Diffraction by a convex polygon with side-wise constant impedance

Bogy

2007

Proc. R. Soc. A.

This paper presents a series of novel representations for the solutions of the Helmholtz equation in a broad class of wedge-like domains including those with curvilinear, non-flat faces. These representations are obtained by an original method which combines ray theory with the probabilistic approach to partial differential equations and uses a specific technique to deal with a need for analytical continuation of the specified boundary function. The main results are reminiscent of the standard Feynman–Kac formula but differ in that the averaging over solutions of stochastic differential equations is replaced by averaging over the trajectories of a new two-scale random motion introduced here. The paper focuses on the development of the solutions and for this reason it includes only a brief outline of numerous applications, consequences, extensions and variations of the method, which include, but are not limited to, problems of diffraction and scattering, problems in three-dimensional domains and problems of wave propagation in non-homogeneous media.

Section: Introductionmentioning

confidence: 99%

“…Here, we complete the development started in Budaev & Bogy (2005c, 2006) and find a new class of explicit probabilistic representations for the solutions of the Helmholtz equation in the wedge-like domains described as G : r O 0; a 1 ðrÞ! q!…”

Section: Introductionmentioning

confidence: 99%

Novel solutions of the Helmholtz equation and their application to diffraction

Bogy

2007

Proc. R. Soc. A.

“…Here we take the next step and extend the probabilistic method in a way which makes it possible to handle boundary conditions with variable coefficients. This is the centerpiece of the paper, and it is based on further development of the idea of analytical continuation introduced by Budaev and Bogy [2005c] and Budaev and Bogy [2006a], who applied it to other problems of interest.…”

Section: Introductionmentioning

confidence: 99%

“…The probabilistic approach has already been successfully applied by the authors to a standard two‐dimensional problem of diffraction by a wedge with constant impedances, as well as to more challenging three‐dimensional problems of diffraction by a plane angular sector [ Budaev and Bogy , 2004] and by an infinite wedge with anisotropic face impedances [ Budaev and Bogy , 2006b]. Most recently, the probabilistic method was applied to diffraction by an arbitrary convex polygon with side‐wise constant impedances [ Budaev and Bogy , 2006a]. This problem with nontrivial geometry does not have known conventional closed‐form solutions, but its probabilistic solution is not considerably more complex than the solutions of the other problems mentioned above.…”

Section: Introductionmentioning

confidence: 99%

Diffraction by a wedge with a face of variable impedance

Bogy

2007

Radio Science

Self Cite

[1] The two-dimensional problem of diffraction by a wedge with a face of variable impedance is explicitly solved by a probabilistic random walk method. The solution admits numerical simulation based on simple scalable algorithms with unlimited capability for parallel processing. The diffracted field is represented as a mathematical expectation of a specified functional on trajectories of random motions determined by the configuration of the problem. The solution is not significantly more difficult than a similar probabilistic solution of the problem with constant impedance faces.Citation: Budaev, B. V., and D. B. Bogy (2007), Diffraction by a wedge with a face of variable impedance, Radio Sci., 42, RS6S02,

“…Surely, to apply formula (16), it is necessary that the boundary values of f ±α (r) be analytic in r. However, in diffraction problems this condition is always fulfilled, and the procedure described above makes it possible to solve many nontrivial problems. For instance, in [3], there are references to a solution of the Maluizhinets problem of diffraction by a wedge, the diffraction problem on an arbitrary polygon, the three-dimensional problem of diffraction on a plane sectorial screen, as well as a more general problem of diffraction on a pyramid of infinite length. Moreover, it is possible to solve the problems mentioned above not only for Dirichlet and Neumann boundary conditions, but also for much more general conditions of impedance type.…”

mentioning

confidence: 99%

How should we improve the ray-tracing method?

St. Petersburg Math. J.

2011

Abstract. The possibility is discussed to improve the ray approximation up to an exact representation of a wave field by the Feynman-Kac probabilistic formula (this formula gives an exact solution of the Helmholtz equation in the form of the expectation of a certain functional on the space of Brownian random walks). Some examples illustrate an application of the solutions obtained to diffraction problems.It would not be an exaggeration to claim that a large (if not a major) part of studies in the theory of wave propagation is related somehow to the ray-tracing method. The contribution of Babich and Buldyrev to the creation of this method was quite substantial.The ray-tracing method is based on the universal "divide and conquer" principle, which is revealed, in the present case, in splitting a difficult problem into two simpler ones. For instance, a solution of the Helmholtz equationis sought in the formi.e., an unknown function ϕ(x) is expressed in terms of two unknown functions u(x) and s(x). It is easily seen that the new unknowns must satisfy the equationwhich does not seem to be simpler than (1) but, in recompense, can be split in the eikonal equationand the transport equationwith the coefficientsdetermined by (3). The procedure described above is feasible because the eikonal equation (3) can be solved exactly with the use of standard methods of Hamiltonian mechanics, which determine s(x) uniquely on a certain (maybe, many-sheeted) domain G that covers the physical domain G. As to the transport equation (4), in distinction to the Helmholtz 2010 Mathematics Subject Classification. Primary 81Q30.