**Problems of wireless telegraphy**

Since the beginning of the 20-th Century, the problem of transmission of information by electric waves is in the
foreground of technical physical interest. Can we understand the astonishing range of wireless signals through the otherwise
proven Maxwell theory? The answer is: Yes and No. Yes, in as far as no other than the known electro-dynamic laws enter into it.
No, in as far as the ionosphere (**Keneley-Heaviside
layer**) during the overcoming of Earth's curvature has an essential role and must be added to
Maxwell's wave propagation as *deus ex machina*.

Unfortunately, we cannot deal in the sequel with the reflection processes at the ionosphere, but must limit
ourselves to the questions of the spreading in the
**homogeneous atmosphere and the likewise assumed to be homogeneous Earth**.
Likewise, we must bypass the, for the technician very important, questions regarding the construction of transmitters and
receivers, since they do not really belong to the topic of partial differential equations. Moreover, we will idealize our
transmitter antenna extremely, indeed to Hertz's dipole.
On the other hand, the questions concerning the spreading belong definitely to our topic and
will yield us a full coverage for the practical usefulness of the earlier developed methods, which we have so far mainly applied
to pretty unrealistic heat conduction and potential problems. Moreover, this is demonstrated by the general electro-dynamic states
of oscillation. They have been collected with some completeness in Frank-Mises, Chapter VIII. When we present among these problems
once more wireless telegraphy, it is because the earlier presentation in the interest of simplification of formulae was more
simplified than is compatible with practical demands. Hence, we will now not place our antenna dipole directly on Earth's surface,
but at some distance from it, pursue in more detail the radiation of the horizontal antenna and establish its identity with the
radiation of the vertical antenna, which occurs with growing distance from its origin, and discuss also the radiation
characteristics with respect to second order terms in 1/*r*, etc. We will sketch the energetic consequences (necessary
energy supply in the case of a prescribed antenna flow, heat loss in the Earth) in a final section. Almost throughout, we will
assume Earth to be plane; we can only treat in the Appendix, which opens up for the method of eigen-functions one more area of
applications, the analytically interesting problem of Earth's curvature.

**6.31
Hertz's Dipole in a Homogeneous Medium and above a Perfectly Conducting Earth:****
**We assume that you know the concepts of electro-dynamics and their combination
through Maxwell's equations (Volume III). Since we are not now dealing, as in the earlier work, with atomic physics, but with the
phenomenological Maxwell theory, we will employ, according to our agreement in Volume III, the system of 4 units *M*
(metre), *K* (kilogram mass), *S* (second), *Q* (charge, measured in Coulomb's unit). In this measurement
system, there exist the dielectric constant and permeability; as usual; we will denote their values in a vacuum by *e*_{0}
and* m*_{0}, when *e*_{0}*m*_{0},=
1/*c*˛. In our measurement system, the factor 4*p*
, which disfigures the usually employed electro-magnetic equations, will be suppressed automatically unless it is conditioned by
the spherical symmetry of a problem.

**6.31.1 Introduction of Hertz's Dipole: **In
the electro-static case, we had derived the dipole's potential by directed differentiation (4.24.3)
from the basic potential *F *= 1/*r*;
the electric field of the dipole followed then from this
potential by one more differentiation. In the electro-dynamic case, the place of *F
*is taken by the function of the spherical
wave

The notation *P* is due to Hertz in his
fundamental work The Forces of electric Oscillations, Collected Works
II, p. 147, which also contains the well known images of the lines of force of the vibrating dipole. In the second form of (1), we
assume that the oscillation is **purely periodic **and **undamped
in time** (realized by a tube transmitter).

When using the abbreviated first form of (1), we must naturally observe that

As we know already, *P *satisfies the vibration
equation (7.4) which for purely periodic processes
becomes the wave equation

However, in the electro-dynamic case, *P *is
not a **scalar**, but a **vectorial**
quantity. Hence, we speak of the **Hertz
vector**** **
It is related to the vector potential in Volume III by
the simple relation

Just as the individual elements composing
have the direction of the current element in question, our
in empty space, i.e., in the absence of Earth, would have for a single antenna the direction of the antenna current. We assume
here the antenna to be **short compared with the wave length**,
i.e., its ends having capacitors, so that the current along the antenna can be treated as equal phased. In the presentation (1),
this vector character of *P *could
be expressed in such a way that the right hand side of (1) has a vectorial constant, which would have the direction of the antenna
and, as we will see below, the dimension of an electric moment (charge×length). However, we will disregard this, in order not to
make the following formulae too unwieldy, i.e., we retain Equation (1), although it is not written vectorially and dimensionally
correctly. We will correct this **beauty spot**
only in 6.36.
However, we can already emphasize here that due to the vector character of
we must , in general, attach to the Laplace operator *D *in
(3) its vector analytical meaning

(cf. Vol. II, (3.10a). This will already apply in 6.32.
Only in this and the following section, where we merely deal with a single Cartesian component *P*_{z}*
*or *P*_{x},
the ordinary *D *is active.

Now, we claim that the field can be derived from by the differentiation:

In the proof, we must show that thereby Maxwell's equations in vacuum

are fulfilled, in which, like in (2), one can replace

Due to (4) and (5a), the left hand sides of Equations (5) become

respectively. Both vanish, the first due to curl grad, the second due to (3) and
(3b). Thus, if we substitute (1) for and determine the
free multiplicative constant there from the strength of the alternating current flowing in the antenna, we have, according to
Maxwell, in (4) the field emitted by this antenna, valid for all distances which are large compared with *l**
= 2**p** */*k*;
for the immediate neighbourhood of the antenna, our representation obviously fails due to too large a schematization of our
antenna model. With Hertz, we call our model an oscillating and pulsating dipole, because the ends of the antenna (in this
description and in reality) carry changing charges with opposite signs. This extreme simplification of the real antenna with a
complicated structure is to serve us only as an example for showing how one may idealize a physical situation, in order to make it
accessible to fertile mathematical treatment.

Now, we proceed from the vacuum to a medium Earth with more **general
electro-magnetic behaviour**: Let it also be homogeneous, but have an arbitrary
dielectric constant *e *and
conductivity *s *; we also let,
for the time being, its permeability be arbitrary and denote it by*m*.
Then, Equations (1) and (3) for *P *formally
remain, but the wave number *k*
is not determined by (2a), but by

Equations (4) must be simultaneously replaced by

As before, you should confirm that in this way the corresponding generalized Maxwell equations

are fulfilled. The oscillation equation, which arose from the wave equation (3) by elimination of the time dependence following (7.4), becomes

**6.31.2 Integral
Representation of Primary Excitation:**** **To
start with, as a preparation for the following sections, we will give Representation (1) of *P
*a form in which it appears as a **superposition
of eigen-functions**. Since we will be concerned with cylindrical co-ordinates *r*,
*j, **z,* we return to the
eigen-functions *u* and eigen-values *l *of
Equations (26.3) and
(26.3a) in the case of independence on the *j*-co-ordinate,
where we will write *m *in place
of *m**p*/*h. *( A
confusion of this *m* with the
earlier employed magnetic constant *m *is
not likely. By the way, the latter will soon disappear from our formulae). Then:

However, while there the *l*-values
were restricted to a discrete spectrum, corresponding to the boundary conditions given on a cylinder mantle of finite radius, *l
*has in an unbounded
medium a continuous spectrum Thus,
also the *m*-values become
continuous and, in general, they are, by (8), complex for prescribed *k*. Moreover, since the boundary conditions for the
cover surfaces of the cylinder drop out, we will replace cos *m**z
*by *e*^{±}^{m }^{z}*.*
However, for *P*, we demand a
representation of the form

where *F*(*l*)*d**l**
*takes the place of the arbitrary amplitude constant, which can be added to every eigen-function. We must then rewrite (1) for
*P, *due to the changed meaning
of *r* (cylinder co-ordinate *r* instead of the spherical co-ordinate *r* in (1))

Specialized to *z* = 0, our Condition (9) becomes

In order to meet this condition, we use the integral representation of an arbitrary function by Bessel functions. It is best to employ there (8a) which with

becomes

The first of these equations becomes identical with (11), if we change the notation as follows:

the second equation then becomes

and represents in this form the **solution of the
integral equation** (11). The integral in (11b) is readily executed by using (19,14)
for *I*_{0} with the bounds ±*p *; in fact,
reversion of the sequence of integrations, yields

The last expression here comes from the lower bound *r *=
0 of the preceding double integral; the vanishing of the term, due to the upper bound ,
could be caused by a slight shift of the integration path into the shaded part of the *w*-plane (cf. Fig.
18), which, however, can be reversed in the sequel. The remaining integral with respect to *w* is
known, i.e., equal to

Hence, we can write instead of (12)

and instead of (11)

Now, we can obtain from here immediately a corresponding representation for *P
*in (10). In fact, we can complement (13a) into a function of *r* and *z*, which satisfies
the differential equation (1), by writing

with the instruction that is always
to be taken with a **positive real part**.
In this way, one ensures convergence of the integral and its vanishing in the limit .
Its agreement with (13a) for *z* = 0 then guarantees also that
is given correctly by (14).

We will still convert the representation (14) below into

with a more accurate description of the now complex integration path. Due to the
asymptotic character of *H*^{1}_{0}, (14a) has the advantage that it clarifies the satisfaction of the
radiation condition, corresponding to (1), which also had been fitted by its factor *e*^{+ikr}* *to
the radiation condition.

**6.31.3
Vertical- and Horizontal Antenna over an Infinitely Well Conducting Earth:****
**Hitherto, we have been occupied with an unbounded space - empty or filled with a
uniform medium with the constants *e, m, s*.
We now come to the half-space *z* > 0, which is bounded at *z* = 0 by an **infinitely
well conducting Earth** .
In Earth, Hence, due to the equality of tangential field
strengths, demanded by Maxwell's theory, there follows the vanishing of
on the positive side of *z.* By (7), this means that

We meet this condition by constructing at the two end points of the dipole, given in the air space, their images with inverted signs.(Fig. 27)

a) **Vertical
antenna at the distance ****h**** above ****z**** = 0**:**
**The two arrows pointing from the negative to the positive charge are for the
reflected and initial dipole
**equally **directed, whence
we write

The parallelogram at the left hand side of the figure shows that pairs of charges of the dipole, equidistant
from *z *= 0, exercise forces on a unit charge, imagined to be in the plane *z* =0, which combine into a resultant
in the direction *z* = 0. However, this means as much as
= 0.

b) **Horizontal
antenna at distance ****h**** above ****z**** = 0:****
**The reflected dipole has the **opposite**
direction as the initial one, whence we set with the same meaning of *R* and *R*' as before

If opposite signs are chosen in (16) and (17), the vector character of *H *appears,
which was suppressed in (1).

The sketch on the right hand side of Fig. 27 shows that now as well two of the
suitably allotted charges of our two dipoles exercise forces on a positive unit charge in the plane *z* = 0, the resultant
of which is perpendicular to *z* = 0.

In Exercise 6.1, these two cases a) and b) are to be checked using (15).

However, in one respect, Cases a) and b) differ fundamentally. In fact, if we pass on to the boundary , we find

Thus, a **vertical antenna resting on
Earth's surface** generates in the case of sufficiently good conductivity of the
ground twice that field which the same antenna free in space would generate in the absence of Earth. In contrast, a **horizontal
antenna, lying** **directly above Earth**,
will in the case of perfect conductivity of the ground be cancelled in its field action by its mirror image. The former made it
possible, at the start of wireless telegraphy, to transfer the formulae and figures, relating to the empty space of Hertz's
initial work, directly to the earthed antenna (Max Abraham). Indeed, one can cut Hertz's lines of force of the oscillating dipole
along a central plane and replace this plane by Earth's surface; the images of the lines of force are then perpendicular to this
plane , but meet our condition (15). The latter, i.e., the vanishing of the field of the horizontal antenna, expressed by (17),
while it is for *h *= 0 a matter of course, it loses for *h* > 0 quickly its significance; in this context,
compare the figures 6.36.
In fact, already *h *< *l *is
sufficient for converting the horizontal antenna into an effective message transmitting apparatus, even in the case of sea water,
which one can view for the usually long comparatively long waves of wireless telegraphy as a very good conductor; this applies yet
more in the case of a badly conducting dry soil. Hence, we conclude from this that for the horizontal antenna the condition of the
subsoil and its distance from it has a bigger role than for the vertical antenna. The start off
in (17) is then no longer sufficient and must be generalized.(6.33)

**6.31.4
Symmetry axis of the fields of electric and magnetic antennae:**** **There
arises at the boundary from the vertical antenna, as we
have just seen, the field of a Hertz dipole of strength 2, from the horizontal antenna the field 0. However, if in the last case,
we let the antenna current grow in the same measure as *h* becomes smaller, we obtain the field of a **quadrupole**.
Indeed, during this limiting process, Fig. 27b above becomes directly the earlier quadrupole
schema on the right hand side. Hence, if we replace the amplitude factors 2 and 0
generally by *A* and *B*, we can write

In Fig. 27b
, the last representation corresponds to the combination of the two single poles, each located on the same vertical line, into a
dipole with vertical axis and their combined displacement in the horizontal direction. At the same time, this means that the
horizontal antenna in the *x*-direction is equivalent to two vertical antennae, displaced with respect to each other in the
*x*-direction and with coherent-opposite current. This will be developed further in Fig.
30. The second formula (18), rewritten in polar co-ordinates *x* = *r* cos*j*,
*y *= *r *sin *j, *is

Hence*, P**
*has a preferred direction in the extension on both sides of the antenna *j
*= 0 and *j *=*p
*; it vanishes in the to it perpendicular directions *j
*= ±*p*
. We will present the direction characteristic of the horizontal antenna, associated with this, in Fig.
29, when we will also compute the constant *B*, which vanishes with increasing
conductivity. In contrast, the field of the vertical antenna is symmetric with respect to the *z*-axis, its directional
characteristic is also a circle. Hence, we see already the preferred suitability of the horizontal antenna for directed
transmission (cf. 6.33).

Both, the bar antennae of vertical and horizontal direction, we call **electric
transmitters**. In the case of a coil through which flows alternating current or
an arbitrary(circular, rectangular, etc.) closed conductor, we speak of a **magnetic
transmitter**, because then the magnetic field is concentrated on the axis of the
coil (the normal to the wires). The usual term is **frame
antenna**. A magnetic alternating flow pulsates towards the central vertical of
the frame just as along the bar axis pulsates an alternating current. While in the case of an electric transmitter the **magnetic
lines of force** run in circles about the bar axis, in the case of a magnetic
transmitter, the electric lines of force are circles about the normal to the frame antenna (at least for distances which are large
compared with the frame dimensions). However, both only apply for the vertically placed electric and magnetic dipole; in the case
of an inclined or horizontal location, the circular symmetry is perturbed by the conducting soil. Speaking generally, the
conditions in the case of a magnetic transmitter can be derived from those of the electric transmitter by exchange of
more details are given in 6.35.
Due to the boundary conditions for at the infinite,
well conducting soil, now also the signs in (16) and (17) change. In fact, one has for the magnetic *P*_{z}*
*(horizontal position of the plane
of the frame)

and for the magnetic *P*_{x}*
*(vertical position
of the frame)

This is proved in Exercise
6.1, The frame antenna of Type (19) has practically no importance; we will deal further
with that of Type (20) in 6.35.
As a transmitter, it displays a pronounced directed action towards the plane of the frame (i.e., for *P*_{x
}, the *y,z*-plane) with the same characteristic as the bar electric antenna, described by (18). Used as a
receiver, it is made to rotate about the vertical; it points, if one sets it to maximal reception, corresponding to its directive
characteristic, with its plane to the source of the signal and is therefore especially suitable for direction detection (cf. 6.34).

**6.32 The
Vertical Antenna over an arbitrary Earth:**** **Let
*e *and *s
*be the electric constants of the soil. As regards its magnetic behaviour, we can assume
that *m *= *m*_{0},
which corresponds to the real situation and somewhat simplifies the following calculations. We set

and call, as in optics, *n *the **complex
refraction index**. Now, denote the wave number *k*, computed in (31.6),
by *k*_{E}* *to distinguish it from *k* for air. Then, by (31.6)
and (31.2a),

As in (31.16),
let *h *be the height of the dipole antenna over the ground .

We must now distinguish three domains:

**I.**** ****Air
space ****z**** > *** h*.
Apart from the primary excitation, which becomes singular at the dipole

Following (31.14) and in analogy with (31.9), we write

Here, F(*l*) is an as yet unknown function, so
to say, the **spectral distribution**
in the *l*-continuum of the eigen-functions. The factor *e*^{-}^{m}^{h},
added to *F*(*l*) in *P*_{sec}*,
*will be convenient in the sequel and as pure function of *l *admissible,
because it affects only the meaning of the function *F*(*l*)
which anyhow is left undetermined.

**II. Air layer ****h ****> ****z****
> 0**. Here too, there exist primary and secondary excitations. The former must be set as in (3),
because, due to *z *< *h*, according to the sign convention in (31.14),
one has to work with inverted signs of the exponent, the latter has, as analytic
continuation of *P*_{sec} , the
same form as in (3)

At the boundary between **I **and **II**, the *P
*field must be continuous which is guaranteed by our settings (3) and (4) for an arbitrary *F*(*l*).

**III. Earth**
Here is no excitation; moreover, the *P*- field -
we will call it *P*_{E }- must be continuous
throughout. In order that it will satisfy the differential equation (31.3),
ruling in Earth, with *k*˛_{E} instead of *k*˛, we give it the form

According to our general rule, we had to choose for *m*_{E
}*z *the positive sign, because *z* < 0. The factor *e*^{-m}^{h
}has been added for convenience. Again, it only affects the meaning of the anyhow freely disposable function *F*_{E}(*l*).
The functions *F*_{E}(*l*) and* F*(*l*)
are determined from the boundary conditions at Earth's surface.

According to Maxwell, we must demand here continuity of the tangential components , for which only

are possible. In fact, the electric lines of force lie in the meridian planes through the dipole axis, the
magnetic ones are circles about this axis, so that
disappear. (Both follow from the fact that *P **= **P*_{z
}is only a function of *r* and *z*.) Now, by (31.4)
and (31.7),

Hence follow the continuity conditions for *z* = 0:

They can be integrated with respect to *r*, when the integration constants are 0 due to the
disappearance of all expressions for Moreover, if we replace, following
(2), in the second of the preceding equations *k*_{E}˛ by *n*˛*k*˛, we obtain simply

On the right hand side of this equation, one must enter the value of *P*_{E}
from (3), on the left hand side the sum of *P*_{prim} and
*P*_{sec}* *from
(4). Thus, one arrives at the two conditions

They are fulfilled by setting

whence follows

Thus, we have shown that indeed (3), (4) and (5) solve our problem and lead to fulfilment of its boundary
conditions; we conclude from the already confirmed uniqueness axiom of physical boundary value problems that there cannot be
another solution. Note yet that Equations (8) can be written more symmetrically with respect to the meaning of *n*, *m*,
*m*_{E}

We give now the primary excitation in its initial elementary form *e*^{ikR}/*R*
with *R*˛* = r*˛* + *(*z - h*)˛ and note that the contribution to *P*_{sec}*
*from the first term of *F* in (8) will differ from *P*_{prim}*
*only in that way that -*h* must be replaced by *+h*, i.e., *R*˛ by *R*˛*
= r*˛* + *(*z + h*)˛. It is then possible to contract (3) and (4) for the domains I and II and obtain as**
****general solution of our problem**
for *z*>0, *z<*0*, *respectively,

In particular, if *h* = 0 and one employs (4) for the then coinciding expressions of *e*^{ikR}/*R
*and *e*^{ikR'}/*R*', then the first line of (9) can be compressed elegantly, i.e., one then
obtains in the author's earlier developed representations

On the other hand, if one specializes the perfectly conducting Earth to the case
, then one can neglect *m*_{E }compared with *n
*and the integrands in both expressions (9) vanish. Thus, the result of the elementary
reflection method is confirmed:

It is recommendable to pursue further the limiting process to large *n*. We replace at the end of the
denominator of the integrand of the first Equation (9) *n*˛*m*
+ *m*_{E}* *by_{ }*n*˛*m*
and set in the numerator for all these results of *l*, on which
alone it depends,

(cf. Fig. 28 regarding the
selected sign of *m*_{E}* *). Thus, the
first Equation (9) becomes

One can arrive at a clear interpretation of the last integral as follows: Corresponding to

set

and compute

Due to the fact that the denominator *m *˛ in (10c) vanishes for *l **=
k, *the integration path in the complex *l-*plane must bypass the point *l
**= k*. The same applies to all the following *l*-integrals. The integrals
from (31.14) onwards contain only the denominator *m* which is
unobjectionable for the convergence*.*

Hence, the doubtful integral in (10c) represents the action of a virtual continuous cover of the half straight
line with dipoles, which extends from the image point *z*
= -*h*, *r* = 0 to Hence,
the approximate formula (10c) can be rewritten

We recall now a quite similar cover of a half-line with virtual source-points, which we employed with the heat
conduction equation in Fig. 15. While we
were there concerned with the exact fulfilment of the simple boundary condition
(of course, *h *is there not the present *h*), we are now dealing with the approximate satisfaction of the
complicated boundary condition, which arises out of the coupling of the air space with the well conducting soil.

The preceding formulae allow to derive the field directly by differentiation. However, we will not write down here the somewhat complicated calculation, since we require them only for the energy considerations in 6.36.

Our integrals in (9) and (10) are not yet uniquely defined due to the square roots

in them. Corresponding to the four sign combinations of *m *and
*m*_{E}, the integral would have four values,
its Riemann-surface be four-leafed. By our sign rule (31.14),
which was concerned with the real part of *m *and can be
transferred to *m*_{E}, one sorts out one of the
four leaves as being **permissible**. In order to ensure
the convergence of the integral, we will demand that the integration path lies at infinity exclusively on this leaf. We achieve
this by linking the **branch points**

to infinity by two (each arbitrarily led) **branching
cuts** which may not be crossed by the integration path. However, with reference to
Fig. 28 below, we do not integrate along the real axes over the branch point *l
**= k*, but bypass it below into the negative imaginary *l*-half-plane
and reach from there infinity about parallel to the real *l*-axis,
i.e., we follow the pass *W*_{1} in Fig. 28 from
and make precise only by this the meaning of the
integrals in (9) and (10).

However, even now, our representations (9) and (10) suffer from a mathematical beauty spot. The integrals with
the fixed starting value *l* = 0 are not integrals on a closed
path in the *l*-plane, which would be much more useful by their
convertibility. We correct this error by using the relation

and the **half-by-pass relation**
(10) in the hint to Exercise 4.2. If we set in the last
relation *r = l**r*, then the preceding equation becomes

Imagine (12) multiplied by an arbitrary function of *l*˛
(indicated hereafter by ··· ) as well as *l**d**l**
*and integrated along *W*_{1}. Then there arises from the subtrahend on the right hand side in (12) with *l*'=*l**e*^{i}^{p}

^{}

Here, *W* ' is the path arising by reflection in the origin in the direction ,
which, apart from the sign, is identical to *W*_{2} in Fig. 28 above. Thus, (12a) is the same as

it now follows from (12), if we denote the integration variable throughout by *l*
and combine the two paths *W*_{1}* *and* W*_{2}* *into the path *W* = *W*_{1
}*+ W*_{2},

We have now reached our goal of replacing the integration, starting at *l**
= *0 and seemingly real in our representations (9) and (10), by a complex integration which is closed at infinity. Imagine now
the transformation (13) having been performed in the integrals in (9) and (10). In particular, we write, for example, the primary
excitation from (31.14) and the first line of (10) in the new
version

The attentive reader will have noticed long ago that our present Fig.
28 agrees with the former Fig. 26
(also with respect to the paths *W*, *W*_{1}* *and the integration variable *l*)
and that our present problem (determination of *P *in the space,
subdivided by Earth's surface with given singularity at the location of the dipole antenna) adds up to the general problem of
Green's function. Here, as there, we have constructed the solution out of eigen-functions which satisfy at infinity the **radiation
condition**. The fact that this condition is also fulfilled in the present case is shown by only
the first Hankel function *H*^{1} appearing in (14) and (14a).

We have assumed here the presence of a time dependence of the preferred form *e*^{-i}^{w
}^{t}.With the time dependence *e*^{+i}^{w
}^{t}, we would have had to make in (12) the transition from *I *to *H *by
employment of the half-loop relation (10a) in Exercise 4.2 and
would have reached thereby a representation which, for example, consists in (14a) of elements of the form

i.e., is also of the radiating wave type.

Consider now the upper section of Fig.
28. Since, as we know, *H*^{1}_{0}(*l**r*)
vanishes at infinity of the positive-imaginary half-plane, we can drag the path *W *into this plane. In the process, it
gets stopped at the branching cuts (11a), which it must circumvent with the loops *Q* and *Q*_{N}.
However, there is one more singularity in the integrand in (14) and the analogous integrals, namely the location where the
numerator *n*˛*m* +*
m*_{E}* *disappears. We will call it

It corresponds to a **pole **of
the integrand and must be circumvented in a loop *P*. We have not shown in Fig. 28 the path linking it to infinity, because
they annihilate each other during the integration.

From the contributions *Q*, *Q*_{E}, *P *to the integral, *Q*_{E}
is to be neglected for large |*k*_{E}|, because at a large distance from the real axis *H*^{1}(*l**r*)
vanishes exponentially. To start with, consider *P *separately, but we will soon convince ourselves that *P *and *Q
*can hardly be separated.

The definition for *p*

yields

for which we can also write

Since , one has approximately

However, we note specially that the strict value (16) or (16a) of *p* is symmetric in *k *and*
k*_{E}.

We now construct the by-pass *P* by the residue method, applied to (14a). For this, we can set in all
factors of the integrand of (14a) *l *= *p*, but must
replace the denominator, which vanishes for *l*=*p*, by

with *l*=*p*, whence

the quantity *K *introduced here is again symmetric in *k* and *k*_{E}.
Thus, one has the contribution of the circuit *P* to (14a)

In the same manner, one finds for *z* < 0 (Earth, Interchange of *k* and *k*_{E}
and reversion of sign of *z*)

Apart from the immediate proximity of the transmitter, i.e., for all distances ,
we can replace *H* by its asymptotic value and obtain

These expressions have all the characteristics of a surface wave, as we have known from Volume II for **water
or seismic Rayleigh waves**.

**1. **They are bound to the surface *z* = 0 and their strength fades from there, towards
Earth quickly due to the factor (*p*˛* - k*˛_{E})^{˝} of *z*, towards the
atmosphere more slowly, but also exponentially for large *z* .

**2.** The propagation along *z* = 0 is given by

i.e., it is linked symmetrically to the material constants of the atmosphere and Earth, as it must be for a surface wave.

**3. **Disregarding for the present absorption in the radial direction, the amplitudes
of (19), (19a) drop with increasing distance from the transmitter like
the intensity like 1/*r*. This too
is a criterion for the essentially two-dimensional spreading of the energy on the surface *z* = 0.

**4. **For the sake of completeness, we mention yet the **exponential
absorption** during advance in the radial direction; it is given by the real part of *ipr*;
by (16b) as well as (1), (2), it amounts to

valid for *z* > 0 as well as for *z *< 0.

For sufficiently large *r*, when the factor *r*^{-˝}* *changes relatively little.
we can obviously interpret (19), (19a) as waves, which have their origin at an infinitely far away point, say, on the negative *x*-axis.
Then, these equations become

where *A *is a slowly changing amplitude factor; they are identical with the so-called
**Zenneck waves** (Ann.
Physik **23**, 846). Indeed, Zenneck has already studied in 1907 graphically and numerically exactly the fields ,
derived from (20), (20a) and discussed the material constants of different kinds of soil (as well as sea and fresh water). This
was the main concern in the author's work in 1909 (Ann. Physik **28**, 665) to show that these fields automatically
are contained in the wave complex, which according to the theory emanates from a dipole antenna. Naturally, nothing has changed
regarding this fact; only the emphasis, which we have placed on it, has changed. At that time, it appeared to be possible to
explain the wireless signals' overcoming of Earth's curvature by the character of the **surface
waves**. Today, we know that it is due to the **ionosphere**
(cf. the introduction to this chapter). The author believes
the recurring interest of the technical literature regarding the **reality of the
Zenneck waves **is pointless.

Moreover, we emphasize today more than we have stressed at that time that the wave complex *P
*is accompanied by the wave complex *Q *in Fig. 28. Speaking generally, this has the character of **space
waves** and, in contrast to (20), is represented by a formula of the type

It is best to represent *P + Q *by a contour integral, which embraces the two, almost coinciding points *l
*= *p* and *l *= *k *and,
depending on their relative position, must be discussed by the saddle-point method. This has been done most completely by H. Ott
(Ann. Physik **41**, 443). However, we must forego here a reproduction of these results in order not to get lost too
much in details.

We will only present here still one general point of view and a special formula, which is convenient for numerical calculations.

The former concerns a kind of similarity law of wireless telegraphy, namely the introduction of the **numerical
distance**. The radial path of the space waves, measured in wave lengths during the time *t*,
amounts to *kr *(apart from a factor 2*p*), the path
covered during the same time by the surface wave equals the real part of *pr*. We form the difference of the two paths and
define by it the quantity

We call the absolute value of *r *the
**numerical distance**; *r
*is a pure number, the absolute value of which is small compared with *kr*. In fact, by (16a),
one has

We now see: For small values of *r*,
the space wave type excels in the expression for the reception strength; the characteristics of the soil do not influence it
noticeably and one can compute (like Abraham)
without noticeable error with infinite soil conductivity. In contrast, for larger *r*,
there occurs a competition between the space- and surface-waves as also the value (21) of *r
*is computed out of the difference of the two propagation possibilities. Then, the material constants of
the soil becomes important, not only its *s*, but also its *e*.
Speaking generally, an equality of *r *is conditioned by equality
of the wave type and reception strength, i.e., *r *leads to a **similarity
law**. The circumstance that , by (21a), *r *for
sea water, due to its relatively large conductivity, becomes much smaller than (at equal absolute distance *r*) that for
fresh water or for a smooth dry soil, explains the favourable reception strength over sea water (the difference in the reception
strength during the day and at night depends, of course, only on the ionosphere.

An expansion in terms of rising powers of *r *led
the author in his first paper in 1909 to a convenient approximation formula, which was subsequently derived in a partly simpler
manner by different authors (B. van der Pol, K.F.Niessen, L.H.Thomas, F.H.Murray); its definite
form is

and it applies to the air space near Earth's surface.

As confirmation of the preceding observations, we state: The first term, which dominates for sufficiently small *r*,
is of the space wave type and corresponds, due to the factors 2, to the concept of an infinitely well conducting soil; the second
term is of the surface wave type and agrees qualitatively in the required approximation as well as quantitatively with the first
equation (19); the third term represents a correction, which arises for larger *r*.
The generalization of (22) to moderate elevations *z* over Earth's surface
is

where

and, for *z* = 0, *t *changes
into *r *and (23) becomes (22).