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

Our investigations in this paper are centred around the mathematical analysis of a ldquomodal waverdquo 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 ldquolong modulated wavesrdquo and the ldquomodulated long wavesrdquo. 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.Die vorliegenden Untersuchungen beziehen sich auf die mathematische Behandlung des ldquorModalwellenrdquo-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 ldquorlangmodulierten Wellenrdquo sowie die ldquormodulierten Langwellenrdquo 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.

Long waves in inviscid compressible atmospheres: Isothermal atmosphere

Solitary waves and cnoidal waves have been found in an adiabatic compressible atmosphere which, under ambient conditions, has winds, and is isothermal. The theory is illustrated with an example for which the background wind is linearly increasing. It is found that the number of possible critical speeds of the flow depends crucially on whether the Richardson number is greater or less than one‐fourth.

Ion-acoustic solutions and shock waves in multicomponent plasmas

The present investigation of ion-acoustic waves is based on the study of the nonlinearity of plasma waves in a dispersive medium. Here the authors study ion-acoustic solitary waves in a warm ion plasma with non-isothermal electrons and then the results for solitary waves in a plasma with isothermal electrons are obtained. Incorporating the previous results obtained from the solitary wave solutions, the authors generalize the effect of negative ions on ion-acoustic waves in plasmas consisting of either a warm or cold ion species. A reflection phenomenon of ions in these waves is also studied. These results can be generalized, but the discussion is limited to a particular model of the plasma.

Long waves in inviscid compressible atmospheres: Isothermal atmosphere

Solitary waves and cnoidal waves have been found in an adiabatic compressible atmosphere which, under ambient conditions, has winds, and is isothermal. The theory is illustrated with an example for which the background wind is linearly increasing. It is found that the number of possible critical speeds of the flow depends crucially on whether the Richardson number is greater or less than one‐fourth.

A geodesic-based triangulation technique for damage location in metallic and composite plates

Lamb-wave-based damage detection methods using the triangulation technique are not suitable for handling structures with complex shapes and discontinuities as the parametric/analytical representation of these structures is very difficult. The geodesic concept is used along with the triangulation technique to overcome the above problem. The present work is based on the fundamental fact that a wave takes the minimum energy path to travel between two points on any multiply connected surface and this reduces to the shortest distance path or geodesic. The geodesics are computed on the meshed surface of the structure using the fast marching method. The wave response matrix of the given sensor configuration for the healthy and the damaged structure is obtained experimentally. The healthy and damage response matrices are compared and their difference gives the time information about the reflection of waves from the damage. A wavelet transform is used to extract the arrival time information of the wave scattered by the damage from the acquired Lamb wave signals. The computed geodesics and time information are used in the ellipse algorithm of triangulation formulation to locate the loci of possible damage location points for each actuator-sensor pair. The results obtained for all actuator-sensor pairs are combined and the intersection of multiple loci gives the damage location result. Experiments were conducted in aluminum and composite plate specimens to validate this method.

Alfvén waves in inhomogeneous magnetic fields: An integrodifferential equation formulation

An integrodifferential formulation for the equation governing the Alfvén waves in inhomogeneous magnetic fields is shown to be similar to the polyvibrating equation of Mangeron. Exploiting this similarity, a time‐dependent solution for smooth initial conditions is constructed. The important feature of this solution is that it separates the parts giving the Alfvén wave oscillations of each layer of plasma and the interaction of these oscillations representing the phase mixing.

Absorption of Electromagnetic Waves in Astrophysical Plasmas

We study parametric Decay Instabilities (PDI) using the kinetic description, in the homogeneous and unmagnetic plasmas. These instabilities cause anomalous absorption of the incident electromagnetic (e.m) radiations. The maximum plasma temperatures reached are functionas of luminocity of the non-thermal radio radiation and the plasma parameters.

Resonant absorption of Alfven waves and the associated phenomenon of magnetic reconnection

It is proposed that the mathematical analysis of the Alfven wave equation in inhomogeneous magnetic fields which explain the resonance absorption of Alfven surface waves near a resonant layer can also be used to show that the magnetic reconnection process can arise near the zero-frequency resonant layer driven by VLF Alfven surface waves. It is suggested that the associated phenomena of resonant absorption and magnetic reconnection can account for the recent observations of intense magnetic activity in the long-period geomagnetic micropulsation range, at cusp latitudes, during flux transfer events.

On the Existence of Hydromagnetic Interface Waves in a Structured Atmosphere

The conditions under which the hydromagnetic interface waves can exist at a magnetic interface is deduced. Using these conditions, it is shown that a slow interface wave with a phase velocity about 5Km/s and a fast interface wave with a phase velocity 6.5 to 8km/s at the photospheric level can exist.

Interaction of flux tubes with sound waves

The Landau damping of sound waves in a plasma consisting of ensemble of magnetic flux tubes is discussed. It is shown that sound waves cannot be Landau damped in general but under certain restricted conditions on plasma parameters the possibility of absorption of these waves can exist. The possibility of radiative damping of the acoustic waves along the magnetic filaments is also discussed. It appears that the most plausible mechanism of damping of sound waves in a plasma consisting of magnetic filaments can be due to scattering of a sound wave by the filaments.

A New Approach to the Problem of Scattering of Water Waves by Vertical Barriers

Using a modified Green's function technique the two well-known basic problems of scattering of surface water waves by vertical barriers are reduced to the problem of solving a pair of uncoupled integral equations involving the “jump” and “sum” of the limiting values of the velocity potential on the two sides of the barriers in each case. These integral equations are then solved, in closed form, by the aid of an integral transform technique involving a general trigonometric kernel as applicable to the problems associated with a radiation condition.