Volume 89, Issue 3
September 1980, pages 133-192
pp 133-157 September 1980
Ramanujan’s unpublished manuscripts which came to light recently show that he had made significant advances towards proving the conjectures, on the congruence properties of the partition function, which were made by him. He also had proved several congruence relations for prime moduli other than 5, 7 and 11. In this paper a complete account of Ramanujan’s work on the congruences for 5 and 7 and their powers and for the prime 13 is given.
pp 159-170 September 1980
We consider the diffraction of impulsive SV waves by a fluid circular cylinder. The cylinder is embedded in an unbounded isotropic homogeneous elastic medium and it is filled with some acoustic fluid. The line source, generating the incident pulse, is situated outside the cylinder parallel to its axis. We investigate the problem by the method of dual integral transformation as developed by Friedlander. The resulting integrals are evaluated approximately to obtain the short-time estimate of the motion near the wave front in the shadow zone of the elastic medium. We also interpret the approximate solution in terms of Keller’s geometrical theory of diffraction.
pp 171-181 September 1980
The combined approach of linearisation and finite difference method is used to solve the improved Boussinesq equation. A three-level iterative scheme having second order accuracy and constant coefficients matrix is devised and used in discussing the dynamics of waves having various initial wave packets. The results are in good agreement with the available results.
pp 183-192 September 1980
Growth and decay of weak discontinuities headed by wave front of arbitrary shape in three dimensions are investigated in an unsteady flow of a relaxing gas. The transport equations representing the rate of change of discontinuities in the normal derivatives of the flow variables are obtained and it is found that the nonlinearity in the governing equations plays an important role in the interplay of damping and steepening tendencies of the wave. An explicit criterion for the growth and decay of weak discontinuities along bicharacteristic curves in the characteristic manifold of the governing differential equations is given and special reference is made of diverging and converging waves under different thermodynamical situations. It is shown that there is a special case of a compressive converging wave, irrespective of the thermodynamical state whether weak or strong, in which the effects of thermodynamical influences and that of wave front curvature are unable to overcome the tendency of the wave to grow into a shock.