PARALLEL-PLANES APPROXIMATION FOR PROPAGATION IN RECTANGULAR TUNNELS Considering, for example, only vertically polarised waves, we shall assume that kx and ky are given by the transverse wave numbers of two infinite slot waveguides of slot widths a and b, respectively, one supporting a TE mode and the other a TM mode as shown in Fig. 1. Indexing terms: Waveguides, Propagation EH Natural propagation of electromagnetic waves in rectangular tunnels is analysed using parallel lossy planes approximations with no restrictions on the electrical properties of the walls. The analysis provides a better understanding of the physical characteristics of the low-order modes as a function of frequency. Numerical results are compared satisfactorily with published measurements. An exact analytical solution for a rectangular guide with lossy walls is not possible because of the difficulty in matching the boundary conditions. Approximate analysis usually makes use of some kind of perturbation theory. The attenuation constant a of any mode in the lossy guide above cutoff is normally obtained using the power loss method. 1 Below cutoff, a is given by the propagation constant of the lossless guide. A better approximation for the propagation constant in imperfect rectangular waveguides has been obtained assuming an arbitrary, but small, surface impedance in the walls, which couples the TE and TM modes, making necessary a definite combination to satisfy the boundary conditions. 1 ' 2 The propagation constant y for the dominant (perturbed TE 10 ) mode in a rectangular guide predicted by this 'surface impedance' method is given by (1) where a, b = width and height of the guide k0 = free-space wave number y0 = TE 1 0 propagation constant in the loss-free waveguide Zs = surface impedance of the walls In the case of road or mine tunnels eqn. 1 fails to predict the attenuation constant at high frequencies since Zs will no longer be small. On the other hand, at low frequencies it may predict a too highly attenuated wave. At high frequencies, the surrounding medium of the tunnel will behave more like a low-loss dielectric than as an imperfectly conducting material if the electrical parameters a and e are assumed to be frequency independent. Under this condition the attenuation rates of the waveguide modes will depend almost entirely on refraction loss, rather than on ohmic loss. For the dominant vertically polarised mode identified as a hybrid E H U , the attenuation constant has been determined supposing that the height and the width of the tunnel are much greater than the wavelength and that the losses are small.3-4 The obtained result is TE € TM 0 1777777777777/ V///////////7 '-k2 c.t 2-1/2 y 1?-k K "Kx 0 Fig. 1 Geometry of waveguides This 'parallel-planes' approach has the merit of being exact when the guide walls are perfectly conducting and also yields the approximate solutions given by eqns. 1 and 2 when the corresponding restrictions are observed. Applying symmetry and boundary conditions to the fields shown in Fig. 1 the following transcendental equations for kx and ky are found: (4) (5) where (6) is the propagation constant of a plane wave in the lossy medium. Eqns. 4 and 5 provide an approximate method for finding the propagation constant in a rectangular tunnel without any restriction on size, frequency or electrical properties of the walls. At very low frequencies it can be shown that kx —> ym and ky—*• 0. Therefore y—> ym, which means that the attenuation in the tunnel will be the same as through the lossy medium, and the propagating mode will have no variations in the transverse section. If now the conditions |y m | P | y t | and \ym\ P \y2\ hold, eqns. 4 and 5 lead to the same propagation constant predicted by the surface impedance method (eqn. 1), and the propagating mode will be a hybrid mode similar to the TE 1 0 mode in a perfectly conducting guide. At high frequencies, where the lossy medium will behave as a low-loss dielectric we can assume that yf. « y2. ~ —founder these conditions eqns. 4 and 5 lead to the same attenuation constant given by eqn. 2, and the propagating mode will be similar to a hybrid E ^ x mode. In order to test the validity of the proposed method, numerical solutions of eqns. 4 and 5 were obtained using the parameters of two tunnels analysed by Deryck5 and assuming that e = 10e0. The calculated results for the attenuation constant as a function of frequency for a concrete and a rock tunnel are shown in Figs. 2 and 3, where it can be seen a better agreement with experimental results than that obtained (er - I) 1 ' 2 where er is the complex relative permittivity of the walls (er = [e — 7'(ff/w)]/e0]) and X is the free-space wavelength. The restrictions inherent in the above methods may preclude the application of eqns. 1 and 2 in a real tunnel of general characteristics when natural propagation is considered in a wide frequency range. In this letter we propose a simple approach in order to determine the propagation constant in a lossy rectangular tunnel in the most general case. A general EH mode in a rectangular guide will have a propagation constant y given by (3) where kx and kv are complex transverse wave numbers. ELECTRONICS LETTERS 11th November 1982 Vol.18 f r e q u e n c y , MHz B5Z771 Fig. 2 Attenuation against frequency in a concrete tunnel I7m,b = 4-9 m, a - 0-1 s/m, e = l(te0 O experimental eqns. 4 and 5 - eqn. 1 - - eqn. 2 No. 23 1011 by Deryck using the power loss method. The approximate solutions for a obtained from eqns. 1, 2 and 6 are also included in Figs. 2 and 3, where their range of validity is clearly appreciated. equatorial and tropical climates is derived from an indirect method by using measurements of attenuation due to rain in the People's Republic of Congo (equatorial and tropical climates). Attenuations calculated from this distribution are compared with experimental data gathered on the Ivory Coast (tropical climate) at a different frequency. Raindrop size distribution: The size distribution of raindrops depends on precipitation rate R (mm/h), and is generally represented by an exponential function:1 n(a) = No exp (— 102 frequency, MHz Fig. 3 Attenuation against frequency in a rock tunnel d = 4m, b = 5 m, a = 0-01 s/m, e •= 10eo O experimental eqns. 4 and 5 eqn. 1 eqn. 2 eqn. 6 Basic integral equation: Considering single scattering, the propagation constant Kv H for vertical or horizontal polarisation is given by the Van de Hulst equation: 1 + 00 I Acknowledgments: This work was supported by the University of Chile under grant DDI 1448 8213. 12th October 1982 References 1 COLLIN, R. E.: 'Field theory of guided waves' (McGraw-Hill, 1960) 2 LEWIN, L. : 'Theory of waveguides' (Halsted Press, 1975) 3 EMSLIE, A. c , LAGACE, R. L., and STRONG, P. F.: 'Theory of the propagation of UHF radio waves in coal mine tunnels', IEEE Trans., 1975, AP-23, pp. 192-205 4 (1) where a is the equivolumetric radius of the oblate spheroidal drop, and No (cm" 4 ) and A (cm" 1 ) parameters, the value of which vary, respectively, with rainfall rates and with regional characteristics of raindrops. W3 B. JACARD A. VALENZUELA O. MALDONADO Department of Electrical Engineering University of Chile PO Box 5037, Santiago, Chile = A o i? LAAKMANN, K. D., and STEIER, w. H.: 'Characteristic modes of hollow rectangular dielectric waveguides', Appl. Opt., 1976, 5, pp. 1334-1340 5 DERYCK, L.: 'Natural propagation of electromagnetic waves in tunnels', IEEE Trans., 1978, VT-27, pp. 145-150 (2) where n(a) d(a) is the number per unit volume (cm3) having radius between a and a + da, k0 (cm" 1 ) is the propagation constant in free space, and fViH(6, <x0, a) is the forward scattering amplitude (with 6 = 0° and a 0 = 90° for terrestrial links) of an oblate spheroidal raindrop for vertical or horizontal polarisation, respectively. Attenuation and phase shift: Using eqns. 2 and 1, one may calculate the attenuation A (dB/km) and phase shift <p (deg/km) of the electromagnetic wave after passing through the rainy medium for vertical or horizontal polarisation as 1 AY H = 8-686 Im (KVH - k0) x 105 <pv<H = (180/TT) Re'(KVJ1-k0) x 10 dB/km 5 deg/km (3) These two quantities can be determined experimentally. Then, setting FvA", 0013-5194/82/231011-02$ 1.50/0 da f 8-686 x X x \mfVH 5 A) = 10 x j ( 1 8 O / 7 r ) x k x R e ^ H for attenuation for p h a s e shift (4) with X = 2n/k0, one can write + OO J RAINDROP SIZE DISTRIBUTION FROM MICROWAVE SCATTERING MEASUREMENTS IN EQUATORIAL AND TROPICAL CLIMATES = No Indexing terms: Radiowave propagation, Microwave measurements, Rain attenuation A raindrop size time averaged distribution is inferred from an indirect method by using measurements of attenuation due to rain in Congo (equatorial and tropical climates). Attenuations calculated from this distribution are compared with experimental data gathered on the Ivory Coast (tropical climate) at a different frequency. Introduction: The influence of rainfall on radiowave propagation through the atmosphere has been extensively investigated in European, North American and Japanese hydrometeorological areas, but very little is known for African equatorial and tropical zones. In this letter, a raindrop size time averaged distribution for 1012 FVH(a, X) exp ( - Ao R ~ pa) da (5) where ^/Rv H are measured data (attenuation or phase shift for each given rainfall data). From eqn. 5, the three parameters No, A o and /? may be determined. This has been done by fitting with a nonlinear minimum-least-square method the experimental data on rain attenuation gathered by Moupfouma 2 on a 33-5 km, 7 GHz terrestrial radio link with horizontal incident polarisation, during 16 months at Brazzaville (Congo). We obtain the following expressions: n(a) = 0-052e~Aa (cm" 4 ), A = 70-4R- 0 2 3 (cm" 1 ) (6a) or, alternatively, using the equivolumetric diameter of the drops, n{D) = 0026e " AD (cm " 4 ), A = 35-2R ELECTRONICS LETTERS 11 th November 1982 Vol.18 No. 23

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