Geodesic
Calculation of exact frequency-dependent rays when the solution of the Helmholtz equation is known
Numerical methods and programming, Tome 16 (2015) no. 4, pp. 586-594.

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A numerical method is proposed for the calculation of exact frequency-dependent rays when the solution of the Helmholtz equation is known. The properties of frequency-dependent rays are analyzed and compared with the classical ray theory and with the method of finite-difference modeling for the first time. In this paper we study the dependence of these rays on the frequency of probing signals and show the convergence of the exact rays to the classical rays with increasing frequency. A number of numerical experiments demonstrate the distinctive features of exact frequency-dependent rays, in particular, their ability to penetrate into shadow zones impenetrable for the classical rays.
Keywords: Helmholtz equation, exact frequency-dependent rays, ray theory, numerical methods. 
@article{VMP_2015_16_4_a12,
     author = {K. G. Gadylshin and M. I. Protasov},
     title = {Calculation of exact frequency-dependent rays when the solution of the {Helmholtz} equation is known},
     journal = {Numerical methods and programming},
     pages = {586--594},
     publisher = {mathdoc},
     volume = {16},
     number = {4},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMP_2015_16_4_a12/}
}
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K. G. Gadylshin; M. I. Protasov. Calculation of exact frequency-dependent rays when the solution of the Helmholtz equation is known. Numerical methods and programming, Tome 16 (2015) no. 4, pp. 586-594. http://geodesic.mathdoc.fr/item/VMP_2015_16_4_a12/