| dc.description.abstract | This thesis consists of four chapters including a chapter on general introduction and deals with some problems in nonlinear wave propagation in multi-dimensions. The problems are: nonlinear wave propagation in a two-dimensional steady transonic flow, nonlinear modulation of three-dimensional wave packets on surface water waves, nonlinear modulation of three-dimensional waves in rotating fluids and propagation of multi-dimensional fast ion-acoustic waves.
In compressible fluid flow containing a sonic transition, the upstream propagating waves perpendicular to the streamlines for which each of the bicharacteristic velocity components vanishes at a sonic point, remain in the transonic region for a time large compared with other waves and this explains the existence of almost trapped waves with weak shocks in a transonic flow. In Chapter 2 of this thesis, we have studied these waves in detail. Under the assumption that a characteristic length in the steady state and the radius of curvature of the wave front are large compared with the extent of the wave in the direction normal to the wave front, and that the angle between the normal to the wave front and the streamlines is small, we have discussed the turning effect of the waves by using a model equation. This model equation has been derived from an approximate equation for the perturbations of a steady solution of the equations of motion of an inviscid, non-conducting polytropic gas. This equation, governing the motion of the perturbations of the steady flow, is a first-order quasi-linear partial differential equation. It completely takes into account the curvature of the wave front, the nonlinear steepening of the pulse and the two-dimensional turning effect and gives the complete history of the pulse as it moves in the transonic region. We have also done a local analysis of the model equation in the neighbourhood of points on the sonic line, assuming the initial wave front to be plane. By this analysis we have clearly shown the interplay of the effects of nonlinear steepening, two-dimensional turning and wave front curvature. We have considered two special cases of the local analysis: (i) When the initial value of the angle ? between the normal to the wave front and direction is zero and (ii) When the initial value of ? is non-zero. We have computed the complete history of these two pulses by numerically integrating the model equation.
In Chapter 3, the problem of nonlinear modulation of a periodic sinusoidal wave has been discussed for two different physical systems. The first system consists of surface water waves which is an isotropic one. The second one comprises waves in rotating fluids which is an example of an anisotropic system.
In the first case we have used the method of operators of multiple scales of Hasimoto and Ono to derive the modulation equations and have derived the 'complete' equations showing the 'full' two-dimensional nature of the modulation of water waves. We have also discussed its equivalence with the simpler system of equations of Davey and Stewartson where they have considered a phase function of the form kx - ?t; k being the wave number and ? the frequency. When there is circular symmetry in the dispersed wave, we have shown that the two coupled equations reduce to the nonlinear Schrödinger equation with an extra term representing the effect of the curvature of the wave front.
In the second case, we have considered waves in rotating fluids. The fact that small disturbances in a uniformly rotating incompressible fluid can propagate as wave motions has long been established. By using the Krylov-Bogoliubov-Mitropolsky method we have derived the modulation equation governing the slow variations of the amplitude of waves propagating in the medium. Even though we are working with a nonlinear theory, it is really worth noticing that the nonlinear term (which generally appears when such techniques are used to other systems) just drops out of the equation. This linear equation shows that the original plane wave solution is always modulationally stable.
In Part A of Chapter 4, we have rederived some of the known results using an approximate equation governing the perturbation of the dependent variables of a first-order quasilinear hyperbolic system which gives the complete history of a disturbance not only at the wave front but also within a short distance behind the wave front. We have deduced the transport equation of Varley and Gumberbatch giving the intensity of discontinuities in the normal derivatives of the dependent variables at a wave front. Whitham’s theory for calculating the complete history of nonlinear sound pulses of arbitrary shapes in a compressible medium has also been derived using the same approximate equation.
In Part B, we have derived a multi-dimensional Korteweg-de Vries equation in the case of fast ion-acoustic waves in a collisionless plasma consisting of hot and isothermal electrons and cold ions with applied constant magnetic field in z-direction. The most important point we have noticed is that the external magnetic field does not contribute anything to the final equation. We have also deduced the equation in the case of a spherical wave front.
A major part of this thesis has been published in the form of papers as given below:
(1) Nonlinear wave propagation in a two-dimensional steady transonic flow, J. Fluid Mech., 82, 17, 1977 (in collaboration with Phoolan Prasad).
(2) On multi-dimensional packets of surface waves, J. Phys. Soc. Japan, 44, 1028, 1978 (in collaboration with Phoolan Prasad).
(3) Nonlinear modulation of periodic waves in rotating fluids. (To be published).
(4) Appendix to ‘Approximation of the perturbation equations of a quasilinear hyperbolic system in the neighbourhood of a bicharacteristic’, J. Math. Anal. Appl., 54, 470, 1975; J. Math. Anal. Appl., 60, 716, 1977.
(5) A note on the Korteweg-de Vries equation for the propagation of fast ion-acoustic waves in multi-dimensions, Phys. Lett., 62A, 483, 1977 (in collaboration with Phoolan Prasad). | |
