounded by a free surface is recovered. For a set of physical parameters the results are analyzed through a number of figures. The convergence and correctness of the method is confirmed by comparing the results with the one available in the literature. It is shown that a pair of thick bottom-mounted barriers have significant impact on reflected and transmitted flexural gravity waves. Resonance effects are observed and verified for different parameters of the current problem. • The oblique scattering of flexural gravity waves by a pair of bottom-mounted rectangular barriers is considered. • Eigenfunction matching technique and multi-term Galerkin method are used. • Characteristics of flexural gravity waves, namely length and phase velocity are discussed. • The reflection and transmission coefficients are calculated numerically and represented graphically. • Results are discussed for resonating reflections in presence of a pair of rectangular barriers. [ABSTRACT FROM AUTHOR]

The interaction of water waves with multiple flexible porous barriers partially immersed in a uniform finite-depth fluid is investigated in the present study. Using linear theory, the physical problem is mathematically modeled as a boundary value problem. The eigenfunction expansion approach is used to generate the velocity potential function in terms of orthogonal eigenfunctions. The Havelock's inversion formula is then used to convert the boundary value problem into a set of Fredholm-type integral equations. By choosing a suitable weight function, the multi-term Galerkin's approximation technique is used to obtain the mathematical solutions of these integral equations in terms of the Chebyshev polynomial. Numerical solutions regarding the reflection and transmission coefficients, hydrodynamic forces, and dissipated wave energy at the barriers are obtained and graphically represented in several figures for various non-dimensional parameters. It is observed from the graphical representations that the permeability and flexibility of the barriers play a crucial role in the modeling of efficient breakwaters. • This study examines the interaction between obliquely incident waves and multiple thin poroelastic barriers in a uniform finite-depth fluid. • Havelock's inversion formula and the multi-term Galerkin approximation technique are employed to solve the BVP. • Graphical representations depict reflection and transmission coefficients, dissipated energy, and hydrodynamic force. • Porous, flexible barriers effectively reduce wave impact by dissipating wave energy through their pores and mitigating wave forces. • Increasing the number of barriers in coastal breakwaters reduces wave effects by altering reflection and reducing transmission coefficients. [ABSTRACT FROM AUTHOR]