Equations of Mathematical Diffraction Theory

Let us consider a given incident wave p arriving from infinity. Diffraction is any change of this wave due to its interaction with some given obstacle. In this book, the diffraction is studied for different bounded and unbounded obstacles as interior and exterior of sphere, round disc, layer of constant thickness etc. Since the main processes that are studied are harmonic in time, i.e., p(x,t)=Re(exp(-iωt)q(x)), this leads to the study of the elliptic problem ∆p+k2p=0 on various domains. In the equation k=ω/c, ω is the angular frequency and c stands for the wave speed. In dependence on the properties of an obstacle (if it is acoustically hard or soft) the equation is equipped with Neumann or homogeneous Dirichlet boundary conditions. The main problems under consideration are: “How do we determine diffraction from the shape of the obstacle?” and “How do we determine the shape of the obstacle from known diffraction?” and “How to implement these methods numerically?”
The authors prefer to use a classical approach for the study of the problems, for example the method of Green’s function, the method of potentials and the Fourier transform. They decided not to be extremely formal and rather clearly explain main ideas. The book is readable and self-contained, assuming only knowledge of fundamentals of real, complex and functional analysis. Basic preliminaries are summarized in the first chapter. It follows that students interested in the subject can read it and as there are different problems studied by different methods it will serve for engineers when solving a given practical problem. An expert in this area would find that there were some original results from the authors of the book.

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