Asymptotic evaluation of highfrequency fields near a caustic: An introduction to Maslovs method
نویسندگان
چکیده
It is well known that the geometrical optics approximation, generally valid for high-frequency fields, fails in the vicinity of a caustic. A systematic procedure of V. P. Maslov that remedies this situation will be reviewed in this paper. Maslov's method makes use of a representation of the geometrical optics field in the phase space M = X x K, where a point m = (x, •c) is a pair of a position vector x • X and a wave vector •c • K. A Lagrangian submanifold of M, A, that lies in the dispersion surface and is a union of the phase space trajectories selected by the initial conditions is constructed. It can be considered as a global representation of the phase. The phase space amplitudes (half densities) satisfy transport equations defined along those trajectories in A. Since trajectories in M never form a caustic, a globally defined amplitude can be established on A. The field on X is related to the resultant field on A by the "canonical operator," an operator introduced by Maslov. It generates an integral form of the solution near a caustic that can be evaluated analytically, numerically, or with uniform asymptotic techniques. Away from the caustic it recovers the geometrical optics field. Alternatively, the phase space field can be projected on a hybrid space Y where some of the space coordinates have been replaced by the corresponding wave vector components. For any caustic point in X, one such hybrid space Y where this projection does not encounter a caustic exists. A geometrical optics field results in Y that is related to the original in X by an asymptotic Fourier transform. The solution in X near a caustic can be represented as the Fourier transform to X of that hybrid space geometrical optics solution. These techniques are illustrated with two simple but revealing problems: continuation of the field through a fold caustic in a linear layer medium and through a caustic with a cusp point in a homogeneous medium.
منابع مشابه
True Amplitude One-Way Wave Equations near Smooth Caustics: An Example
True amplitude one-way wave equations provide full waveform solutions that agree with leading order WKBJ solutions of the full, two-way wave equation away from caustics and other anomalies of the ray field. Both a proof of this fact and numerical examples are available in the literature. Near smooth caustics, ray-theoretic solutions fail for both the one-way and two-way wave equations; the WKBJ...
متن کاملRegeneration and treatment of sulfidic spent caustic using analytic hierarchy process
Background: Sulfur compounds must be removed from petroleum because they contribute to environmental pollution. A strong alkaline solution such as caustic soda is used to remove these compounds. This spent caustic has high values for chemical oxygen demand (COD) concentration, pH and total sulfur. In this study, the regeneration and treatment methodology of sulfidic spent caustic was inve...
متن کاملEvent horizon deformations in extreme mass-ratio black hole mergers
We study the geometry of the event horizon of a spacetime in which a small compact object plunges into a large Schwarzschild black hole. We first use the Regge-Wheeler and Zerilli formalisms to calculate the metric perturbations induced by this small compact object, then find the new event horizon by propagating null geodesics near the unperturbed horizon. A caustic is shown to exist before the...
متن کاملWave Propagation near a Smooth Caustic
A caustic is an envelope of a family of rays of geometrical optics. At a caustic, the usual equations of geometrical optics are not valid. We give a formal asymptotic series, valid both near and away from a smooth caustic, which satisfies the differential equation exactly. In the case of high-frequency oscillations, the solution is represented in terms of the Airy function and its derivative. T...
متن کاملAbsorbed Dose Calculation In Irregular Blocked Radiation Fields: Evaluation of Clarkson’s Sector Integration Method for Radiation Fields Commonly Used in Conventional Radiotherapy
Introduction: Irregular/blocked fields are routinely used in radiotherapy. The doses of these fields are usually calculated by means of equivalent square method that is inherently prone to uncertainty. On the other hand, Clarkson’s sector integration method is a dose calculation method which offers far better accuracy in dose calculation of irregular fields. The Scatter Air Ratio (SAR) of an in...
متن کامل