Second Time Derivative Research Articles

Comparison between theory and experiment of nonlinear propagation for a-few-cycle and ultrabroadband optical pulses in a fused-silica fiber

Wave-propagation equations, including effectively the second derivative in time under the condition of a small difference between the group and phase velocities and the first derivative in position /spl xi/ in the group velocity coordinate, are derived based on the slowly evolving wave approximation. These can describe ultrabroadband optical pulse propagation with not only self-phase modulation (SPM), but also induced-phase modulation (IPM) in the monocycle regime in a fiber. It is shown that linear dispersion effects can be rigorously included in the numerical calculations. Calculations including SPM in a single-mode fused-silica fiber with the Raman effect are performed and compared with experimental results. Also, calculations including IPM in the fused-silica fiber are compared with experimental results. The effects of each term in the calculations on spectra are analyzed and it is shown that inclusion of the Raman effect and the dispersion of the effective core area is important for obtaining better agreement with experiments. It is shown that inclusion of more than third-order dispersion terms is necessary for calculations of monocycle pulse propagation.

The second time derivative analysis of chemiluminescence emission profiles and its application to the accurate determination of oxidative induction times

The technique of second time derivative (STD) analysis is developed and applied to the chemiluminescence (CL) profiles of two polypropylene (PP) formulations and a 5% w/w blend of polybutadiene (PBD) in PP to assess this novel method of analysis as a means of reliably determining the oxidative induction time (OIT) of polymers. It is proposed that the STD technique, when used in conjunction with the integrated CL profile, can enable evaluations of the OIT to be made that are less subjective than those made using the conventional extrapolation method. This is particularly so in systems that exhibit a gradual onset towards autoacceleration and/or convoluted CL profiles. Chemiluminescence profiles of the PBD–PP blend that were obtained at different temperatures were subjected to STD analysis, and Arrhenius plots of the data were made. The results are consistent with the notion that the PBD and PP phases oxidize almost independently. The activation energies for the oxidation of the PBD and PP phases were calculated to be 200 ± 31 kJ mol−1 and 146 ± 9 kJ mol−1, respectively. The higher activation energy for the PBD phase is partly attributed to the greater partitioning of thermal stabilizer in this phase. © 2001 John Wiley & Sons, Inc. J Appl Polym Sci 79: 1986–1993, 2001

THE BLOWUP OF RADIALLY SYMMETRIC SOLUTIONS FOR 2-D QUASILINEAR WAVE EQUATIONS WITH CUBIC NONLINEARITY

For a special class of quasilinear wave equations with small initial data which satisfy the nondegenerate assumption, the authors prove that the radially symmetric solution develops singularities in the second order derivatives in finite time while the first order derivatives and the solution itself remain continuous and small. More precisely, it turns out that this solution is a "geometric blowup solution of cusp type", according to the terminology posed by S. Alinhac[2].

High-Order Generalized LorenzN-Cycle Schemes for Semi-Lagrangian Models Employing Second Derivatives in Time

Abstract Having recently demonstrated that significant enhancement of forecast accuracy in a semi-Lagrangian model results from the application of high-order time integration methods to the second-derivative form of the equations governing the trajectories, the authors here extend the range of available methods by introducing a class of what they call “generalized Lorenz” (GL) schemes. These explicit GL schemes, like Lorenz’s “N-cycle” methods, which inspired them, achieve a high formal accuracy in time for linear systems at an economy of storage that is the theoretical optimum. They are shown to possess robustly stable and consistent semi-implicit modifications that allow the deepest (fastest) gravity waves to be treated implicitly, so that integrations can proceed efficiently with time steps considerably longer than would be possible in an Eulerian framework. Tests of the GL methods are conducted using an ensemble of 360 forecast cases over the Australian region at high spatial resolution, verifying at ...

On small-time expansion of nonlinear free surface problems

The small-time expansion of the 2-D problem of a heaving semi-submerged circular cylinder, starting from rest, in analyzed. The first three terms of the small-time expansion of the outer solution (away from the cylinder) are developed. A logarithmic singularity is found for the second derivative in time of the surface elevation at the intersection point. This singularity is of second order in the heaving velocity of the cylinder, and the inner expansion developed for the linear wavemaker problem therefore does not cover this case. The problem of the horizontal motion of the semi-submerged cylinder is analyzed as well. For the case with the initial condition for the velocity potential chosen as zero on the free surface, the logarithmic singularity found for the wavemaker is recovered. Selecting the initial condition as a vanishing normal velocity on the free surface leads to a solution where the singularity appears in the fifth derivative in time of the free-surface elevation. The solution of this problem shows the characteristic behaviour of formation of asymmetry, which finally would lead to formation of lee waves. The inner solution of the problem of the heaving cylinder is discussed, showing that the oscillatory behaviour reported for the wavemaker problem appears in the present solution as well.

On the detection of motion and the computation of optical flow

A method for the detection of motion in image sequences is presented. In this method, the intensity history at each pixel is convolved with the second derivative in time of a temporal Gaussian smoothing function. The zero crossings in a single frame of the resulting function indicate the positions of moving edges. Intensity changes in time due to illumination effects do not produce zero crossings; thus, they are not interpreted as motion by the present method. It is also shown that the spatial and temporal derivatives of this function can be used to compute the component of the optical flow that is normal to the zero-crossing contours. This computation is also insensitive to nonconvective temporal and spatial variations in the image intensity that are caused by illumination effects.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The second order Langevin equation and numerical simulations

A stochastic equation, differing from the Langevin equation by the addition of a second order derivative in the fictitious time, is studied for numerical simulations of euclidean field theories. The simplest discretization in the time direction (step size ϵ) is shown to lead to O(ϵ 2) errors, to be compared with O(ϵ) errors in the simplest, Euler-discretized Langevin equation. It was essentially because of this that the hyperbolic equation was found, from extensive numerical experiments on the XY model, to be far more efficient than the Langevin equation: for equal systematic errors, sweep-sweep correlation times were found to be much less in the hyperbolic case. Relations to other algorithms are also discussed.

Formation of singularities for wave equations including the nonlinear vibrating string

AbstractUnder very general assumptions, the authors prove that smooth solutions of quasilinear wave equations with small‐amplitude periodic initial data always develop singularities in the second derivatives in finite time. One consequence of these results is the fact that all solutions of the classical nonlinear vibrating string equation satisfying either Dirichlet or Neumann boundary conditions and with sufficiently small nontriviai initial data necessarily develop singularities. In particular, there are no nontrivial smooth small‐amplitude time‐periodic solutions.

Nonlinear Propagation of Wave-Packets on Fluid Interfaces

The method of multiple scales is used to derive two partial differential equations which describe the evolution of two-dimensional wave-packets on the interface of two semi-infinite, incompressible, inviscid fluids of arbitrary densities, taking into account the effect of the surface tension. These differential equations can be combined to yield two alternate nonlinear Schro¨dinger equations; one of them contains only first derivatives in time while the second contains first and second derivatives in time. The first equation is used to show that the stability of uniform wavetrains depends on the wave length, the surface tension, and the density ratio. The results show that gravity waves are unstable for all density ratios except unity, while capillary waves are stable unless the density ratio is below approximately 0.1716. Moreover, the presence of surface tension results in the stabilization of some waves which are otherwise unstable. Although the first equation is valid for a wide range of wave numbers, it is invalid near the cutoff wave number separating stable from unstable motions. It is shown that the second Schro¨dinger equation is valid near the cutoff wave number and thus it can be used to determine the dependence of the cutoff wave number on the amplitude, thereby avoiding the usual process of determining a new expansion that is only valid near the cutoff conditions.