ome boundary value problems for helmholtz equation arising in diffraction theory

Abstract

In recent years, the problem of wave diffraction by semi?infinite obstacles with edges, in particular half plane and strip under mixed boundary conditions (Dirichlet on the top and Neumann condition on the bottom surface) have gained importance while modelling noise reduction problems. One of the most powerful techniques of solving such wave diffraction problems, namely Wiener–Hopf technique, has reduced the problem to a system of Wiener–Hopf equations and there exists no direct way of handling such equations. However the solution to the above problem has been obtained by adopting ad?hoc procedures. With a view to provide an alternate method of solution to the above and allied problems, we have proposed a method based on integral transform technique (Laplace and Kontorovich–Lebedev) to solve the half plane diffraction problems and these form the first part of the thesis (Chapters 1, 2, 3, 4 and 5). The method has been illustrated by re?deducing the solution of the well?known classical Sommerfeld’s diffraction problem and later successfully applied to the problem of diffraction by a half plane under mixed boundary conditions; and the final scattered field has been expressed in closed form involving an incomplete integral. The far?field results have also been presented. The method has been applied to solve the problem of diffraction by a metallic half plane, with a view to examine the role of the impedance factor (which in general is small) in the final scattered field. Due to the impedance type of boundary conditions, the problem reduces to that of solving difference?type equations for the determination of the transformed field. A simple method of solving these equations has been presented by making use of the integral representation of the modified Bessel function. Finally the scattered field has been obtained in the case of plane wave incidence, with the assumption that the impedance factor is small, so that its higher powers could be neglected. Also the dependence of the impedance factor in the final scattered field is clearly exhibited which was masked in the solutions obtained earlier through other approaches. The above method has been extended to treat the problems of diffraction by a metallic half plane with different face impedances and by an acoustically penetrable or electro- magnetically dielectric half plane. In both these problems, we once again encounter the difference?type equations for the determination of the transformed field and they are solved as described earlier. The final form of the field has been obtained in closed form except for two infinite series involving Bessel functions and their derivatives with respect to their order. In the second part of the thesis, we consider problems of diffraction by a strip of finite width under various types of mixed boundary conditions. First the problem of diffraction by a strip of finite width under mixed boundary conditions is considered. An application of the Wiener–Hopf technique has reduced the problem to a system of coupled integral equations. These integral equations are solved up to first order by assuming the width ? of the strip large by the method of successive approximations. The zeroth?order equations and their solution have also been presented for the sake of completeness (the author acknowledges Dr. A. Chakrabarti for providing the pre?print). Using the higher?order solution the scattering coefficient has been computed and the additional terms are shown to be of order 1/?, where ? is the width of the strip. The above procedure yields neat results when applied to the problem of diffraction by an acoustically penetrable or electromagnetically dielectric strip of finite width. In this case the system of integral equations obtained are uncoupled and the solution is obtained up to first order. Scattering coefficient is computed and involves Whittaker functions. The above method has been formally extended to the problem of diffraction by a strip of finite width in the presence of a fluid medium and the scattered field obtained in a form suitable for far?field calculations. However, the physical problem at hand needs more care due to the presence of the edges in the medium. A section on Mathematical Preliminaries is provided to supplement the work reported in Chapters 1 to 8. A General Introduction is presented in the beginning surveying the previous work in connection with problems of diffraction by a half plane and a strip. The work reported in this thesis is partly based on the following papers and the papers which are being communicated for publication.

A note on Sommerfeld’s diffraction problem, V. V. S. S. Sastry and A. Chakrabarti, J. Math. Phys., V 20, n 10, pp. 2123. Cylindrical pulse diffraction by a half plane under mixed boundary conditions, A. Chakrabarti and V. V. S. S. Sastry, Can. J. Phys., V 57, n 9, pp. 1324. Cylindrical pulse diffraction by a metallic half plane, A. Chakrabarti and V. V. S. S. Sastry, Can. J. Phys. (in press).