| dc.description.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). | |
