The algorithm of modeling wave distribution in layered random environment containing a stream of discrete inhomogeneities has been developed and realized. Dependences of dispersed and absorbed signal energies on the parameters describing inhomogeneous structure of random layered environment and inhomogeneity flux were investigated. Numerical calculations of angular spectrum of dispersed signal were carried out. Frequency spectrum of a signal passed through the flux of discrete inhomogeneities was analyze

Fedotov, I. V.

Electronic archive of Tomsk Polytechnic University

IntroductionThere is quite a number of physical problems demanding accurate calculation of the problems of radiation transport theory subject to multiple scattering. Solving a number of applied problems of radio physics, optics and acoustics modeling the process of wave distribution in layered discrete media including a flux of discrete inhomogeneities is of interest. Questions of electromagnetic waves distribution in random inhomogeneousdiscrete media at multiple scattering approximation arethe least developed theoretically.In papers [1–3] the methods of studying the equation systems made for solving the problem of reflectance and absorptances defining for an arbitrary number oflayers are examined. This resulted in obtaining generalschemes in the form of which the solutions may be presented. However, these solutions have very bulky formand if a number of layers exceeds 3 they are almost useless for analyzing the influence of one or another layeron absorptance or refraction coefficient.For example, L.M. Brekhovskih [3] suggests proceeding from Maxwell equations provided homogeneousisotropic medium for revealing general ratios when studying reflection and refraction of electromagnetic waves. He examines the analytical solution of the problemof wave reflection and passing for a random layer number. The solution of assigned task is acceptable for two,three layers but increasing the number of layers the formula bulkiness and respectively computational complexity multiply growths.The aim of the given work is to develop numericalmodel of the process of signal multiple interaction withrandom discrete inhomogeneities of layered mediumand flux.List of symbolic notations N, V are the profiles of concentration and velocity ofdiscrete inhomogeneities flux;e6s is the unit isotropic vector of discrete inhomogeneity reradiation;e6v is the velocity vector of discrete inhomogeneitiesflux;e6i is the distribution vector of discrete inhomogeneity flux;D is the length of particle free propagation in inhomogeneous flux;x is the random number uniformly distributed in therange from 0 to 1;l is the average length of wave free distribution in themedium;f0 is the carrier frequency of incident wave;Fi is the angle of particle outlet from the media;S is the quantity of particles.1. Solution techniqueIn the given paper the problem of electromagnetic wave distribution in discrete random layered medium is solved by the MonteCarlo method [4–10]. The numericalmodel of wave distribution in layered discrete medium including the flux of discrete inhomogeneities is developed.Signal source with isotropic directional pattern issupposed to be on the surface of layered medium including inhomogeneous scatterer flux. The length of incident wave is supposed to be less than layer size and fluxcharacteristic dimensions. Scattering occurs incoherently at statistically independent discrete inhomogeneities. Irradiator of electromagnetic signal is representedas particle source with proper radiation pattern. Theoriginal particle coordinates are specified in the point ofirradiator location. The mode of wave interaction withdiscrete inhomogeneities is determined according to thespecified scattering and absorption coefficients.In the case of fulfillment of dispersion condition thedirection of wave distribution changes according to thespecified indicatrix of discrete inhomogeneity reradiation. The solution of the problem of wave distribution inrandom discrete medium comes to defining the function of particle distribution in coordinate space.The wave radiated by the source is modeled by particles uniformly distributed in wave front space and oriented along the direction of signal e6i distribution. Moving discrete inhomogeneities are assigned statisticallyindependent scatterers.Profiles of concentration N and velocity V of discrete flux inhomogeneities are shown in Fig. 1 and problemgeometry is in Fig. 2.Determining free propagation length of particle D innonuniform flux the ratios are used [4]:Bulletin of the Томsк Pоlytеchnic University. 2007. V. 311. № 242UDC 519.713MODELING OF WAVE DISTRIBUTION IN LAYERED INHOMOGENEOUS ENVIRONMENT CONTAINING A FLUX OF DISCRETE INHOMOGENEITIES I.V. FedotovTomsk Polytechnic UniversityEmail: tenshi3@mail.ruThe algorithm of modeling wave distribution in layered random environment containing a stream of discrete inhomogeneities has beendeveloped and realized. Dependences of dispersed and absorbed signal energies on the parameters describing inhomogeneous structure of random layered environment and inhomogeneity flux were investigated. Numerical calculations of angular spectrum of dispersedsignal were carried out. Frequency spectrum of a signal passed through the flux of discrete inhomogeneities was analyzed.(1)The wave is supposed to be distributed along axisX(Fig. 1). The interaction type of the wave with the medium is determined according to the sections of absorption and scattering. In the case of fulfillment of wavescattering condition the direction of particle distribution changes according to the specified indicatrix of reradiation. The indicatrix represents vector scatteringpattern [3]. All calculations are carried out in dimensionless quantities.Fig. 1. Profiles of velocity and discrete inhomogeneities concentrationFig. 2. Problem geometry, e6i is the vector of propagation and e6vis the vector of velocity of discrete inhomogeneity fluxThe velocity of discrete inhomogeneity flux is determined by the formula(2)V0 is the velocity of discrete inhomogeneity flux alongaxis Z, X0 is the specified value of coordinate X.The profile of flux concentration is determined bythe expression(3)where: N0 is the concentration of turbulent flux alongthe axis of the flux(4)where Xm is the size of flux along axisX.The length of particle free path in the flux is determined as(5)To specify frequency spectrum of incident wave thefunction is applied [5](6)where x1, x2 are the random numbers uniformly distributed in the range from 0 to 1, А determines the position ofspectrum centre, ϑ is the width of frequency spectrum.Owing to discrete inhomogeneity transfer in the fluxthe Doppler shift of frequency scattered signal occurs [7, 9](7)where ϕ(V) is the function characterizing profile inhomogeneity of flux velocity.To model a unit isotropic vector of reradiation thefollowing algorithm is used [6]:(8)at(9)Unit vector of flux velocity determines moving direction of discrete inhomogeneity, according to the statement of the problem it is directed along axis Z.The coordinates of scatterer r, with which the waveinteracts, are determined from the expression [7]:(10)Modeling random paths of particles propagation innonhomogenious medium the resultant array of a quantity of absorbed and dissipated energy in the medium isaccumulated in cells [7, 9]. Doppler frequency shiftacquired by the wave as a result of multiple interactionwith discrete inhomogeneities is computed and accumulated. In the case of particle spillover the medium thestorage of its propagation direction occurs in PC.2. Electromagnetic wave propagation in layered environment containing the flux of discrete inhomogeneities Let us consider at the beginning the results of comparative calculations carried out using the algorithm described above and with the help of algorithm descried in thearticle [11] and package MCNP. Geometry and statements of problem from the work [11] were used for comparison. Monochromatic source of roentgen radiationwith photon energy 100 keV, iron plate 5×5×1 cm wereused in calculations. The results of calculations both ofdirect and scattered radiation given in Fig. 3, 4 coincidewithin the bounds of calculation statistical error..o ir r D e= + ⋅G G G2 22 3 1.γ γ+ ≤11 12 23 32 23 32 2 22 311 2 ,,,1 2 ,1 2 ,( ) /Uy xy Uy Uxxγγγγγγ γ= − += −=== −= −0* ( ) ( ) ,s i vf e e e V fϕ= − +G G G0 1 22 ln( ) cos(2 ) ,f x x Aπϑ= − +( ) / ( ).Z D N Zλ =21 122 2( ) ; 0 / 2;( ) ; / 2 ,mm mX X X XX X X X Xρρ= − ≤ ≤= − ≤ ≤0( ) exp( ),cN X N μ ρ= −20 0( ) exp( ( ) ),vV X V X Xμ= − −ln .D l x= − ⋅Mathematics and mechanics. Physics43Fig. 3. The results of direct radiation calculations Fig. 4. The results of scattered radiation calculationsIn the given Figures the graphs of relative intensityof recorded radiation from the coordinate along centralsection of plate image on detector surface are given. Theresults obtained using the technique [11] are displayedby full line, triangles – MCNP, and squares – the considered algorithm. It should be mentioned that the algorithm considered in the given article stores the propagation of scattered and absorbed signal dependence overthe whole volume of the modeled medium but not onlyin the plane of detector plate.Thus, the comparison of the results of the carriedout calculations with the data obtained by the othertechniques [11] showed proper work of the proposed algorithm. Now let us pass to the main problem solution.It is supposed that particle interaction with the fluxof discrete inhomogeneities occurs according to isotropic scattering indicatrix. All the results are presented inrelative units and normalized to the maximum for eachFigure. Medium is characterized by the presence of parallelplate layers and flux in the form of parallelepiped.The source of electromagnetic signal with isotropicradiation indicatrix placed on the medium surface withcoordinates X=0, Y=50, Z=50 is considered in the firstexperiment. Initial data describing average length of freepropagation and wave absorbtance in the medium arepresented in Fig. 5 and 6. Profiles of velocity and concentration of discrete inhomogeneities in the flux areshown in Fig. 7, 8.Fig. 5. Wave absorptance in the mediumFig. 6. Average length of wave free propagation in the medium Fig. 7. Profile of discrete inhomogeneities velocity in the flux Fig. 8. Profile of discrete inhomogeneities concentration in the fluxBulletin of the Томsк Pоlytеchnic University. 2007. V. 311. № 244Frequency spectrum of incident signal is presentedin Figure 9. As it is seen from the Figure the frequencyspectrum has the form of gaussoid with maximal valueby the frequency 7, 4 and minimal by the frequency 2,57. Frequency characteristic of the signal escaped fromthe flux is presented in Fig. 10.The results of calculation of the absorbed signal energy and energy of the signal scattered in the mediumcorrespondingly are presented in Figures 11, 12.Fig. 9. Spectrum of incident signal Fig. 10. Frequency and angular spectrum of the signal escapedfrom the flux Fig. 11. Distribution of the absorbed signal energy in the mediumThe coordinates of radiator position are changed byX=10, Y=80, Z=50 in the next experiment. The rest initial parameters of the model remain unchanged andcorrespond to the initial parameters of the previous experiment (Fig. 5–9). Thus, the source of radiation istransferred inside the flux. It is necessary for checkingwork correctness of the algorithm with the flux. Havingplaced the source of radiation in the flux we get rid ofparticles with small quantity of interactions in the fluxwhich have a very low frequency shift.Frequency and angular spectrum of the signal escaped the flux is shown in Fig. 13. Fig. 14 demonstratespropagation of scattered signal energy in the medium.Fig. 12. Distribution of the scattered signal energy in the mediumFig. 13. Frequency and angular spectrum of the signal escapedthe flux Fig. 14. Propagation of scattered signal energy in the medium Mathematics and mechanics. Physics453. Analysis of numerical simulation results Absence of absorbed signal in the flux, Fig. 11 maybe explained by the fact that there is a very low coefficient of absorption in the flux in comparison with the medium, in which we can be sure looking at Fig. 5. But theresult (Fig. 11) is significantly influenced by the lengthof free range of a particle in the medium. The length offree range in the flux is insignificant (in comparisonwith the medium, Fig. 6) owing to which particles donot practically penetrate inside the flux but interact only on the surface of the flux. Layers are also seen in thisFigure as they have comparatively high absorbtance. InFig. 11 and 12 particles getting in object shadow regionis clearly observed. Such result is explained by the factthat beams are propagated chaotically in the mediumand therefore they may be equiprobabally reradiated inany direction including shadow region.Paying attention to frequency characteristics of thesignal escaped the flux (Fig. 10) it is seen that frequency spectrum form is held. But in this case maximal valueof frequency is 62,9 and minimal is 58,6. Such changeof spectrum width is explained by significant velocity ofthe flux (Fig. 7) and high concentration of scatterers inthe flux (Fig. 8). We can also observe that energy maximum is not practically shifted by frequency. It is explained by the fact that the majority of particles having interacted with the flux underwent only single or doublescattering and accumulated insignificant frequencyshift. As the source is on the boundary of the flux andmedium then only certain particles with high energypenetrate the flux and interact in it and the rest of themeven though interacting with the flux then only on itssurface.Having transferred the source inside the flux in thesecond experiment it was suggested to obtain the spectrum with considerably wider range of frequencies andinsignificant influence of particles with low amount ofinteractions in comparison with the previous experiment. Frequency and angular spectrum of the signal escaped the flux obtained as a result of modeling is presented in Fig. 13. The analysis of the obtained resultshows significant decreasing the influence of particleswith low amount of interaction on frequency spectrumof the signal escaped the flux. Profile of flux concentration is presented in Fig. 1. According to this fact theprofile of discrete inhomogeneity concentrations is unsteady in simulated flux, Fig. 8. Energy distribution observed in Fig. 14 is explained by inhomogeneous concentration of discrete inhomogeneities in the flux,Fig. 8. Such result is stipulated by lower concentrationof discrete inhomogeneities inside the flux in comparison with concentration at approaching to the flux edges.The analysis of the results of program operation on thebasis of the developed algorithm allows making conclusion about its proper work.ConclusionThe algorithm of modeling multiple interaction ofelectromagnet wave with random threedimensionaldiscrete medium including the flux of discrete inhomogeneities was developed. The developed algorithm wastested for the task of roentgen radiation scattering andthe results of carried out calculations were comparedwith the data obtained by other methods. The problemof threedimensional visualization of model input andoutput parameters was solved. Frequency spectrum ofthe signal passed through the flux was analyzed. Thecorrectness of algorithm operation at simulating electromagnet signal propagation in the flux was checked.The results of the carried out numerical experiments demonstrate electromagnet signal reflection from the object. Particles getting in object shadow region was clearly observed in one of carried out experiments. In futurethe algorithm is supposed to be improved for decreasingtime of calculation and comparing the results of calculations with the experimental data.Bulletin of the Томsк Pоlytеchnic University. 2007. V. 311. № 246REFERENCES1. Apresyan L.A., Kravtsov Yu.A. Theory of radiative transfer. – Moscow: Nauka, 1983. – 216 p.2. Isimaru A. Propagation and scattering of waves in random inhomogeneous media. – Moscow: Mir, 1981. – V. 1. – 280 p., V. 2. – 317 p.3. Brekhovskih L.M. Waves in layered media. – Moscow: Nauka, 1973.– 505 p.4. Ermakov S.M. Method of MonteCarlo and related questions. –Moscow: Nauka, 1971. – 327 p.5. Ermakov S.M., Mikhailov G.A. 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Modeling of wave distribution in layered inhomogeneous environment containing a flux of discrete inhomogeneities

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