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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

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7, was calculated from Riccati's differential equation [6, 7] as follows:

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(85)

(86)

(87)

(88)

δε

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3

5

9

15

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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)):

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2

(2p1)"

n=0 (2n)!

1

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(2n)!
n=1 2"(n!) (2p)

P is the numerical distance described first by Sommerfeld [11]:

(91)

(92)

(93)

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e

The numerical distance, p1, is related to the spherical-earth theory parameters K, and and can be computed therefrom if it is convenient.

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(94)

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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,

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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

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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,

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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.

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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,

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=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

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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.
V2=the operator of Laplace.

BOULDER, March 14, 1956.

U. S. GOVERNMENT PRINTING OFFICE: 1956

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