Renuka Ravindran

Coupled amplitude theory of nonlinear surface acoustic waves

The nonlinear propagation characteristics of surface acoustic waves on an isotropic elastic solid have been studied in this paper. The solution of the harmonic boundary value problem for Rayleigh waves is obtained as a generalized Fourier series whose coefficients are proportional to the slowly varying amplitudes of the various harmonics. The infinite set of coupled equations for the amplitudes when solved exhibit an oscillatory slow variation signifying a continuous transfer of energy back and forth among the various harmonics. A conservation relation is derived among all the harmonic amplitudes.

A mathematical analysis of nonlinear waves in a fluid filled visco-elastic tube

SummaryOur investigations in this paper are centred around the mathematical analysis of a “modal wave” problem. We have considered the axisymmetric flow of an inviscid liquid in a thinwalled viscoelastic tube under certain simplifying assumptions. We have first derived the propagation space equations in the long wave limit and also given a general procedure to derive these equations for arbitrary wave length, when the flow is irrotational. We have used the method of operators of multiple scales to derive the nonlinear Schrödinger equation governing the modulation of periodic waves and we have elaborated on the “long modulated waves” and the “modulated long waves”. We have also examined the existence and stability of Stokes waves in this system. This is followed by a discussion of the progressive wave solutions of the long wave equations. One of the most important results of our paper is that the propagation space equations are no longer partial differential equations but they are in terms of pseudo-differential operators.ZusammenfassungDie vorliegenden Untersuchungen beziehen sich auf die mathematische Behandlung des „Modalwellen”-Problems. Die achsensymmetrische Strömung einer nichtviskosen Flüssigkeit in einem dünnwandigen viskoelastischen Rohr, unter bestimmten vereinfachenden Annahmen, wird betrachtet. Zuerst werden die Gleichungen des Ausbreitungsraumes im Langwellenbereich abgeleitet und eine allgemeine Methode zur Herleitung dieser Gleichungen für beliebige Wellenlängen bei nichtrotierender Strömung angegeben. Eine Operatorenmethode mit multiplem Maßstab wird verwendet zur Herleitung der nichtlinearen Schrödinger-Gleichung für die Modulation der periodischen Wellen, und die „langmodulierten Wellen” sowie die „modulierten Langwellen” werden aufgezeigt. Weiters wird die Existenz und die Stabilität der Stokes-Wellen im System untersucht. Anschließend werden die progressiven Wellenlösungen der Langwellengleichungen diskutiert. Eines der wichtigsten Ergebnisse dieser Arbeit ist, daß die Gleichungen des Ausbreitungsraumes keine partiellen Differentialgleichungen mehr sind, sondern Ausdrücke von Pseudo-Differentialoperatoren.

A new theory of shock dynamics part I: analytic considerations

A new theory for the calculation of the successive positions of a shock is suggested here. For one-dimensional problem, it requires integration of a finite number (say, 4 or 5) of ordinary differential equations derived from an infinite system of compatibility conditions. Numerical results in Section 2 show that error is less than 0.1 percent.

A New Theory of Shock Dynamics Part II: Numerical Solution

A new theory for the calculation of the successive positions of a shock was suggested in Part I. For one dimensional problems, it required integration of a finite number (say, 4 or 5) of ordinary differential equations derived from an infinite set of compatibility conditions. Numerical results here show that the error can be made less than 0.1 percent.

Shock propagation in gas dynamics : explicit form of higher order compatibility conditions

SummaryWith the use of tensor analysis and the method of singular surfaces, an infinite system of equations can be derived to study the propagation of curved shocks of arbitrary strength in gas dynamics. The first three of these have been explicitly given here. This system is further reduced to one involving scalars only. The choice of dependent variables in the infinite system is quite important, it leads to coefficients free from singularities for all values of the shock strength.

Long time behaviour of the solution of a system of equations from new theory of shock dynamics

The new theory of shock dynamics gives a system of n + 2 equations, the solution of which determines the propagation of a discontinuity in the initial condition. For the model equation ut + u ux = 0, the long time behaviour of the solution is investigated for various confined initial data. The suitability of using Pade approximants for the solution is examined. Special exact solutions are obtained.

A study of simple shearing flows in polar fluids

SummaryIn the present paper we have made a study of simple shearing flows in three polar fluids — (i) the model ofCondiff andDahler, (ii) the model ofEringen, and (iii) the model ofStokes. We have investigated in detail the components of stress, equations governing the motion and boundary conditions in the above fluids in flows induced by motion of boundaries or a pressure gradient. By means of tables, we have studied the common features of these fluids and related the fluid parameters involved in each case.

A CASE STUDY WITH THE NEW THEORY OF SHOCK DYNAMICS

Abstract A new theory of shock dynamics (NTSD) with analytic considerations and a numerical solution has already been discussed in Part I and Part II. This paper deals with the numerical simulation of the system of ordinary differential equations of the new theory of shock dynamics for the equation u t + ( u 2 2 ) x = 0 when ux(x, 0) = φ′(x)

Arrow’s impossibility theorem

In this article a number of voting models are considered and the drawbacks of each are indicated. The aim is to develop a voting method, which is based on individual preferences and which finally represents the choice of society. Arrow’s theorem addresses this problem.

The characteristic rule for shocks

The characteristic rule has so simplified the solution of problems of shock propagation that many researchers have used it, far beyond its realms of validity. With the help of a simple example we show that it fails to even come close to the exact solution in most cases.