3-D kinematical conservation laws (KCL): Evolution of a surface in R-3 - in particular propagation of a nonlinear wavefront

3-D KCL are equations of evolution of a propagating surface (or a wavefront) Omega(t), in 3-space dimensions and were first derived by Giles, Prasad and Ravindran in 1995 assuming the motion of the surface to be isotropic. Here we discuss various properties of these 3-D KCL.These are the most general equations in conservation form, governing the evolution of Omega(t) with singularities which we call kinks and which are curves across which the normal n to Omega(t) and amplitude won Omega(t) are discontinuous. From KCL we derive a system of six differential equations and show that the KCL system is equivalent to the ray equations of 2, The six independent equations and an energy transport equation (for small amplitude waves in a polytropic gas) involving an amplitude w (which is related to the normal velocity m of Omega(t)) form a completely determined system of seven equations. We have determined eigenvalues of the system by a very novel method and find that the system has two distinct nonzero eigenvalues and five zero eigenvalues and the dimension of the eigenspace associated with the multiple eigenvalue 0 is only 4. For an appropriately defined m, the two nonzero eigenvalues are real when m > 1 and pure imaginary when m < 1. Finally we give some examples of evolution of weakly nonlinear wavefronts.

Surface explosive instability for high frequency waves

High frequency three-wave nonlinear 'explosive' interaction of the surface modes of a semi-infinite beam-plasma system under no external field is investigated. The conditions that favour nonlinear instability, keep the plasma linearly stable. The beam runs parallel to the surface. If at least one of the three wave vectors of the surface modes is parallel to the beam, explosive interaction at the surface takes place after it has happened in the plasma bulk, provided the bulk waves propagate almost perpendicular to the surface and are of short wavelength. On the other hand if the bulk modes have long wavelength and propagate almost parallel to the surface, the surface modes can 'explode' first.

Periodic Rayleigh waves of finite amplitude on an isotropic solid

Existence of a periodic progressive wave solution to the nonlinear boundary value problem for Rayleigh surface waves of finite amplitude is demonstrated using an extension of the method of strained coordinates. The solution, obtained as a second-order perturbation of the linearized monochromatic Rayleigh wave solution, contains harmonics of all orders of the fundamental frequency. It is shown that the higher harmonic content of the wave increases with amplitude, but the slope of the waveform remains finite so long as the amplitude is less than a critical value.

Surface explosive instability for high frequency waves

High frequency three-wave nonlinear 'explosive' interaction of the surface modes of a semi-infinite beam-plasma system under no external field is investigated. The conditions that favour nonlinear instability, keep the plasma linearly stable. The beam runs parallel to the surface. If at least one of the three wave vectors of the surface modes is parallel to the beam, explosive interaction at the surface takes place after it has happened in the plasma bulk, provided the bulk waves propagate almost perpendicular to the surface and are of short wavelength. On the other hand if the bulk modes have long wavelength and propagate almost parallel to the surface, the surface modes can 'explode' first.

On Multi-Dimensional Packets of Surface Waves

By using the method of operators of multiple scales, two coupled nonlinear equations are derived, which govern the slow amplitude modulation of surface gravity waves in two space dimensions. The equations of Davey and Stewartson, which also govern the two-dimensional modulation of the amplitude of gravity waves, are derived as a special case of our equations. For a fully dispersed wave, symmetric about a point which moves with the group velocity, the coupled equations reduce to a nonlinear Schrödinger equation with extra terms representing the effect of the curvature of the wavefront.

Trapped waves in the neighborhood of a sonic-type singularity

A model equation is derived to study trapped nonlinear waves with a turning effect, occurring in disturbances induced on a two-dimensional steady flow. Only unimodal disturbances under the short wave assumption are considered, when the wave front of the induced disturbance is plane. In the neighbourhood of certain special points of sonic-type singularity, the disturbances are governed by a single first-order partial differential equation in two independent variables. The equation depends on the steady flow through three parameters, which are determined by the variations of velocity and depth, for example (in the case of long surface water waves), along and perpendicular to the wave front. These parameters help us to examine various relative effects. The presence of shocks in a continuously accelerating or decelerating flow has been studied in detail.

Propagation of quasi-simple waves in a compressible rotating atmosphere

A class of self-propagating linear and nonlinear travelling wave solutions for compressible rotating fluid is studied using both numerical and analytical techiques. It is shown that, in general, a three dimensional linear wave is not periodic. However, for some range of wave numbers depending on rotation, horizontally propagating waves are periodic. When the rotation ohgr is equal to $$\sqrt {(\gamma - 1)/(4\gamma )}$$ , all horizontal waves are periodic. Here, gamma is the ratio of specific heats. The analytical study is based on phase space analysis. It reveals that the quasi-simple waves are periodic only in some plane, even when the propagation is horizontal, in contrast to the case of non-rotating flows for which there is a single parameter family of periodic solutions provided the waves propagate horizontally. A classification of the singular points of the governing differential equations for quasi-simple waves is also appended.

A characteristic based numerical method with tracking for nonlinear wave equations

Many physical problems can be modeled by scalar, first-order, nonlinear, hyperbolic, partial differential equations (PDEs). The solutions to these PDEs often contain shock and rarefaction waves, where the solution becomes discontinuous or has a discontinuous derivative. One can encounter difficulties using traditional finite difference methods to solve these equations. In this paper, we introduce a numerical method for solving first-order scalar wave equations. The method involves solving ordinary differential equations (ODEs) to advance the solution along the characteristics and to propagate the characteristics in time. Shocks are created when characteristics cross, and the shocks are then propagated by applying analytical jump conditions. New characteristics are inserted in spreading rarefaction fans. New characteristics are also inserted when values on adjacent characteristics lie on opposite sides of an inflection point of a nonconvex flux function, Solutions along characteristics are propagated using a standard fourth-order Runge-Kutta ODE solver. Shocks waves are kept perfectly sharp. In addition, shock locations and velocities are determined without analyzing smeared profiles or taking numerical derivatives. In order to test the numerical method, we study analytically a particular class of nonlinear hyperbolic PDEs, deriving closed form solutions for certain special initial data. We also find bounded, smooth, self-similar solutions using group theoretic methods. The numerical method is validated against these analytical results. In addition, we compare the errors in our method with those using the Lax-Wendroff method for both convex and nonconvex flux functions. Finally, we apply the method to solve a PDE with a convex flux function describing the development of a thin liquid film on a horizontally rotating disk and a PDE with a nonconvex flux function, arising in a problem concerning flow in an underground reservoir.

Growth and characterization of single crystal cored fibers of organic nonlinear optical materials

The recent development of several organic materials with large nonlinear susceptibilities, high damage threshold and low melting points encouraged researchers to employ these materials in fiber form to efficiently couple diode laser pumps and obtain enhanced second harmonic generation (SHG). In this paper we report the growth of single crystal cored fibers of 4-nitro-4'-methylbenzylidene aniline, ethoxy methoxy chalcone and (-)2-((alpha) -methylbenzylamino)-5- nitropyridine by inverted Bridgman-Stockbarger technique. The fibers were grown in glass capillaries with varying internal diameters and lengths and were characterized using x-ray and polarizing microscope techniques. The propagation loss at 632.8 nm and 1300 nm were measured and SHG was studied using 1064 nm pump.

Interannual variations of Indian summer monsoon in a GCM: External conditions versus internal feedbacks

The potential predictability of the Indian summer monsoon due to slowly varying sea surface temperature (SST) forcing is examined. Factors responsible for limiting the predictability are also investigated. Three multiyear simulations with the R30 version of the Geophysical Fluid Dynamics Laboratory's climate model are carried out for this purpose, The mean monsoon simulated by this model is realistic including the mean summer precipitation over the Indian continent. The interannual variability of the large-scale component of the monsoon such as the "monsoon shear index" and its teleconnection with Pacific SST is well simulated by the model in a 15-yr integration with observed SST as boundary condition. On regional scales, the skill in simulating the interannual variability of precipitation over the Indian continent by the model is rather modest and its simultaneous correlation with eastern Pacific SST is negative but poor as observed. The poor predictability of precipitation over the Indian region in the model is related to the fact that contribution to the interannual variability over this region due to slow SST variations [El Nino-Southern Oscillation (ENSO) related] is comparable to those due to regional-scale fluctuations unrelated to ENSO SST. The physical mechanism through which ENSO SST tend to produce reduction in precipitation over the Indian continent is also elucidated. A measure of internal variability of the model summer monsoon is obtained from a 20-yr integration of the same model with fixed annual cycle SST as boundary conditions but with predicted soil moisture and snow cover. A comparison of summer monsoon indexes between this run and the observed SST run shows that the internal oscillations can account for a large fraction of the simulated monsoon variability. The regional-scale oscillations in the observed SST run seems to arise from these internal oscillations. It is discovered that most of the interannual internal variability is due to an internal quasi-biennial oscillation (QBO) of the model atmosphere. Such a QBO is also found in the author's third 18-yr simulation in which fixed annual cycle of SST as well as soil moisture and snow cover are prescribed. This shows that the model QBO is not due to land-surface-atmosphere interaction. It is proposed that the model QBO arises due to an interaction between nonlinear intraseasonal oscillations and the annual cycle. Spatial structure of the QBO and its role in limiting the predictability of the Indian summer monsoon is discussed.

Synthesis of nonlinear controller to recover an unstable aircraft from poststall regime

Dynamics of the aircraft configuration considered in this paper show a unique characteristic in that there are no stable attractors in the entire high angle-of-attack flight envelope. As a result, once the aircraft has departed from the normal flight regime, no standard technique can be applied to recover the aircraft. In this paper, using feedback linearization technique, a nonlinear controller is designed at high angles of attack, which is engaged after the aircraft departs from normal flight regime. This controller stabilizes the aircraft into a stable spin. Then a set of synthetic pilot inputs is applied to cause an automatic transition from the spin equilibrium to low angles of attack where the second controller is connected. This controller is a normal gain-scheduled controller designed to have a large domain of attraction at low angles of attack. It traps the aircraft into a low angle-of-attack level flight. This entire concept of recovery has been verified using six-degrees-of-freedom nonlinear simulation. Feedback linearization technique used to design a controller ensures internal stability only if the nonlinear plant has stable zero dynamics. Because zero dynamics depend on the selection of outputs, a new method of choosing outputs is described to obtain a plant that has stable zero dynamics. Certain important aspects pertaining to the implementation of a feedback linearization-based controller are also discussed.

Diaphragmless shock wave generators for industrial applications of shock waves

The prime focus of this study is to design a 50 mm internal diameter diaphragmless shock tube that can be used in an industrial facility for repeated loading of shock waves. The instantaneous rise in pressure and temperature of a medium can be used in a variety of industrial applications. We designed, fabricated and tested three different shock wave generators of which one system employs a highly elastic rubber membrane and the other systems use a fast acting pneumatic valve instead of conventional metal diaphragms. The valve opening speed is obtained with the help of a high speed camera. For shock generation systems with a pneumatic cylinder, it ranges from 0.325 to 1.15 m/s while it is around 8.3 m/s for the rubber membrane. Experiments are conducted using the three diaphragmless systems and the results obtained are analyzed carefully to obtain a relation between the opening speed of the valve and the amount of gas that is actually utilized in the generation of the shock wave for each system. The rubber membrane is not suitable for industrial applications because it needs to be replaced regularly and cannot withstand high driver pressures. The maximum shock Mach number obtained using the new diaphragmless system that uses the pneumatic valve is 2.125 +/- 0.2%. This system shows much promise for automation in an industrial environment.

A Nonlinear Voltage Regulator With One Tunable Parameter for Multimachine Power Systems

This paper proposes a nonlinear voltage regulator with one tunable parameter for multimachine power systems. Based on output feedback linearization, this regulator can achieve simultaneous voltage regulation and small-signal performance objectives. Conventionally output feedback linearization has been used for voltage regulator design by taking infinite bus voltage as reference. Unfortunately, this controller has poor small-signal performance and cannot be applied to multimachine systems without the estimation of the equivalent external reactance seen from the generator. This paper proposes a voltage regulator design by redefining the rotor angle at each generator with respect to the secondary voltage of the step-up transformer as reference instead of a common synchronously rotating reference frame. Using synchronizing and damping torques analysis, we show that the proposed voltage regulator achieves simultaneous voltage regulation and damping performance over a range of system and operating conditions by controlling the relative angle between the generator internal voltage angle delta and the secondary voltage of the step up transformer. The performance of the proposed voltage regulator is evaluated on a single machine infinite bus system and two widely used multimachine test systems.

Analysis for the Reflected Waves Transmitted on to the Channel by Tall Grounded Objects Using a Special Case

During lightning strike to a tall grounded object (TGO), reflected current waves from TGO are transmitted on to the channel. With regard to these transmitted waves, there seems to be some uncertainties like: 1) will they get reflected at the main wavefront; and 2) if so, what would be their final status. This study makes an attempt to address these issues considering a special case of strike to a TGO involving equal channel core and TGO radii. A macroscopic physical model for the lightning return stroke is adopted for the intended work. Analysis showed that the waves transmitted on to the channel merges with the main wavefront without any sign of reflection. Investigation revealed that: 1) the nonlinear spatio-temporal resistance profile of the channel at the wavefront is mainly responsible for the same; and 2) the distributed source provides additional support. The earlier findings are not limited to the special case of TGO considered. In spite of considering equal TGO and channel core radii, salient features of the model predicted remote electromagnetic fields agree well with the measured data reported in literature.

Analysis for the Reflected Waves Transmitted on to the Channel by Tall Grounded Objects Using a Special Case

During lightning strike to a tall grounded object (TGO), reflected current waves from TGO are transmitted on to the channel. With regard to these transmitted waves, there seems to be some uncertainties like: 1) will they get reflected at the main wavefront; and 2) if so, what would be their final status. This study makes an attempt to address these issues considering a special case of strike to a TGO involving equal channel core and TGO radii. A macroscopic physical model for the lightning return stroke is adopted for the intended work. Analysis showed that the waves transmitted on to the channel merges with the main wavefront without any sign of reflection. Investigation revealed that: 1) the nonlinear spatio-temporal resistance profile of the channel at the wavefront is mainly responsible for the same; and 2) the distributed source provides additional support. The earlier findings are not limited to the special case of TGO considered. In spite of considering equal TGO and channel core radii, salient features of the model predicted remote electromagnetic fields agree well with the measured data reported in literature.