Random walk approach to wave propagation in wedges and cones

Budaev, Bair V.; Bogy, David B.

doi:10.1121/1.1605413

Cited by 14 publications

(43 citation statements)

References 19 publications

(13 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…McLaughlin and Keller [24] note that Buslaev's result agrees with Keller's geometrical theory of diffraction for the field of a surface diffracted ray. An interesting alternative to these methods may be the solution of the wave equation by analytic continuation of a random walk proposed in [25].…”

Section: Discussion Generalization and Conclusionmentioning

confidence: 99%

Diffraction in the semiclassical approximation to Feynman's path integral representation of the Green function

Schaden

Spruch

2004

Annals of Physics

View full text Add to dashboard Cite

We derive the semiclassical approximation to Feynman's path integral representation of the energy Green function of a massless particle in the shadow region of an ideal obstacle in a medium. The wavelength of the particle is assumed to be comparable to or smaller than any relevant length of the problem. Classical paths with extremal length partially creep along the obstacle and their fluctuations are subject to non-holonomic constraints. If the medium is a vacuum, the asymptotic contribution from a single classical path of overall length L to the energy Green function at energy E is that of a non-relativistic particle of mass E=c 2 moving in the two-dimensional space orthogonal to the classical path for a time s ¼ L=c. Dirichlet boundary conditions at the surface of the obstacle constrain the motion of the particle to the exterior half-space and result in an effective time-dependent but spatially constant force that is inversely proportional to the radius of curvature of the classical path. We relate the diffractive, classically forbidden motion in the ''creeping'' case to the classically allowed motion in the ''whispering gallery'' case by analytic continuation in the curvature of the classical path. The non-holonomic constraint implies that the surface of the obstacle becomes a zero-dimensional caustic of the particle's motion. We solve this problem for extremal rays with piecewise constant curvature and provide uniform asymptotic expressions that are approximately valid in the penumbra as well as in the deep shadow of a sphere.

show abstract

Section: Discussion Generalization and Conclusionmentioning

confidence: 99%

Diffraction in the semiclassical approximation to Feynman's path integral representation of the Green function

Schaden

Spruch

2004

Annals of Physics

View full text Add to dashboard Cite

show abstract

“…0 where dj 1 t has a positive imaginary part. Additional information about the motion j 1 t comes from the observation [7] that it has a drift along the lines…”

Section: Waves Radiating Into the Exterior Of A Circular Cylindermentioning

confidence: 99%

“…(2.25). The mathematical expectation was simulated by the averaging of 2000 random motions approximated by the discrete random walks [6,7] with the spatial step 1 ¼ 0:05 corresponding to the time increment Dt ¼ 1 2 ¼ 0:0025: The computations were stable and the difference between the results was under 0.015 at all considered observation points.…”

Section: Waves Radiating Into the Exterior Of A Circular Cylindermentioning

confidence: 99%

“…The governing equations for these problems are similar to those employed in Refs. [6,7] for the analysis of waves in wedge-shaped and conical domains, but in order to accommodate boundary conditions formulated on nonplane surfaces we have to invoke a specific technique which makes it possible to 'steer' the random walks into a preferable, albeit still random, direction. This steering technique constitutes an important step towards the extensions of the random walk method to problems of wave propagation in domains with non-trivial geometry such as deformed spherical or cylindrical domains with surfacebreaking cracks or cavities.…”

Section: Introductionmentioning

confidence: 99%

“…In Ref. [7] the random walk method was applied to problems of wave propagation in wedge-shaped and conical domains. Then these results were used in Ref.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations