Wave Propagation and Diffraction

This introductory chapter presents some of the methods that are useful for solving the problems of wave diffraction theory: method of separation of variables, method of power series, method of spline functions, and method of an auxiliary boundary. We also consider some algorithms for the numerical inversion of the Laplace transform, which is often used to solve the wave diffraction problems. Finally, we give a brief account of the method of multiple scales that is often used to study the propagation of transient waves.

This chapter deals with spectral methods in the theory of wave propagation. The main focus is given to the Fourier methods in application to studying the Stokes (gravity) waves on the surface of an inviscid fluid. A spectral method for calculating the limiting Stokes wave with a corner at the crest is considered as well. We also briefly consider the evolution of narrow-band wave trains on the surface of an ideal finite-depth fluid. Finally, a two-parameter method for describing the non-linear evolution of narrow-band wave trains is described by the example of the Klein–Gordon equation with a cubic nonlinearity. The problem is reduced to a high-order nonlinear Schrödinger equation for the complex amplitude of wave envelope. This equation is integrated numerically using a split-step Fourier technique to describe the evolution of quasi-solitons.

Refraction refers to any change in the wavefront configuration as a result of depth variation. This chapter deals with the numerical and analytical approachs of investigating the refraction of surface gravity waves that can be realised in the form of a computational programme and allows the distribution of wave fronts, rays and heights to be constructed and analysed for the case of the transition of regular waves from deep to shallow waters. The elaborated technique is based on the ray method and is implemented for two different cases—when the bottom surface is given either in analytical or tabular form. The refraction of waves with allowance for caustics is investigated as well. Based on the nonlinear theory of propagation of surface gravity waves, we also outline an approach describing the nonlinear refraction of waves and present the results demonstrating the effect of nonlinearity.

Wave interaction with rough bottom surfaces (topography), offshore drilling platforms and wave energy collectors is accompanied by the diffraction of waves. Here, an account is given of our research results in the field of wave diffraction. Specific aspects and methods used to solve the problems of the wave-diffraction theory are described in brief. The problem of wave diffraction by a submerged elliptical cylinder with elliptical lower face under an arbitrary incidence of plane waves is discussed. Wave diffraction by a submerged compound cylinder is also considered. Horizontal wave force is calculated as an example illustrating the dependence on the degree of submersion of the obstacle. Both the decrease of the submersion depth and the increase of the wave number are shown to favour the approach to the resonance range and thus worsen the reliability of the results. Scattering of magnetoacoustic waves by a cylinder is studied, the effect of magnetoelastic waves on the scattered field is demonstrated. Wave diffraction in a multiconnected domain formed by a set of vertical cylinders is analysed. The mutual influence of the cylinders is considerable for \( l/2a < 2 \), where l is the distance between the cylinders and 2a is the diameter. Interaction between diffracted fields is studied and the maximum lateral wave force that can also act outside the frontal cylinder is calculated. An exact analytical solution to the problem of wave diffraction by an asymmetrically inhomogeneous cylinder is found, the effect of asymmetry on the total scattering cross-section is analysed. An efficient numerical-analytical method is also considered, i.e. an auxiliary surface of simple shape (spherical or cylindrical) is introduced that surrounds the scatterer. In this case, the boundary-value problem can be subdivided into two problems, interior and exterior ones. The former is solved analytically in an infinite domain, the latter is solved numerically in a finite domain with a complex boundary. Furthermore, we study the wave diffraction by a prolate body of revolution consisting of a cylinder and spherical end cups. To solve the problem, a new finite-element algorithm is proposed.

The approach that is based on the repeated use of the method of images is used to solve the problems of stationary acoustic, electromagnetic, and elastic wave scattering and diffraction by cylindrical and spherical obstacles in a semi-infinite domain. The solution is written in terms of an infinite series of multiply diffracted fields. Explicit approximate asymptotic solutions are found and investigated for the case of distant scattered fields in the longwave approximation. The known solutions for point obstacles are obtained as special cases described by the first terms of the series.

This chapter deals with some aspects of the initial-boundary-value problems of the initiation, generation and propagation of tsunami waves. The generation of tsunami waves by bottom movements is considered. We formulate an appropriate initial-boundary-value problem and analyse the effect of the sharpness of vertical axisymmetric bottom disturbance and the disturbance duration on the generation of tsunami waves. The propagation of nonlinear waves on water and their evolution over a nonrigid elastic bottom are investigated. Some aspects and indeterminacy of the formulation of the initial-boundary-value problems dealing with the initiation and generation of tsunami waves are considered. We consider some typical types of tsunami waves that demonstrate the indeterminacy of their initiation in time because of the indeterminacy in the physical trigger mechanism of underwater earthquakes. Based on the three-dimensional formulation, evolution equations describing the propagation of nonlinear dispersive surface waves on water over a spatially inhomogeneous bottom are obtained with allowance for the bottom disturbances in time. We use the Laplace transform with respect to the time coordinate and the power series method with respect to the spatial coordinate to find a solution to the nonstationary problem of the diffraction of surface gravity waves by a radial bottom inhomogeneity that deviates from its initial position. The propagation and stability of nonlinear waves in a two-layer fluid with allowance for surface tension are analysed by the asymptotic method of multiscale expansions.