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Oblique wave scattering by a pair of partially immersed porous barriers, placed (i) on one side of (Position I), (ii) just above (Position II) and (iii) on the two sides (Position III) of a thick rectangular bottom-standing rigid obstacle, is studied here. The mathematical formulation and solution procedure of the corresponding boundary value problem are discussed using Havelock's expansion and Galerkin's approximation technique. During the solution procedure, a set of first-kind Fredholm-type integral equations is solved approximately by choosing suitable basis functions (Chebyshev and Gegenbauer polynomials). To address the singularities of order half and one-third that appear at the submerged edges of the thin barriers and the corners of the thick obstacle, appropriate basis polynomials with suitable weight functions are chosen in Galerkin's approximation. The reflection and transmission coefficients, dissipating wave energy, and dimensionless wave force are computed for various values of different parameters and depicted graphically in quite a number of figures. The numerical results for some special cases are compared with those available in the literature and excellent agreement has seen to have been achieved thus validating the correctness of the numerical method presented here. Based on numerical results it is observed that Position III of the barriers is more effective than Position I and Position II in resisting wave loads on marine areas and attenuating wave energy. [ABSTRACT FROM AUTHOR]

The focal point of the current study lies in investigating oblique wave scattering within the framework of linear potential theory, with particular attention to scenarios involving asymmetric trenches of both finite and infinite depths. By employing the eigenfunction expansion method, the physical problem undergoes a transformation into an equivalent boundary value problem. This newly formulated problem is characterized by a system of four weakly singular integral equations, which pertain to the horizontal component of velocity across the gaps situated above the edges of the trenches. The solution to these integral equations is achieved through the utilization of a multi-term Galerkin approximation method. This approach involves expansions using ultraspherical Gegenbauer polynomials as basis functions, coupled with the appropriate weight functions tailored to address the one-third singularity. Graphical representations are employed to depict the numerical evaluations of reflection and transmission coefficients across various non-dimensional parameters. These visualizations offer insight into the behavior and dependencies of these coefficients under different conditions. To validate the accuracy of the current model, it is compared against previously published results available in the literature. [ABSTRACT FROM AUTHOR]

The main objective of this article is to study how a finite crack interacts with horizontally polarized shear (SH) waves in a nonlocal isotropic medium when a magnetic field is present. The study aims to understand the effects of various parameters, including the magnetic field, non-locality, frequency of SH waves, on the behavior of the crack. Here, we are concerned with Griffith's theory of a brittle fracture crack mechanism. The article starts by formulating the governing equations for the propagation of SH waves in the given medium. These equations are converted into dual integral equations using Fourier transformation. Subsequently, an Abel transformation is applied to obtain a Fredholm integral equation of the second kind. Daubechies wavelet family has been used to find an approximate solution for the integral equation. The article aims to determine specific physical quantities including stress intensity factor (SIF), shear stress, and crack energy related to the behavior of the crack. These parameters are essential in understanding how the crack responds to the external conditions. The application of Daubechies wavelets in this context is a relatively new and innovative approach. The study performed a comprehensive analysis of the effects of various factors on the crack behavior. The findings are presented in graphical form to help illustrate the conclusions. To assess the accuracy of the proposed method, the obtained results are compared with known results from the literature for local media. This is an important step in demonstrating the reliability and effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]