C ERL. /63-XTS-/C 
 
 ESSA TR ERL 163-ITS 105 
 
 A UNITED STATES 
 DEPARTMENT OF 
 COMMERCE 
 PUBLICATION 
 
 
 ESSA Technical Report ERL 163-ITS 105 
 
 U.S. DEPARTMENT OF COMMERCE 
 Environmental Science Services Administration 
 Research Laboratories 
 
 round Wave Propagation 
 in an Ionized Atmosphere 
 
 LWith Arbitrary Variation 
 of the Conductivity With Altitude 
 
 J. RALPH JOHLER 
 
 BOULDER, COLO. 
 MARCH 1970 
 
ESSA RESEARCH LABORATORIES 
 
 The mission of the Research Laboratories is to study the oceans, inland waters, the 
 lower and upper atmosphere, the space environment, and the earth, in search of the under- 
 standing needed to provide more useful services in improving man's prospects for survival 
 as influenced by the physical environment. Laboratories contributing to these studies are: 
 
 Earth Sciences Laboratories: Geomagnetism, seismology, geodesy, and related earth 
 sciences; earthquake processes, internal structure and accurate figure of the Earth, and 
 distribution of the Earth's mass. 
 
 Atlantic Oceanographic and Meteorological Laboratories and Pacific Oceanographic 
 Laboratories: Oceanography, with empnasis on ocean basins and borders, and oceanic 
 processes; sea-air interactions: and land-sea interactions. (Miami, Florida) 
 
 Atmospheric Physics and Chemistry Laboratory: Cloud physics and precipitation; 
 chemical composition and nucleating substances in the lower atmosphere; and laboratory 
 and field experiments toward developing feasible methods of weather modification. 
 
 Air Resources Laboratories: Diffusion, transport, and dissipation of atmospheric con- 
 taminants; development of methods for prediction and control of atmospheric pollu- 
 tion. (Silver Spring, Maryland) 
 
 Geophysical Fluid Dynamics Laboratory: Dynamics and physics of geophysical fluid 
 systems; development of a theoretical basis, through mathematical modeling and computer 
 simulation, for the behavior and properties of the atmosphere and the oceans. (Princeton, 
 New Jersey) 
 
 National Severe Storms Laboratory: Tornadoes, squall fines, thunderstorms, and other 
 severe local convective phenomena toward achieving improved methods of forecasting, de- 
 tecting, and providing advance warnings. (Norman, Oklahoma) 
 
 Space Disturbances Laboratory: Nature, behavior, and mechanisms of space disturb- 
 ances; development and use of techniques for continuous monitoring and early detection 
 and reporting of important disturbances. 
 
 Aeronomy Laboratory: Theoretical, laboratory, rocket, and satellite studies of the phys- 
 ical and chemical processes controlling the ionosphere and exosphere of the earth 
 and other planets. 
 
 Wave Propagation Laboratory: Development of new methods for remote sensing of the 
 geophysical environment; special emphasis on propagation of sound waves, and electromag- 
 netic waves at millimeter, infrared, and optical frequencies. 
 
 Institute for Telecommunication Sciences: Central federal agency for research and serv- 
 ices in propagation of radio waves, radio properties of the earth and its atmosphere, nature 
 of radio noise and interference, information transmission and antennas, and methods for 
 the more effective use of the radio spectrum for telecommunications. 
 
 Research Flight Facility: Outfits and operates aircraft specially instrumented for re- 
 search; and meets needs of ESSA and other groups for environmental measurements for 
 aircraft. (Miami, Florida) 
 
 ENVIRONMENTAL SCIENCE SERVICES ADMINISTRATION 
 
 BOULDER, COLORADO 80302 
 
-r^ENTQ^ 
 
 *'fl¥C£ stW* iS> 
 
 U.S. DEPARTMENT OF COMMERCE 
 Maurice H. Stans, Secretary 
 
 ENVIRONMENTAL SCIENCE SERVICES ADMINISTRATION 
 Robert M. White, Administrator 
 RESEARCH LABORATORIES 
 Wilmot N. Hess, Director 
 
 ESSA TECHNICAL REPORT ERL 163-ITS 105 
 
 Ground Wave Propagation 
 in an Ionized Atmosphere 
 With Arbitrary Variation 
 of the Conductivity With Altitude 
 
 J. RALPH JOHLER 
 
 Q, 
 O 
 
 o 
 a 
 a> 
 a 
 
 «/> 
 
 INSTITUTE FOR TELECOMMUNICATION SCIENCES 
 BOULDER, COLORADO 
 March 1970 
 
 For sale by the Superintendent of Documents, U. S. Government Printing Office, Washington, D. C. 20402 
 
 Price 35 cents. 
 
FOREWORD 
 
 The mathematical techniques presented in this paper were 
 developed for the Defense Atomic Support Agency as part of the work 
 performed under interagency work order 804-69. This paper deals 
 with the specific task of estimating arbitrary altitude variations in the 
 ion content of the atmosphere in the nuclear environment and of develop- 
 ing mathematical techniques for calculating ground waves in such an 
 environment. 
 
 The listing of the FORTRAN computer program is available 
 from the author upon request. 
 
TABLE OF CONTENTS 
 
 Page 
 
 FOREWORD ii 
 
 ABSTRACT iv 
 
 1. INTRODUCTION 1 
 
 2. THEORETICAL CONSIDERATIONS 7 
 
 3. DISCUSSION AND CONCLUSIONS 18 
 
 4. REFERENCES 20 
 
 5. APPENDDC. Computer Program 23 
 
 in 
 
ABSTRACT 
 
 The atmosphere can apparently be ionized to such a degree 
 that the ground wave is disturbed, as a result of, for example, a 
 nuclear event, and ionizing radiation of sufficient intensity to overcome 
 the ion -ion recombination processes may cause the ions to accumulate 
 near the surface of the ground. Gamma radiation from the fallout of a 
 nuclear weapon may produce such ionization for a considerable time 
 after the nuclear event. The problem is similar to that of terrestrial 
 radio wave propagation via the ionosphere, except that in the former 
 case the ionization has a maximum near the surface of the ground and 
 decreases exponentially with altitude. In other words, we have an up- 
 side down ionosphere to consider. A new theoretical approach to such 
 ground wave problems based on methods of numerical analysis has 
 been developed for computer application. The computer computation 
 concept is discussed. and sample computations based on the fallout 
 model profile are presented. A particular fallout model based on data 
 given in the U. S. Atomic Energy Commission" s handbook on "The 
 Effects of Nuclear Weapons" is considered in some detail, and the 
 effect of an ionization profile (ion number density as a function of 
 altitude) on the propagation of the ground wave is ascertained. A 
 physically possible but not likely extra attenuation of 12 dB at 100 km 
 distance is noted. 
 
 IV 
 
GROUND WAVE PROPAGATION IN AN IONIZED ATMOSPHERE WITH ARBITRARY 
 VARIATION OF THE CONDUCTIVITY WITH ALTITUDE 
 
 J. Ralph Johler 
 
 1. INTRODUCTION 
 
 In an earlier paper (Johler, 1969a) ground wave propagation in 
 a uniformly ionized atmosphere was considered in some detail. It was 
 found that uniform ionization of the atmosphere in the vertical direction 
 could affect ground wave propagation if the ionization were sufficiently 
 intense. Some nuclear events apparently provide sources which could 
 cause a disturbance of the ground wave. In this paper, we take a more 
 realistic and more complicated approach by introducing a profile or 
 variation of the atmospheric conductivity with altitude. For example, 
 gamma radiation from the fallout of a nuclear weapon may produce 
 intense ionization near the surface of the ground for a considerable 
 time after the nuclear event. The ionization will decrease rapidly with 
 altitude. This problem is similar to that of terrestrial radio wave 
 propagation via the ionosphere, but in the latter case the ionization has 
 a maximum near the surface of the ground and decreases approximately 
 exponentially with altitude. Thus fallout produces an upside-down 
 ionosphere. 
 
 The basic unit for measuring fallout (AEC, 1962, p. 3 7 5) is the 
 roentgen > which is the quantity or "dose" of gamma radiation or x-rays 
 that will give rise to 2. 08(10 9 ) ion pairs/ cc under standard atmospheric 
 conditions at sea level. An interesting case is cited by AEC (1962, 
 p. 462), which gives the radiation level for the BRAVO test explosion 
 (15 Mt yield). At a distance of 190 km and a time of 96 hours after the 
 explosion, the level of 3000 roentgens was noted. The roentgen can be 
 converted into an average production rate, 
 
roentgens _ nnt * n 9, roentgens „ non . 9 , . , , ., ,. 
 
 q = £^| 2.08(10 9 ) = 98 - | 6Q0 2.08(l0 9 )ions/cc/sec. (1.1) 
 
 The unit dose rate may be greater than the average does rate. Refer- 
 ring again to AEC (fig. 920, p. 423), we can conclude that the production 
 rate q should be increased by a factor of 96/ 8. 2, or approximately 12. 
 Thus the unit time dose rate deduced from 3000 roentgens at 96 hours 
 is 3000/ 8. 2, or 366 roentgens per hour at one 1 hour and 1. 5 roentgens 
 per hour at 96 hours. The average dose rate is, of course, between 
 these values or 3000/ 96 or 31. 25 roentgens per hour. (For references 
 of this type see AEC, fig. 9. 16a). The profile in figure 1, which is 
 lbased on figure 9.131 in the AEC handbook, gives the dose rate variation 
 with altitude of a uniform distribution of radiation sources over an 
 infinite plane. Thus a profile of ion density and conductivity as a 
 function of altitude can be deduced from the formulas 
 
 **■-<[$ 
 
 N. = -?■ , (1.2) 
 
 and 
 
 a = 2-^ N±_ i2±_t£^ . (1.3) 
 
 m+ V +j0 e m e 
 
 Here, e 2 / m+ is the ratio of electronic charge squared to the mass of 
 the positive ion. The permittivity is given by e ~ 8. 85419(10" ), and 
 the positive ion-neutral collision frequency v +j0 is given in figure 1- 
 The collision frequency V +i0 is multiplied by the mass ratio of neutral 
 mass m to the sum of positive ion and neutral mass to account 
 for the redistribution of velocity after collision (Johler and Berry, 1965). 
 This point is often overlooked because, in the case of electron-neutral 
 
 -2- 
 
10 
 8 
 
 10 
 8 
 
 6 
 
 _ 5 
 i 
 
 o 
 
 <D 4 
 
 10 
 
 BRAVO ESTIMATE BASED 
 ON 3000 ROENTGENS AT 
 270 km AND 96 HRS. 
 
 10' 
 
 Figure 1, 
 
 0.2 
 
 0.4 
 
 0.6 0.8 
 
 ALTITUDE, km 
 
 1.0 
 
 1.2 
 
 Theoretical profile of atmospheric conductivity, 
 positive ion density* and collision frequency. Ion 
 densities are based on fallout measurements 
 270 km from ground zero and 96 hours after the 
 BRAVO nuclear test. This is the case F n = 1, 
 with an average dose rate and a nominal ion -ion 
 recombination coefficient a 
 
 i» o 
 
 3(10" a ). 
 
 -3- 
 
collisions, the mass ratio times the collision frequency in the hydro dynamic 
 
 equation is 
 
 m, + m V *' ° ~ V *' ° ' 
 
 since 
 
 Thus, 
 
 and 
 
 m # < < m . 
 
 m <— 5. 313(10" 6 ) for oxygen, 
 
 m. - 9. 108(1 0" 31 ) in kilograms. 
 On the other hand, 
 
 
 m+ + m + »• 2 + » ° ' 
 
 if we assume positive ions and neutrals have the same mass. Thus, 
 strictly speaking, the effective collision frequency is half the actual 
 collision frequency in the case of ion-ion collisions in which each of the 
 colliding particles have the same mass. This means that the drift 
 velocity change after a collision is redistributed equally between the 
 two colliding particles. Since the ion collision frequency is now more 
 precisely known, the mass ratio described above should be taken into 
 
 account in the equation of motion of the ions, as was done by Johler and 
 Berry (1965). 
 
 The conductivity cj reflects the effect of the mass ratio on the 
 resultant ion velocity change. The number density N + is critically 
 
 3 / 
 
 dependent upon the value of the ion -ion recombination rate, a t cm / sec, 
 near the ground. Considerable uncertainty as to the value of a t seems 
 
 -4- 
 
to exist in the literature. The lowest possible value is the two -body- 
 ion -ion recombination rate used at high altitude, ot.j ~ 3(1 0~ 8 ) (Knapp, 
 19^9, private communication, G.E. Tempo, Santa Barbara, California); 
 the highest value quoted for the three-body ipn-ion recombina- 
 tion rate is a ± ~ 3(10" ). This difference, by a factor of 100, between 
 these values means that the profile in figure 1 would change from one 
 that would have little effect on the propagation of the ground wave to 
 one that would seriously alter it. Until recently the three -body ion-ion 
 recombination rate most quoted in the literature a t ~ 2. 65(10" ) (see, 
 for example, Loeb, 1961) was apparently based on researches by 
 Thomson (1906, 1924), Thomson and Thomson (1933), Rutherford (1913) 
 and Natanson (1959). Experimental verification of the theory by Sayers 
 (1938) gave the value quoted above, which agreed with theory „ More 
 recently, Bruechner (1964) quotes a value of 3„ 95(10" 7 )cm 3 / sec for 
 a t . This is based on a corrected theory of ion-ion recombination that 
 accounts for an error in the Thorns on -Natanson theory by retaining the 
 more rigorous expression for the value of the average energy in the 
 analysis. Brueckner concludes: "The disagreement (with the Thomson- 
 Natanson theory) may be due to the simplifying assumptions of the 
 Thomson theory as modified in this paper to include dissociative col- 
 lisions, such as the failure to consider the internal degrees of freedom 
 of the ions and the dispersion in energy and momenta of the colliding 
 particles. The changes which result from a more complete theory than 
 previously used suggest, however, that the pre vious agreement between 
 theory and experimen t has be en fortuitous" . More recently experi- 
 mental evidence has been obtained in support of Brueckner' s theory 
 (Hirsh, 1967), indicating that a considerably lower value of a i should 
 be used near the ground. 
 
 Taking into account the uncertainty in 04 and the production 
 rate q^we can translate the profile shown in figure 1 simply by using 
 an ion number density multiplication factor F n . Thus a change in the 
 
 5 
 
production rate or the ion-ion recombination rate merely modifies the 
 number density N + in (1.2). Physically possible values, according 
 to this author' s estimates, range from F n = 0. 1 to F n = 20 in a 
 nuclear environment. This would correspond to ground level production 
 rates q/q = 0. 01 to 400, where q is the average production rate 
 for the BRAVO profile (fig. 1). Thus, in the lower limit, F n = 0. 1 , 
 the techniques described in this paper recovered the numerical values 
 calculated from the classical ground wave theory within a fraction of a 
 decibel in the amplitude at a frequency of 100 kHz and distances of 
 100 km or less. The particular case of BRAVO given by AEC would be 
 represented approximately by F n = 1 or 2, depending on the time. 
 Therefore, (1.2) becomes 
 
 N + =F n / ■£*. , (1.4) 
 
 where 
 
 q e = 1.8(10 7 ) f (1.5) 
 
 and 
 
 a fcQ = 3(10" 8 ) . (1.6) 
 
 This paper solves the electromagnetic theory problem of a 
 ground wave signal propagating from an electrical point source dipole 
 to distances less than approximately 300 km and over finitely conduct- 
 ing ground with an arbitrary variation with altitude of the conductivity 
 of the atmosphere. Numerical values for F n between 0. 1 and 40 are 
 considered at a frequency of 100 kHz. 
 
 -6- 
 
2. THEORETICAL CONSIDERATIONS 
 
 The problem of ground wave propagation at medium and low 
 frequencies in an ionized atmosphere ordinarily concerns short distances, 
 say, 50 to 100 km. It is therefore more practical from a theoretical 
 point of view to regard the earth as flat and solve the problem in terms 
 of a Cartesian coordinate system. 
 
 An ionized atmosphere can be accommodated in the analysis of 
 the ground wave with a system of boundary conditions as depicted in 
 figure 2. Thus, the profile in figure 1 is approximated as a function of 
 altitude with a set of plasma slabs. This is the method used by Johler 
 and Harper (1962). At the low altitudes that pertain to the ground wave, 
 atmospheric reflections are a consequence of the presence of ions that 
 exhibit isotropic behavior at medium and low frequencies. Therefore, 
 advantage can be taken of the simplification resulting from zero 
 terrestrial magnetic field. A detailed treatment of the effects of ions 
 is discussed in an earlier paper (Johler and Berry 1965). 
 
 The scheme for calculating the reflection process is depicted in 
 figure 2. Each slab of the system of slabs contains a reflection coeffi- 
 cient Ti at the top, a reflection coefficient Rj at the bottom, and a 
 transmission coefficient U i connecting it to the adjacent slab. As the 
 thickness of each slab Z i is decreased and the number p of slabs is 
 increased, convergence can be obtained on a reflection coefficient T # , . 
 The phase of the reflection coefficient T. # is referenced to the height 
 h. This reflection coefficient is a function of the slab reflection and 
 transmission coefficients T i , R t , U t , and T within the system of 
 slabs. The first slab of thickness h contains the source dipole and 
 hence must be treated separately to accommodate the spherical waves 
 emanating from such an electrical point source. The analysis pre- 
 sented below then relates the bottom of the slab system at height h 
 (fig. 2) to the ground and to a source dipole (transmitter). 
 
 -7- 
 
>K, 
 
 Tp-i 
 R P-i 
 
 l» 
 
 P-l 
 
 Zi. 
 
 i-l 
 
 >Kp-l 
 
 / W ////// / Ground ////// 
 
 K_ 
 
 Figure 2. Theoretical treatment of the profile based on the concept 
 of continuous stratification of the altitude variation of a 
 atmospheric ion content or conductivity. 
 
 -8- 
 
The reflection coefficient T,, implies vertically polarized 
 waves characteristic of a vertical electric dipole source. A similar 
 reflection coefficient T Bm , for horizontal dipole sources could also be 
 used. Finally, since an isotropic ionosphere is assumed, the abnormal 
 components given by Johler and Harper (1962) are zero (T. B = T B , = 0). 
 
 If 8 (fig. 2) or <pj is an angle of incidence for a single semi- 
 infinite slab, then 
 
 rHP^ 
 
 cos e - ^- l I i 
 
 T " " . (2.1) 
 
 cose + ^i.(^ sine ) 2 
 
 Similarily, the ground reflection coefficient for vertical polarization 
 can be written 
 
 k-i / , f k.! 
 
 , s o . '--! / i i pi- sinO 
 R = k - 2 V Vk - 2 J — . (2.2) 
 
 •♦feV l -(fe*0 
 
 The wave number of the ground, assumed to be of infinite extent in the 
 negative z -direction (fig. 2) can be written 
 
 k -2 =-Je 2 - i - 1 - , (2. 3) 
 
 c V e uo 
 
 where e 2 an d ° are £k e dielectric constant and conductivity of the 
 ground, respectively. The frequency f is given by f = uo/2TT (hertz). 
 The speed of light is c - 2. 997925(l0 8 )m/ s, and e is the permittivity 
 of space, e = 8. (10 12 ). In a particular slab along the profile, 
 
 ■9- 
 
, _ u> / , . o 
 1 " c V X " 1 7> • (2.4) 
 
 where 
 
 a N » e2 , N + e 3 N. e s 
 
 1 *n,(v; >0 + iui) m+K^+iuu) nx.(v: )0 + iuj) ' U * ^ 
 
 / _ m _ 
 
 ^& o - _i_ » a - e, -r , - , 
 
 rru + m„ 
 
 and where the ratio of charge squared of the electron, e, to m (m, , or 
 for the ion m^ _.) , is given by 
 
 e 2 
 -^-= 2. 81 78(1 0- 8 ) (for electrons) , 
 
 and 
 
 e 2 
 = 4. 83 10(10 _12 ) (for ionized oxygen) 
 
 Thus, the number densities N, , N + , and N_ depend upon the ionizing 
 sources. At the very low altitudes that pertain to the ground wave, the 
 electrons can be neglected, and the positive ions (profile illustrated in 
 fig. 1) provide the important contribution to the finite conductivity of 
 the atmosphere. The appropriate collision frequency as a function of 
 altitude for use in the conductivity calculation is also given in figure 1 
 as a function of altitude. 
 
 Since the bottom slab of wave number k_ x contains the source 
 of spherical waves, the propagation in this slab is not as simple, 
 Sommerfeld (1909, 1926) and Weyl (1919), and more recently Brekhovskikh 
 (i960), quickly recognized that the ground wave emanating as a spherical 
 wave from an electrical point source can be expressed as a sum of plane 
 homogeneous waves and plane inhomogeneous waves that could be con- 
 veniently used to satisfy the boundary condition for the spherical wave 
 both at the source and at the plane boundaries illustrated in figure 2. 
 
 10 
 
Thus it is not necessary here to make approximations for the boundary 
 conditions at the source as is often done in mode theory (see Budden, 
 1961, for an extensive treatment of excitation factors for modes). Based 
 on Sommerfeld 1 s (1909) basic approach all waves emanating from the 
 dipole can be included in the analysis. This is especially important in 
 dealing with the problem of the ionized atmosphere or where the 
 ionosphere is depressed abnormally. 
 
 For convenience, let (3 = k_ x , the wave number of the lowest 
 plasma slab that contains the source and the observer, as shown in 
 figure 2. It is well known that the vertical electric field E z , for 
 example, can be calculated from the Hertz vector for a vertical dipole 
 source given by 
 
 « °; =[ iw l° 1 ^^ exp( ; ipD) z (2.6) 
 
 z L -4tt i (3 J D 
 
 by the operation 
 
 E * = [ p2+ !?J n " • (2-7) 
 
 where 
 
 D 2 = x 2 + y 2 + z 2 = r 2 + z 2 , (2. 8) 
 
 and where I I is the dipole current moment in ampere -meters, and 
 l-i is the permeability, \l Q - 4tt(10~ 7 ) henry/ m, and z is the unit 
 vector in the z -direction. 
 
 The factor exp(-i|3D)/D can be written as a Fourier integral 
 to show the equivalent plane wave structure of the spherical wave 
 source: 
 
 11- 
 
sEi ^ £L =^n^-i( Bx x + B 7y+B , B ,]3i^ J (2 . 9) 
 
 -00 -00 
 
 where 
 
 3 Z =[p 2 - 2 -3 y 2 ]" > (2.10) 
 
 and 
 
 3 X = p sin cos cp , 
 
 (3 y = P sin 9 sincp , 
 
 3 Z = p cos 6 , 
 
 where 9 and <p are the vertical and horizontal spherical coordinate 
 angles respectively. Bessel' s function of argument z and order can 
 be written 
 
 2TT 
 
 J (z)=y I exp[±iz costcp-cpj)] dcp • (2.11) 
 
 o 
 
 Neglecting the reflections from regions above the source, (T ## = in 
 fig. 2), one then finds after considerable ado the integral form of the 
 ground wave (see, for example, Brekhovskikh i960, pp. 242-244 for 
 integrals of this type) to be 
 
 E z =[-f I «^ k ^f] I Jo(3rsin0){exp[-ip(z-z o ) cos9] 
 
 + ^ e (0)exp[-i0(z+z o )cos9] j sin 3 9 d9 . (z £ z ) (2.12) 
 
 The integration contour is taken from to tt/ 2 and thence from tt/ 2 
 to the infinity of the imaginary axis. It is apparent that the solution can 
 
 -12- 
 
be split into two integrals: (a) the finite integral from to n/ 2 comp^ 
 rising the homogeneous plane waves and (b) the infinite integral from 
 n/ 2 to io° comprising the inhomogeneous plane waves. Thus, 
 
 TT 
 
 E z =[-3 2 Io^^] J {exp[-ip( Z -z )cos9] 
 
 + 7? e exp [-ip(z + z ) cos 9]} J o (0r sin 9) sin 3 9 d9 
 
 00 
 
 + [-ip 2 Io^l^j J (exp[-3( Z -z ) sinh9] 
 
 o 
 + R[ exp [-0(z+z o )sinh9] \ J ($rcosh9) cosh 3 9d9 (z £ z ) 
 
 (2.13) 
 
 where, in the second integral, 
 
 -isuxhe-l^i-^coshe)* 
 
 7J,'(9) = ===- • (2-14) 
 
 ■J 1 -(b CMh! )' 
 
 -i sinh9 + r^ 
 
 The height of the transmitter z or the receiver z above ground 
 appears in the exponential functions, where it is assumed z ^ z . The 
 second integral is the most difficult to evaluate,but it converges prim- 
 arily as a consequence of the exponential functions. In fact, the greater 
 the altitude of the receiver, the more rapid the convergence. As the 
 receiver altitude is reduced to zero, if the transmitter altitude is 
 assumed to be zero, the computation depends upon the Bessel function 
 J (z) for convergence. 
 
 -13 
 
-1 
 
 If we assume that the complete reflection coefficient T,, at the 
 top of the bottom slab in figure 1 has been evaluated, T, e = F(T, T x ,Ri 
 Ui , T 2 ,R 2 ,U 2 • • • ). .Assigning the appropriate ion density to (3 = k_ x 
 (thus (3 may be a complex number), we can obtain the complete solution 
 to the problem by analogy to ground wave theory (2. 13): 
 
 n 
 2 
 
 E z =\_-flo*<^-~\ I { ex P L-iP(z-Zo) cos 6] 
 
 o 
 
 + E. exp[-i$(z + z )cos9] + T,, exp[ -ip(2h - z - z ) cos 9] 
 
 + B m T,. exp[-i3(2h-z+z )cos6]]- |l -7?. T, . exp(-2i (3h cos 9)} 
 
 X J Or sin 9) sin 3 9 d9 
 
 00 
 
 + [-ip 2 I -t^] \ |exp[-3(z-z )sinh9] 
 
 o 
 
 + R[ exp[-0(z+ z ) sinh 9] + T,'. exp [ -£(2h-z-z ) sinh 9] 
 
 i 
 + El Tj, exp[-p(2h-z+ z ) sinh9]}{l - J?.' Ti.exp(-2ph sinh 9)] 
 
 XJ (P r cosh 9) cosh 3 9 d9 , (2.15) 
 
 (z* z ) 
 
 where R B and i?^ are given by (2. 2) and (2. 14) respectively. For a 
 single semi-infinite slab, T ee is given by (2. l),and T, . can be 
 written 
 
 -l 
 
 -14- 
 
-isinh - - =i - 
 ki ' 
 
 A-Of 
 
 cosh 8 j 
 
 
 
 
 -i sinh 8 + — — 
 
 h-Ct 
 
 \ 2 
 cosh 8 j 
 
 T.\ = . (2.16) 
 
 -i sinh 9 + 7=1. 7 1 - f ^=i- cosh 8 }" 
 k i W V k i y 
 
 The second integral representing the inhomogeneous waves 
 converges for the same reasons as in the case of the ground wave (2. 13). 
 In the complex z -plane, J (z) grows without bounds and may cause the 
 second integral (2.15) to diverge if Im z is sufficiently great to over- 
 come the exponential functions. This requires a shift of the integration 
 contour to keep the argument z of J (z) real. Thus, if 
 
 z cosh | = (x + iy) cosh 8 = A exp(icp) cosh 8 , (2. 17) 
 
 the required integration contour of the second integral (2. 15) in its 
 complex 5 = 5 + i ? plane is given by 
 
 %' + iS* = cosh" 1 [exp (-icp) cosh 8} , (2.18) 
 
 5' + i I" = In [exp (-icp) cosh. 8 + J exp (-2 icp) cosh 2 6 -1 } , (2.19) 
 
 provided that 
 
 ■,/c' , .-//» r exp (-icp) sinh 8 ~l ._ 
 
 d(§ +!§)=[ - " F T/ ., j d8 . (2.20) 
 
 J exp(-2 icp) cosh 2 8 -1 
 
 The integration contour then passes from zero to the point P, in 
 figure 3. At P, , 
 
 >i +i6x' , 
 
 -15- 
 
Complex 
 0-Plane 
 = 0'+i0" 
 
 Figure 3. Map of complex 8 -plane, illustrating integration contour. 
 
 where we may choose 
 
 B"i + iBi = In [exp(-icp) + Vexp(-2icp) - 1 ] 
 
 -1 
 
 = cosh [exp(-icp)] . 
 
 -16- 
 
The contour path is then from Pi to P 3 , where, at P 3 , 
 
 
 63 + i9 3 = £n[exp(-icp) cosh 9 + J exp(-2 icp) cosh 2 9 -1], and 8 is a 
 
 TT 
 
 select value ^ — . The contour must now be connected from P 3 to P 2 , 
 and P 2 in the complex 6 + i 9 plane is given by 
 
 02 + iej ~-\ + !(§' + i5 tf ) . 
 
 or 
 
 and fi 
 
 e£ = £' , 
 
 provided 9 in (2. 19) is equal to zero. 
 
 Finally, the contour extends to the infinity of the imaginary axis along 
 
 , TT 
 a contour 9 < — . The break in the contour between Pi and P 2 can 
 2 
 
 be taken at a convenient point P 3 corresponding to a zero of the Bessel 
 function J (|3 r sin 9 ), so that 9 ^ 9 . Note that the region between 
 
 TT 
 
 the original contour to — + i<» and this new contour is analytic. 
 Alternatively, the points P x and P 3 can be projected to the 9 axis, 
 and the contour from to P , P to P 2 , and P 2 to 9 = i» can 
 also be used. Finally, a contour is followed a short distance to P 4 
 along the real axis, then along a straight line to Pi , and along an 
 identical contour to P 3 , P 2 , and the infinity of the imaginary axis. 
 All integration contours tested with the computer program gave identical 
 results within specified accuracy. 
 
 It has been demonstrated (Johler, 1969c) that integrals of the 
 type (2. 15) involving Bessel functions of real argument (or with relatively 
 small imaginary parts in the argument) can be evaluated as an alternating 
 series in which each term is calculated from a Gaussian quadrature. The 
 zeroes of the Bessel function determine the limits of integration and the 
 turning point for the sign change of the resultant alternating series. 
 Since this technique is the same as the one discussed in an earlier paper 
 
 -17- 
 
(Johler, 1969c) it will not be repeated here. A complete computer 
 program for (2. 15) is discussed in the appendix. This program uses the 
 technique of continuous stratification illustrated in figure 1. The 
 computer subroutine for calculating T # . was, of course, required to 
 accommodate complex angles of incidence <p t = 6. The theoretical 
 basis of this subroutine is discussed elsewhere (Johler 1967). The 
 complexity of the terrestrial magnetic field has, obviously,been re- 
 moved from the program, since this analysis is concerned with an 
 isotropic plasma. 
 
 3. DISCUSSION AND CONCLUSIONS 
 
 The ground wave was calculated from (2. 12) and compared with 
 the classical ground wave computer calculation (Johler, 1969b; Johler and 
 Morganstern, 1965) over both plane and spherical ground. At distances 
 greater than about 200 km the spherical correction may of course be- 
 come significant. However, at the distances we are concerned with 
 here the plane ground error was a fraction of a decibel in the amplitude. 
 The computer program described in section 5 which evaluates (2. 13) 
 was then used in conjunction with the profile given in figure 1, together 
 with the independent variable F n . Note that the production rate is 
 related to the square of this factor, q « F|, according to (1.2). Values 
 of F n from 0. 1 to 40, which in this author 1 s estimate represent all 
 physically possible values resulting from fallout, were then introduced. 
 
 The most notworthy feature of the results obtained in figure 4 
 is a decrease in the amplitude of the propagated wave relative to the 
 normal ground wave for values of F fi of interest to this problem. This 
 is physically a consequence of the fact that the propagated energy is 
 slightly absorbed by the ions near the surface of the ground. If the 
 atmosphere were uniformly ionized, the wave would be much more 
 
 -18- 
 
T3 ' 
 
 !> 
 
 
 O 
 
 
 o 
 
 <1> 
 
 v. 
 
 -> 
 
 (U 
 
 a 
 
 T) 
 
 5 
 
 (\l 
 
 -n 
 
 
 c 
 
 I 
 
 c E 
 
 o s; 
 
 
 o 
 
 o 
 
 m Q- 
 
 b o 
 
 o 
 o 
 
 - b 
 
 Xt 
 
 
 .«g 
 
 
 h 
 
 
 S i 
 
 •M 
 
 M 
 
 a 
 
 o 
 
 o 
 
 o 
 
 T3 
 
 r- 1 
 
 G 
 
 
 
 
 m 
 
 Pi 
 
 O 
 
 X) 
 
 CD 
 O 
 
 .s 
 
 d 
 
 <u 
 
 s 
 
 6 
 
 CO 
 • H 
 
 T3 
 
 m 
 
 
 O 
 
 (d 
 
 a 
 
 ■ti 
 
 o 
 
 fl 
 
 •rH 
 
 4-> 
 
 cd 
 
 o 
 
 n 
 
 ^M 
 
 3 
 
 (d 
 
 o 
 
 
 o 
 
 CO 
 
 f— 1 
 
 ri 
 
 
 
 m 
 
 TJ 
 
 O 
 
 i—4 
 
 
 o 
 
 >^ 
 
 •rH 
 
 
 u 
 
 CD 
 
 •H 
 
 3 
 
 4-> 
 
 & 
 
 o O 
 
 CD 
 
 <t a) 
 
 rH 
 
 - 1 
 
 cd 
 
 i—i 
 
 +-> 
 
 
 id 
 
 ■rH 
 
 
 -M 
 
 ■ 
 
 u 
 
 f-i 
 
 o 
 
 a) 
 
 > 
 
 i—i 
 
 TJ 
 
 •r< 
 
 <u 
 
 +-> 
 
 4-> 
 
 rH 
 
 cd 
 
 
 
 3 
 
 £ 
 
 o 
 
 CD 
 
 cd 
 
 rH 
 
 •H 
 
 o 
 
 <+-l 
 
 
 O 
 
 a; 
 
 u 
 
 £ 
 
 Ph 
 
 O 
 
 CD 
 
 0) 
 
 
 XI 
 
 
 5 
 
 CD* 
 
 • H 
 
 • 
 
 r— 1 
 
 •r-i 
 
 a 
 
 
 a 
 
 •% 
 
 £ 
 
 < 
 
 rH 
 
 CD 
 
 rH 
 
 3 
 
 60 
 
 siaqpag 
 
 
 -19- 
 
severely absorbed as F n was increased (Johler 1969b). Since the 
 atmosphere contains only a thin film of ionization near the ground, the 
 wave propagates through the air but is concentrated near the ground. 
 This is true at least for values of F B up to 20. At this point, the 
 attenuation has increased to 12 dB relative to the normal ground 
 waves, which would be equivalent to increasing the production rate by a 
 factor of 400 with respect to the original BRAVO model, F n = 1. The 
 possibility of such extreme production rates does not seem likely, but 
 if it were possible to increase the number density of the BRAVO profile 
 to F n = 40 there would be a tendency toward excitation of a trapped 
 wave in the ionized atmosphere. This would cause an enhancement of 
 the field. Also, the attenuation of the normal ground wave as a function 
 of distance at a steady 12 dB / 100 km would be replaced by a standing 
 wave as a function of distance. It would therefore seem that the 12 dB 
 attenuation at a distance of 100 km is a reasonable maximum attenuation 
 for physically possible values of F n applied to the BRAVO model. 
 
 4. REFERENCES 
 
 AEC (1962), The effects of nuclear weapons, U. S. Atomic Energy 
 
 Commission (U. S. Govt. Printing Office, Washington, D. C. 
 20402). 
 
 Brekhovskikh, L. M. (i960), Waves in Layered Media (Academic 
 Press, New York, N. Y.). 
 
 Brueckner, K. A. (1964), Ion-Ion recombination, J. Chem. Phys. 40 , 
 439-444. 
 
 Budden, K. G. (1961), The Wave-Guide Mode Theory of Wave Propaga- 
 tion (Prentice -Hall, Inc., Englewood Cliffs, N. J.). 
 
 Hirsh, N. M. (1967), VLF studies in simulated sub-D region, 
 
 G. C. Dewey Corporation, 331 East 38th St. , New York, N.Y. 
 
 •20- 
 
Johler, J. R. (1967), Theory of propagation of low frequency terres- 
 trial radio waves --mathematical techniques for the interpreta- 
 tion of D -region propagation studies, ESSA Tech. Rept. 
 IER 48-ITSA 47 (U. S. Govt. Printing Office, Washington, 
 D. C. 20402). 
 
 Johler, J. R. (1969a), Ground wave propagation in a normal and an 
 ionized atmosphere, ESSA Tech. Rept. ERL 121 -ITS 85 
 (U. S. Govt. Printing Office, Washington, D. C. 20402). 
 
 Johler, J. R. (1969b), Loran C,D phase corrections over irregular, 
 inhomogeneous terrain -simplified computer methods, ESSA 
 Tech. Rept. ERL 116-ITS 83 (U. S. Govt. Printing Office, 
 Washington, D. C. 20402). 
 
 Johler, J. R. (1969c), Mutual impedance of loop antennas over finitely 
 conducting ground, ESSA Tech. Rept. ERL 122-ITS 86 (U. S. 
 Govt. Printing Office, Washington, D. C. 20402). 
 
 Johler, J. R.,andJ. D. Harper, Jr. (1962), Reflection and transmis- 
 sion of radio waves at a continuously stratified plasma with 
 arbitrary magnetic induction, J. of Res. NBS 66D (Radio Prop.) 
 No. 1, 81-99. 
 
 Johler, J. R.^and L. A. Berry (1965), On the effect of heavy ions on 
 LF propagation, with special references to a nuclear environ- 
 ment, NBS Tech. Note 313 (U. S. Govt. Printing Office, 
 Washington, D. C. 20402). 
 
 Johler, J. R. , and J. C. Morgenstern (1965), Propagation of the 
 
 ground wave electromagnetic signal with particular reference 
 to a pulse of nuclear origin, Proc. IEEE 53, No. 12, 2043-2053. 
 
 Loeb, L. B. (1961), Basic Processes of Gaseous Electronics 
 (University of California Press, L. A.). 
 
 -21- 
 
Natanson, G. L. (1959), The theory of volume recombination of ions, 
 Zhurnal Technicheskoe Fiziki 29, No. 11, 1373-1380 (English 
 trans. Soviet Phys. Tech. Phys. 4, 1263-1269, 1959). 
 
 Rutherford, E. (1913), Radioactive Substances and Their Radiation 
 (Cambridge at the University Press). 
 
 Sayers, J. (1938), Ionic recombination in air, Proc. Roy. Soc. 
 
 i 
 
 (London) A 169, 83. 
 Sommerfeld, A. (1909), Uber die Ausbreitung der Wellen in der 
 
 drahtlosen Telegraphie, Ann. Physik 28, 665. 
 Sommerfeld, A. (1926), Uber die Ausbreitung der Wellen in der 
 
 drahtlosen Telegraphie, Ann. Physik 81, 11 35. 
 Thomson, J. J. (1906), Conduction of Electricity Through Gases 
 
 (Cambridge at the University Press). 
 Thomson, J. J. (1924), Recombination of gaseous ions, the chemical 
 
 combination of gases and monomolecular reactions, Phil. Mag. 
 
 s_6, 47, No. 278, 337-379. 
 
 Thomson, J. J., and G. P. Thomson (1933), Conduction of Electricity 
 Through Gases II (Cambridge at the University Press). 
 
 Weyl, H. (191 9), Ausbreitung elektromagnetischen Wellen uber 
 einem ebenen Leiter, Ann. Physik 60, 481-500. 
 
 -22- 
 
APPENDIX 
 Computer Program 
 
 The computer program GIDSPAC3 shown in the computation plan, 
 figure 5, calculates the vertical electric field E by means of (2. 15). 
 The program has two independent variables, the distance DIST and the 
 profile multiplier F n = PROMLT. The amplitude JEJ V/m in decibels, 
 the phase Arg (E) in radians, and the phase correction in radians, 
 
 cp c = r - Arg(E) (radians) ( 5 » 1) 
 
 are given, where the field, E, is defined by 
 
 E = |e| exp[iuot - i - r -icp c ] . (5. 2) 
 
 Homogeneous waves EZHA, EZHP, PHICl and inhomogeneous waves 
 EZIHA, EZIHP, PHIC2 are given separately for convenience, together 
 with the total field, EZA, EZP, PHIC3. Appropriate comments explain 
 each step in the program. A feature of the program is the shift in the 
 integration contour shown in figure 3, which tests the precision of the 
 numerical analysis and interpolation routines used. The shift can be 
 controlled. Thus, in (2.17), <p = AK1P, and with AK1P= 0, 
 the contour 0, P 4 ,P ,P 2 to i« is followed. With AKlP = AKlP, 
 the contour 0, P 4 , P x , P 3 , P 2 to i« is followed. Another example 
 would be AKlP = AKlP/ 2, which would follow a contour between the 
 above contours up to the point P 2 . The contour from P 2 to i® is 
 never disturbed since it has been optimized according to (2. 18). 
 
 -23- 
 
Ion and Electron Density Profile Dota 
 
 PROVIDE 
 
 SUBROUTINE CALCULATION 
 
 PROCEDURE FOR SLABS ABOVE 
 
 BOTTOM SLAB, FIGURE 2 USING 
 
 JOHLER UNO HARPER (1962) TECHNIQUE 
 
 TO RETURN 
 
 T ee »«» T ee F0R COMPLEX 6 
 
 SUBROUTINE PROVIDES 
 COLLISION FREQUENCIES 
 
 Ground Constant Data 
 
 INTEGRATE TO F> TO P Q 
 ALTERNATIVE 3 
 
 INTEGRATE P TO P, 
 
 INTRODUCE 
 COMPUTATION and CONTROL DATA 
 
 COMPUTE 
 
 WAVE NUMBER, K-i, 
 
 OF LOWEST PLASMA SLAB 
 
 PROVIDE 
 
 GAUSSIAN WEIGHTS »ND ABSCISSAS 
 
 FOR NUMERICAL INTEGRATION OF 
 
 EO. (2.15) IN SUBROUTINE USING 
 
 JOHLER (19690 METHOD 
 
 OETERMINE 
 
 4>-AKIP, THE PHASE ANGLE 
 
 Ell. (2. 17) 
 
 CHOOSE 
 ALTERNATIVE INTEGRATION PATHS.ofFIG.3- 
 
 (1) 0- P r P 3 - P 2 -(ia>) 
 
 (2) 0- P 4 -P,-Pj-Pj-(ia>) 
 
 (3) 0-P,-P.-P,-(i«o) 
 
 COMPUTE 
 WAVE NUMBER, K- 2 , OF GROUND 
 
 BEGIN DISTANCE DO LOOP, 
 COMPUTE K_i r, EO.'S (2.4,2.5) 
 
 COMPUTE ANCLES OF INCIDENCE,©, 
 
 REAL OR COMPLEX a INTEGRATION 
 
 INTERVAL LIMITS 
 
 CALCULATE ZEROES OF 
 
 BESSEL FUNCTION FOR INTEGRATION 
 
 INTERVALS IN SUBROUTINE 
 
 TEST CONTROL DATA 
 
 INTEGRATE to P„ To P, 
 ALTERNATIVE 2 
 
 INTEGRATE TO P, 
 ALTERNATIVE I 
 
 INTEGRATE P.to P, 
 
 COMPLETE SUMMATION OF 
 HOMOGENEOUS WAVES E0.I2.I5) INTEGRAL TO n WITH 
 GROUND REFLECTION COEFFICIENTS EO. (2.2) 
 
 CLOSE CONTOUR, 
 INTEGRATE P, TO i co 
 
 COMPLETE SUMMATION OF INHOMOGENEOUS WAVES 
 
 EO. (2. 15) INTEGRAL TO co 
 
 WITH GROUND REFLECTION COEFFICIENTS E8.(2 .14) 
 
 COMPUTE TOTAL FIELD, e., EO. (2.15) 
 
 ACCURACY TEST FOR INHOMOGENEOUS WAVES 
 
 TEST MAXIMUM NUMBER OF DISTANCES 
 
 1 
 
 PRINT OUTPUT 
 
 
 DATA : DIST. - DISTANCE, kn 
 
 
 PROMLT- F„ 
 
 
 FREO. ■ f^ - f, kHz 
 
 
 TOTAL FIELD HOMOGENEOUS 
 
 INHOMOGENEOUS 
 
 EZ A - I E r I EZHA 
 
 EZTHA 
 
 EZP- Arg E r EZHP 
 
 EZIHP 
 
 PHIC3 -<k PHICI 
 
 PHIC2 
 
 Figure 5. Conceptual plan of program GIDSPAC3. 
 
 GPO 858 -372 
 
 - 24 - 
 
PENN STATE UNIVERSITY LIBRARIES 
 
 i mini mi i