Title:

Some mixed boundaryvalue problems for twophase media in elastodynamics and electromagnetic theory

In this thesis we are concerned with studying some stress wave propagation problems in solids consisting of two dissimilar isotropic homogeneous elastic halfspaces (or semi infinite cylinders) bonded together along their common plane boundary as well as some electromagnetic wave propagation problems in waveguides filled with two different dielectric materials. The problems considered are (i) the torsional oscillations of a rigid circular disk symmetrically situated at the interface of two different infinite elastic halfspaces and is in welded contact with them, (ii) the diffraction of normally incident plane harmonic torsion stress waves by an immovable rigid circular disk securely attached to the interface of two different elastic halfspaces, (iii) problem (i) with the infinite halfspaces replaced by semi infinite cylinders, (iv) problem (ii) with the same restriction as in (iii), and finally (v) the diffraction of a normally incident electromagnetic field with no electric component in the direction of propagation, i. e. a TEwave, (or with no magnetic component in this direction, i,e. a TMwave) by a very thin perfectly conducting circular disk situated centrally in a circular waveguide filled with two different dielectric materials on either side of the disk. Using Hankel transforms in the solution of Problems (i) and (ii) and applying the mixed boundary conditions at the interface yield a pair of dual integral equations. The solution of these dual integral equations is reduced to that of a Fredholm integral equation of the second kind for an auxiliary function. The iterative solution of the integral equation at low frequencies is used to give approximations for some quantities of physical interest such as the stress at the diskelastic medium interface, the torque acting on the disk and the forward and backscatter coefficients. The results obtained when the parameters of the two different materials are set equal are in agreement with the known results whenever they exist. For problems (iii) and (iv) the solution is represented in the form of a FourierBessel series when the cylindrical surface is rigidly clamped (case (a) ) and in the form of a Dini series when this surface is stressfree (case (b) ). The mixed boundaryvalue problem is formulated as a pair of dual relations of FourierBessel series (case (a) ) or Dini series (case (b) ). The solution of the dual relations is then reduced to that of a Fredholm integral equation of the second kind for an auxiliary function. The iterative solution of the integral equation when the frequency and the ratio of the disk radius to that of the medium are small is employed as in problems (i) and (ii) to find approximate expressions for some relevant quantities. In problem (v) the electromagnetic diffracted field arising from a TE[01] mode incident field is given in terms of a Hertz vector. The mixed boundaryvalue problem for the determination of this Hertz vector is formulated as a pair of FourierBessel series similar to one obtained in the solution of problem (iv). The equivalent circuit for the discontinuity in the waveguide is described. We also attempt to present a mathematical formulation for the problem of the diffraction of a general TE incident field. The problem of the diffraction of a TM[01]mode is reduced to a Fredholm integral equation of the second kind which can be solved iteratively in the low frequency range when the ratio of disk radius to that of the waveguide is small compared with unity.
