The functions H (z) are Hankel functions [6, 7]. It is often more convenient to use modified Hankel functions h(z) for which values with complex argument (z=x+iy) have been tabulated 7, was calculated from Riccati's differential equation [6, 7] as follows: (85) (86) (87) (88) δε 3 5 9 15 Values of 75,0 and 78, are given in table 44 [13]. In the plane-earth theory, the vertical electric field intensity at the surface of the earth was found to be as follows: E=2E,,F2 (volts per meter). The following computational formulas may be used to evaluate F2 (eq (31)): 2 (2p1)" n=0 (2n)! 1 (2n)! P is the numerical distance described first by Sommerfeld [11]: (91) (92) (93) e The numerical distance, p1, is related to the spherical-earth theory parameters K, and and can be computed therefrom if it is convenient. (94) d=the distance, meters, along the surface of a plane or spherical earth. dmiles=dX0.62136996×10−3. P1the numerical distance of Sommerfeld [11]. the conductivity and permittivity parameter, for a vertical dipole source, the permittivity of the air at the surface of the earth, esu. A value of 1.000676 was assumed. ←2=the permittivity of the earth, esu. A value of 15 was assumed. K=the permittivity, mks, farads per meter μα 440=a universal constant, henrys per meter, the permeability of free space. A value of 4X10-7 was assumed. 8 c=a universal constant, meters per second. A value of 2.997951X10 was assumed [14]. A value of 2.997529X10 is frequently assumed [17]. σ = the conductivity of the medium, specifically, the earth, mhos per meter. a=the radius of the spherical earth, meters. A value of 6.36739X10 6 assumed. a=the "effective radius" of the spherical earth, meters. k'the "effective radius factor" of the spherical earth, meters. was a=the parameter associated with the vertical lapse of the permittivity of the atmosphere, h2 f=the frequency, cycles per second. w=the frequency, radians per second. h1 = the altitude of the source above the surface of the earth, meters. h2 the altitude, meters. miles=hX0.62136996×10−3. the phase, radians. t=the time, seconds (microseconds). : 'the phase in free space or over infinitely conducting (σ= ∞) earth, o'=k1d, = radians. the phase of the secondary factor, F, radians. F=Flexp[i¢c]. t. the phase of the secondary factor, microseconds. [tch-t]=the phase of the secondary factor, F, aloft, relative to the surface value. |E=the amplitude of the electric field intensity. 7=the index of refraction of the atmosphere at the surface of the earth, n=√€1. = the index of refraction of the atmosphere at some point aloft. k2 the wave number of the earth, radians per meter, =the source vector, n=2(t), [2]=(t—nd/c), i=dā/dt amperes per square meter. E the electric intensity, volts per meter. i=the source, amperes per square meter. = E, the scalar vertical electric field intensity over spherical earth, volts per meter. E the free space vertical electric field intensity, cylindrical coordinates, volts per meter. E, the vertical electric field intensity over plane earth, volts per meter. 2 F, the secondary factor, computed from spherical-earth theory. F= the secondary factor, computed from plane-earth theory or free space, cylindrical coordinates. Fthe secondary factor, free space. E=the field intensity in free space, volts per meter; E,,-E; in cylindrical coor dinates. Io the amplitude of the source current, amperes. H=the magnetic field intensity, amperes per meter. f(h) the "height gain" factor of the source. = A value of unity (1) was assumed, i. e., it was assumed that the source (transmitter) was on the surface of the earth. fs (h2)=the "height gain" factor of the observer (receiver). 0=the angular distance from the source over spherical earth, radians, @=d/a. r=the distance from the center of spherical earth, meters. r=a+h2. |E (decibels)=20 log10|E|. tg=group delay, secondary factor, seconds (microseconds). v=phase velocity, meters per second. =group velocity, meters per second. v=signal velocity, meters per second. BOULDER, March 14, 1956. U. S. GOVERNMENT PRINTING OFFICE: 1956 |