Showing papers on "Multiple-scale analysis published in 1976"
TL;DR: In this paper, the Schrodinger equation is derived with coefficients that depend, in a complicated way, on the shear, and the stability of Stokes waves over any shear can be examined by using the results derived here.
Abstract: The method of multiple scales is used to examine the slow modulation of a harmonic wave moving over the surface of a two dimensional channel. The flow is assumed inviscid and incompressible, but the basic flow takes the form of an arbitrary shear. The appropriate nonlinear Schrodinger equation is derived with coefficients that depend, in a complicated way, on the shear. It is shown that this equation agrees with previous work for the case of no shear; it also agrees in the long wave limit with the appropriate short wave limit of the Korteweg-de Vries equation, the shear being arbitrary. Finally, it is remarked that the stability of Stokes waves over any shear can be examined by using the results derived here.
32 citations
TL;DR: In this paper, the problem of Love wave propagation in slowly varying layered media with a general geometry is solved by using the method of multiple scales, and a first-order uniformly valid solution is obtained for the modulation of the amplitude as a function of the scale x 1 = ex, where e is a measure of amplitude of the geometrical variation of the layer.
Abstract: The problem of Love waves propagating in slowly varying layered media with a general geometry is solved by using the method of multiple scales. A first-order uniformly valid solution is obtained for the modulation of the amplitude as a function of the scale x1=ex, where e is a measure of the amplitude of the geometrical variation of the layer. This solution is particularly suited for computational procedures.
4 citations
TL;DR: In this paper, the authors derived a solution of the damped optical Bloch equations of a two-level atomic system due to a strong pulsed field and found the time dependence of the oscillations influenced by detuning and power broadening.
Abstract: The method of multiple scales is used to derive a solution of the damped optical Bloch equations of a two‐level atomic system due to a strong pulsed field. The time dependence of the oscillations of the atomic inversion influenced by detuning and power broadening is found. The population inversion consists, in general, of three terms: a quasisteady term, quasisteady term that decays with time, and an oscillatory term that also decays with time. In the limit of constant fields, the solution of Torrey for damped systems and that of Rabi for undamped systems are recovered. For an adiabatic switching of the field, the solution for undamped systems reduces to that of Crisp in the adiabatic following limit. An equation describing the field envelope is derived for an arbitrary amount of detuning. At exact resonance, this equation reduces to a pendulum equation, in agreement with previous analyses.
4 citations
TL;DR: In this article, a model problem illustrating the techniques needed to analyze weakly nonlinear oscillator systems wherein two small divisors (resonances) may occur separately or simultaneously is discussed.
Abstract: This paper concerns a model problem illustrating the techniques needed to analyze weakly nonlinear oscillator systems wherein two small divisors (resonances) may occur separately or simultaneously. The method of multiple scales in combination with singular perturbation methods is used to construct a uniformly valid asymptotic solution to the proposed model equation. It is shown that matching is necessary in order to establish a composite solution uniformly valid for the entire phase plane. The model equation does not encompass the class of problems where occurrence of two simultaneous small divisors leads to Kolmogorov-Arnol'd-Moser amplitude instability.
2 citations
TL;DR: In this paper, a nonlinear Schrodinger equation for the temporal and spatial amplitude and phase modulations of TM waves in a perfectly conducting guide containing a non-linear isotropic medium was derived.
Abstract: The method of multiple scales is used to derive a nonlinear Schrodinger equation for the temporal and spatial amplitude and phase modulations of TM waves in a perfectly conducting guide containing a nonlinear isotropic medium. This equation is used to show that monochromatic waves are stable if the mechanism producing the nonlinearity is an electric or magnetic polarization and unstable if the nonlinearity is due to electrostriction or magnetostriction. It is also used to determine the amplitude dependence of the cutoff frequencies.
2 citations