Abstract The problem concerning water wave scattering by a floating breakwater has been an interesting subject of study under linearized water wave theory in the past few decades due to the reason that not only do these structures allow better water circulation to have smaller impact on sediment transport and fish migration, but also their construction is cost effective compared with the bottom founded structures. Modern study observes that thick breakwaters, with their larger and bulkier structure, are more effective at reducing wave energy compared with thin breakwaters. However, recent research reveals that breakwaters that combine both thin and thick parts offer enhanced performance by providing effective wave attenuation and reducing the construction costs. We may mention that these problems are primarily studied when the depth of water is uniform. However, in practice, the bathymetry of the water region is not uniform. This motivated us to study the problem of water wave scattering by a -shaped floating breakwater (which is a combination of thick and thin parts) in the presence of a rectangular submarine trench. For analytical solution of the corresponding boundary value problem, the geometric symmetry of the problem simplifies the mathematical difficulty considerably. For this, the domain of definition is divided into smaller subdomains and by suitable matching conditions at the various boundaries of the subdomains, the corresponding boundary value problem is formulated in terms of two Fredholm-type integral equations. The multi-term Galerkin approximation technique, involving Chebyshev polynomials and ultraspherical Gegenbauer polynomials as basis functions, is employed to solve the integral equations. It is important to note here that the choice of basis functions depends on the types of singularities at the corner of the trench and at the submerged sharp edge of the breakwater. The solutions of the integral equations are then employed to evaluate reflection and transmission coefficients. Graphical illustrations show that the length, width of the trench and the width of the breakwater have definite impact on the reflected wave, transmitted wave and the force on the breakwater. It is observed that the presence of trench produces oscillations in the reflection and transmission coefficients. When the width of the trench is equal to the width of the breakwater, reflection and transmission coefficients show smooth uniform oscillations. However, when the width of the trench and the width of the breakwater are unequal, nonuniform, nonsmooth oscillations in the reflection and transmission coefficients are observed. It is interesting to note that for a particular width of the breakwater, the reflected wave of certain wavelength shows resonating behaviour. Also, with the increase of submerged length of the side wall of the -shaped breakwater, the horizontal force on the breakwater enhances and the vertical force diminishes. Additionally, it is observed that when the breakwater is present in a water region with a trench, the horizontal force is more than the vertical force. However, when the breakwater is present in uniformly deep water without a trench, the vertical force is more than the horizontal force.
DAS et al. (Thu,) studied this question.