This paper presents a numerical method of tracing of sound or other refracted rays through a medium with arbitrarily varying refractive index. The method uses a radial basis function (RBF) network to define the refractive index of the medium, allowing continuous gradients to be determined analytically and the ray path to be solved using standard numerical ordinary differential equation (ODE) solution techniques.
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Technical Brief
References
1.
Reynolds
, O.
, 1873
, “On the Refraction of Sound by the Atmosphere
,” Proc. R. Soc. London
, 22
(148–155
), pp. 531
–548
.2.
Blokhintzev
, D.
, 1946
, “The Propagation of Sound in an Inhomogeneous and Moving Medium I
,” J. Acoust. Soc. Am.
, 18
(2
), pp. 322
–328
.3.
Humphreys
, W. J.
, 1964
, Physics of the Air
, Dover Publications
, New York
.4.
Johnson
, S. A.
, Greenleaf
, J. F.
, Samayoa
, W. A.
, Duck
, F. A.
, and Sjostrand
, J.
, 1975
, “Reconstruction of Three-Dimensional Velocity Fields and Other Parameters by Acoustic Ray Tracing
,” Ultrasonics Symposium
, Los Angeles, CA, Sept. 22–24, pp. 46
–51
.5.
Sharma
, A.
, Kumar
, D. V.
, and Ghatak
, A. K.
, 1982
, “Tracing Rays Through Graded-Index Media: A New Method
,” Appl. Opt.
, 21
(6
), pp. 984
–987
.6.
Fletcher
, A.
, Murphy
, T.
, and Young
, A.
, 1954
, “Solutions of Two Optical Problems
,” Proc. R. Soc. London, Ser. A
, 223
(1153
), pp. 216
–225
.7.
Morgan
, S. P.
, 1958
, “General Solution of the Luneberg Lens Problem
,” J. Appl. Phys.
, 29
(9
), pp. 1358
–1368
.8.
Sands
, P. J.
, 1971
, “Inhomogeneous Lenses, IV. Aberrations of Lenses With Axial Index Distributions
,” J. Opt. Soc. Am.
, 61
(8
), pp. 1086
–1091
.9.
Born
, M.
, and Wolf
, E.
, 1999
, Principles of Optics I
, 7th ed., Cambridge University
, Cambridge, UK
.10.
Sakamoto
, T.
, 1993
, “Analytic Solutions of the Eikonal Equation for a GRIN-Rod Lens 1. Meridional Rays
,” J. Mod. Opt.
, 40
(3
), pp. 503
–516
.11.
Sakamoto
, T.
, 1995
, “Analytic Solutions of the Eikonal Equation for a GRIN-Rod Lens 2. Skew Rays
,” J. Mod. Opt.
, 42
(8
), pp. 1575
–1592
.12.
Broomhead
, D. S.
, and Lowe
, D.
, 1988
, “Radial Basis Functions, Multi-Variable Functional Interpolation and Adaptive Networks
,” Royal Signals and Radar Establishment, Malvern, UK, RSRE
Memo 4148.13.
Duchon
, J.
, 1977
, “Splines Minimizing Rotation-Invariant Semi-Norms in Sobolev Spaces
,” Constructive Theory of Functions of Several Variables
, Springer
, Berlin
, pp. 85
–100
.14.
Dormand
, J. R.
, and Prince
, P. J.
, 1980
, “A Family of Embedded Runge-Kutta Formulae
,” J. Comput. Appl. Math.
, 6
(1
), pp. 19
–26
.15.
Shampine
, L. F.
, and Gordon
, M. K.
, 1975
, Computer Solution of Ordinary Differential Equations: The Initial Value Problem
, W. H. Freeman
, San Francisco, CA
.16.
Kierzenka
, J.
, and Shampine
, L. F.
, 2001
, “A BVP Solver Based on Residual Control and the Matlab PSE
,” ACM Trans. Math. Software
, 27
(3
), pp. 299
–316
.17.
Cai
, Y.
, Fan
, W.
, Zhou
, Y.
, Fu
, X. Q.
, Fang
, H.
, and Jin
, T.
, 2012
, “Study on the High Precision Acoustic Measurement Techniques for Determining Temperature Field Around Seafloor Hydrothermal Vent
,” China Ocean Eng.
, 26
(4
), pp. 723
–732
.18.
Wiens
, T.
, 2015
, “Evaluation of Fast RBF Acoustic Tomography Measurement of Flame Temperature and Velocity Field
,” Combustion Institute–Canadian Section Spring Technical Meeting
, Saskatoon
, Canada
.19.
NIST
, 2015
, National Institute of Standards and Technology, Boulder, CO, “Fire Dynamics Simulator
,” http://fire.nist.gov/fdsCopyright © 2016 by ASME
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