6.33 The Horizontal Antenna over an Arbitrary Earth: In the case of an antenna, oriented horizontally along the x- axis , one has to set the Hertz-vector directly equal to Px. However, we have already noted at the end of 6.31.3 that this yields the required result only in the case of infinite soil conductivity and therefore we will first prove this conclusion.
With , Equations (31.4) and (31.7) yield
must be continuous when crossing z = 0. The preceding equations for suggest therefore the continuity of Px, whence obviously follows also the continuity of . However, then the formulae for would imply the equality of k² and k²E, which is a contradiction.
It is resolved by setting up the Hertz vector in two components:
then (1) becomes
Hence, one has for z = 0
Moreover, one has, by (31.4) and (31.7), the magnetic components
The continuity of yields
and then that of
Thus, we have the two conditions (5) and (7) for Px, which we can give the form
as, after the already obtained determination of Px, two conditions (6) and (4) for Px:
Px is computed following the scheme of 6.32, We distinguish there three domains:
In I and II, Px is composed of a primary and a secondary excitation, which is to be set up like in (6.32.4); in III, we only have to use the secondary excitation of (32.5). Conditions (8) yield instead of (32.7a, b)
and, by setting both brackets equal to zero,
The expression for F has already been written so that its second term vanishes for or, what is the same thing, for , and that there remains only the first term, i.e., F=-l/m. Substituting (11) into (32.3, 4, 5), we find as general representation of the Px-field, analogous to (32.9) with R²=r²+(z-h)², R'²=r²+(z+h)²,
In particular, if h = 0, then R' = R and (12) simplifies to
On the other hand, if one specializes to the case then also and the two integrals in (12) vanish so that (12) is reduced to
in agreement with (31.17).
The integration in (12) and (12a) must be executed along W1 in Fig. 28. We can employ instead again with advantage the path W = W1 + W2 closed at infinity. If we interchange at the same time I0 with ½H10, we obtain thus for vanishing h, but finite kA, as in (32.14a),
Turning now to the determination of Px, we consider first the second condition (9). Since Px and PxE do not depend singly on x and y, but only on , one finds, in agreement with (31.17),
correspondingly, It then follows from (9) that also Px must contain the factor cos j, whence we conclude that we must not compose Px as before out of the eigen-functions but out of the next higher eigen-functions with subscript 1
Taking into consideration that there is to be no primary excitation with Px, we thus set with two functions still to be determined
The first condition (9) yields then immediately
The second yields
In fact, the terms outside the integrals in (12), which generated the right hand side of this equation, vanish for z = 0. If one still extends the integrand on the right hand side in the numerator and denominator by m - mE and notes that
as well as that, by (19.52), I0'(r) = -I1(r), one can contract (13) into
Then, we derive a further equation between F and FE, namely
Hence, this yields with (13a) and (32.3)
Now, the final representation of Px becomes, by (13),
The integration path is W1 of Fig. 28 or, if we replace I1 by ½H11, W = W1 + W2 . Since the numerator in (15) agrees with the one of Pz in 6.32, there exist also in Pz , induced by the horizontal antenna, surface waves corresponding to the pole P in Fig. 28, which superimpose also here on present space waves or fuse with them, respectively. Assuming which yields that,, so that Pz vanishes. Thus, our induced vertical component depends greatly on the condition of the soil and could not appear in the older treatment by the elementary method, reproduced in 6.31.
In principle, the radiation of the horizontal antenna differs from that of the vertical antenna by the directional action, which is caused by the factor cos j in (15). The same factor also appears in the field components, decisive for the radiation, and appears quadratically in the radiated energy. We will see below that, in general, the component Px, free of cos j, does not contribute significantly to long distance transmission, whence we can disregard it in the following discussion.
Fig. 29 then represents by the solid line the directional characteristic of the horizontal antenna. One plots, in order to obtain it, as a measure of the radiated energy in a polar diagram, where M is the maximum of , radiated in the direction j = 0 . The curve is symmetric with respect to the direction j = ±p/2, in which there does not occur radiation; in the forward direction j=0 and in the backwards direction j = p, the radiation is the same. If one combines the horizontal antenna coherently with a vertical antenna and arranges it so that the vertical antenna by itself would yield the same radiation M as the horizontal antenna would yield for j = 0 (its polar diagram would be a circle with radius M), one obtains as the total characteristic of both of them the curve
It is represented in Fig. 29 above by the broken curve and displays a one-sided stronger directional action than the two-sided solid curve.
The bottom of Fig. 29 shows a sketch of an arrangement by which such a combination of horizontal and vertical antennae were realized in 1906 by Marconi for the transatlantic traffic (Clifden Station in Ireland). The preferred, one-sided transmission in the direction, indicated by the arrow, caused general amazement and led to the subject for the dissertation (Munich, 1911, Jahrbuch für drahtlose, Telgraphie 5 , 1912, 14. 158) of H. von Hörschelmann, in which the preceding theory was developed (Marconi had mainly designed with the instinct of an ingenious practician). However, Clifden's installation was rather unwieldy; it was replaced later on by a more convenient combination of two or several vertical antennae (Fig. 30).
You have here a horizontal antenna of length l with current flowing through it at each time as well as their input and output through Earth. The last are equivalent to two coherent, but in opposite phase oscillating vertical antennae, visualized by two masts. Their effect at a distance is given by a formula of the type
where P1 is the Hertz vector of the individual mast. We will now show that our theory, in rough approximation, indeed leads to a formula of this kind.
We set in (15) h = 0, since we are concerned with action at a distance and, as in (32.10b),
Moreover, we take into consideration
Then, the first equation (15) becomes
so that one has under the integral sign the primary excitation eikR/R. Indeed, we have obtained the form of (16); the meaning of the antenna length l there thus becomes
Hence, |l| has, in view of the meaning of k, the order of magnitude of the wave length l, but depends otherwise strongly on the condition of the soil; in the limit , one obtains |l| = 0, as has already been noted.
The same method of approximation also yields an order of magnitude estimate of Px. We start from the first equation (12) and set there h = 0 and Thus
The ratio of the two quantities z/r, i.e., near Earth's surface, is very small at large distances from the transmitter. By (16c) and (16a), -Px/Pz is in the same ratio, whence
This has already been emphasized and now been proved.
Our result is most strange: The primary excitation Px only serves to cause the secondary excitation Pz. The transmission into the distance is performed alone by Pz. Only in the immediate neighbourhood of the transmitter, Px is effective and exceeds here even Pz due to the singularity, initially prescribed for Px. At large distance, the field has during transmission with a horizontal antenna the same character as during transmission with a vertical antenna, disregarding, of course, the dependence on j, which indicates the primary origin from a horizontal antenna. In both cases, at large distance, the signals are suitably received with a vertical antenna; a horizontal antenna would be unsuitable as a receiver, because the horizontal component of the induced field is also in the case of moderate conductivity of the soil small compared with the vertical component.
These results are generally known in practice, but can hardly be understood without our theory involving the condition of the soil.
We note yet that our approximations (16a) and (16c) are to be viewed as first terms of an expansion in rising powers of the numerical distance r. Just as we had to complement in (32.23) for the vertical antenna the term eikR/R by terms which increasingly depended on r, also our equations (16.a) and (16.c) must be corrected by terms depending on r.
6.34 Errors during taking bearings of an electric horizontal antenna: As is known in navigation, taking bearings means finding the direction from which a signal reaches a receptor. As the ideal receptor for wireless signals is recommended the frame antenna, which was described in 6.31 and will be examined in detail in 6.35. Imagine its being installed at the location of reception rotatably about a vertical axis. Let the location of the reception, as in the case of navigation at sea, lie near Earth's surface. Assume that the transmitter is a horizontal antenna. We then expect, corresponding to the direction characteristics of Fig. 29, not only maximal reception at all points of the x-axis in the case of an x-directed antenna, but moreover at each point x, y of Earth's surface maximal reception in the r-direction, from which the signal is emitted, and no reception in the j-direction. In practice, the conditions are not that simple. The reason for this is that the horizontal antenna, apart from its main radiation of order 1/r, emits also radiation components which fade with 1/r².
In order to understand this, we must continue the approximation of the field by one more step than happened in the last equations of 6.33. In fact, Equations (33.16) and (33.16c) would yield and from there for h = 0 and z = 0, due to Px = 0, just a field .We will now study more closely. We take Px from Equation (33.12c), which has already been specialized to h=0, Pz from Equation (33.15), where we set h = 0 and can replace I1 by ½H11. Thus, we obtain without omitting anything
and, after simple manipulations,
By (32.14), the last integral is nothing else but the P-field of a vertical antenna, divided by n². Since we want to set z = 0, we can use for the last Equation (32.22). In fact, it is sufficient to employ the first term of this representation, so that we can write simply
It is now expedient to go to polar co-ordinates. Neglecting terms with (kr)-3, we obtain from (2)
and from (31.4)
We must now still estimate Px. With the approximation, given in (33.16c), one would find directly for z = 0 that Px = 0; however, a more accurate computation yields, also with omission of terms of order (kr)-3,
Thus, (3), (4) and (5 ) yield
or also finally, since
Hence, we conclude: For j = 0, the field of the horizontal antenna has here the r-direction, i.e., a frame antenna, erected rotatably in the extension of the transmitting antenna, displays in the direction of the transmitting antenna its greatest reception strength, as we have expected from the start.
In contrast, for j = ±p/2,
A receiver antenna, placed rotatably on the line perpendicular to the transmitter antenna, displays misinformation. In the orientation of largest reception strength, it does not point to the location of the transmitter antenna, but in a, to it perpendicular, to the transmitter antenna parallel direction. Indeed, the reception strength is very small, of the order of 1/r², which, by the way, also explains that this reception strength was given in Fig. 29 for the direction characteristic, which only takes account of terms with 1/r, as being 0.
In general, we can call correct indication and incorrect indication. The latter, as in (7), arises only from terms with 1/r².
For arbitrary j, the relative incorrect indication in our approximation is, according to (6),
It grows in the approach to j = p/2 where it becomes infinite; naturally, this only means that here the correct indication vanishes corresponding to in (7).
The practician makes a mistake when he reduces such erroneous indications to errors in the construction of the receiver antenna. More so, they lie in the nature of the matter, namely in the addition of correction terms to the with 1/r radiating main field of the arrangement in question. Certain other miss indications such as night effects, which are due to reflections at the ionosphere, are not being discussed here.
6.35 The Magnetic or Frame Antenna: One can employ a frame antenna not only for direction finding, but also as directed radiating transmitter. The plane of the loop is in both cases placed perpendicularly to Earth's surface and indicates by its normal a definite horizontal direction, which we will use as x-axis. In the case of a rectangular shape, the loop comprises two pairs of coherent vertical and horizontal antennae with current flowing in opposite directions, similar to the scheme in Fig. 30.
Independently of the shape of the loop, we have called it in 6.31.4 a magnetic antenna. Our, say in the y,z-plane located, frame antenna is equivalent to a magnetic dipole, pointing in the x-direction; its primary action can be described by a Hertz vector Due to the presence of Earth, one reforms it into a vector with a more general structure.
The connection between our present and the electro-magnetic field in vacuum is the same as in (31.4), but with an interchange of Indeed, Maxwell's equations (31.5) are invariant to this interchange. Hence, we set for the vacuum as counter equation to (31.4)
and for Earth as counter equation of (31.7)
where, as before,
Then, Equation (31.3) again holds for P.
The boundary condition for z = 0 forces us here, as in the case of the horizontal antenna, to understand by a two-components vector
In fact, the components demand the continuity of the tangential components of
Thus, we have two conditions (4) and (6) for two conditions (2) and (5) for for the determination in terms of the already known Conditions (4) and (6) are exactly the same as the conditions in (32.7) for the vertical antenna, whence we can transfer the earlier representation (32.9), etc., to our It yields in the form of (32.14a), specialized for h = 0,
In the case of , we start of, as for the electric horizontal antenna, with (33.13) involving cos j. However, now the two there prescribed functions F and FE are the same due to (3); by (5), their common value is
Thus, we obtain from (33.15), for h = 0 and using H instead of I,
However, it is not difficult to see that here, in contrast to the electric horizontal antenna, can be neglected in comparison with so that we will only take into account in the following discussion of the field and its direction.
Then, we have, by (1),
Now, it is a fact that we have obtained the first line of (7) from the representation (32.14a) and it was given for a small numerical distance by the approximation (32.23). Transferring this to (7), we find
This result agrees with (31.20) in the case of infinite soil conductivity. Now, by (9) and (10),
whence follows for the radiated energy E, the square root of which is plotted as function of j, the direction characteristic in the earlier sense:
where M is the maximum of (proportional to 1/r) in the direction j = ±p/2.
We compare (11) with the solid curve in Fig. 29 of the directional characteristic for the horizontal antenna. In our approximation, both are identical apart from the interchange of sin j and cos j, corresponding to the remark at the start of this section regarding the current in the frame and horizontal antennae. The interchange of sin j and cos j obviously arises from the fact that our horizontal antenna had the direction of the x-axis, while the frame antenna was placed perpendicularly to this axis.
Thus, the frame antenna emits on both sides maximally in its plane (j = ±p/2) just as the horizontal antenna emits on both sides maximally in its direction (j = 0 and j = p). Correspondingly, the frame antenna receives maximally when its plane is in the direction of the arriving wave. Since, as we have assumed above and maintained throughout our entire calculation, this plane is the x, y plane, the signal arrives at maximal reception from the y-direction with predominant electric z-component (perpendicular to Earth's surface) and magnetic x-component (perpendicular to the plane of the frame). The electric z-component induces then in the frame an electric current or, as we might say, the magnetic x-component excites the magnetic dipole of the frame. The frame acts as a magnetic receiver just as we have treated it before as a magnetic transmitter.
By the way, when we find directions with the frame antenna, we do not set up for maximal reception, but, as with all zero-methods of the physics of measurement, because it is sharper, for minimal reception. Then, the frame has not the y, z-. but the x, z-position. Its normal then points in the y-direction, i.e., in the direction of the arriving signal.
6.36 Radiation Energy and Earth's Absorption: As we will now discuss certain energetic questions, we leave the field of superposition of capable field strengths and turn our attention to the quadratic magnitude of the energy flux
We can no longer manage here with the complex concepts of fields with omission of the time factor e-iwt, but must multiply with each other the real field components. However, the thus conditioned complication drops out when we take the mean in space and time. These mean values become even simpler than our representation so far, since, due to the orthogonality of the eigen-functions, the Bessel functions drop out and are replaced by more of less elementary functions.
In the first place, it is important for us to consider the total flux of energy, integrated over a horizontal plane in the air space:
Depending on whether we place the horizontal plane above (z > h) or below (z < 0) the dipole antenna, we will denote the energy flux, defined by (1), by S+ or S-. In the process, S- and S+ are referred to the positive x-direction. The energy, which effectively enters Earth in the negative z-direction, is therefore given by -S-, at first taken for the plane z = 0. (Effective entering means here more entering than leaving. The reflected exiting radiation is naturally automatically taken care of in S-.) However, it is readily seen that instead of it any plane and especially the plane could be employed for the evaluation of S- (the space between the two of them is absorptionless and no noticeable energy emits through the infinitely far away section of the cylinder.) Since all the energy, which effectively enters Earth, is transformed there into Joule energy, -S- represents at the same time the total amount of thermal Earth absorption in unit time. On the other hand, S+ represents with the total amount of the radiation, transferred to the air in unit time above the plane z = h. We will call it the useful radiation. Hence
is that energy, which must be fed into the antenna in unit time, as far as we can disregard Ohm's and possibly any other losses in the antenna. Hence, is the performance fed into the antenna. (The letter W in (1a) may remind the reader of Watt.) We will mainly deal with it in the sequel.
A. In the case of the vertical antenna, we know that Denoting the hitherto complex expressions for by Er and Hj and adding the time dependence, Equation (1) becomes
By the averaging in time, the terms with e±2iwt drop out and there remains, using S again to denote this averaged value,
Since the field is independent of j, we can write
In order to compute S+, we take Er and Hj from (32.3), in order to find it from (32.4). The expressions thus obtained differ only by the ± sign with Pprim in (32.3) and (32.4):
where F(l) and F(l) are given by (32.8). It will soon be seen that the use in (4) and (6) of an integration variable other than l and the replacement in (4) and (6) of is very useful. Due to (3) and (4), (2) becomes the triple integral
Here, the simplifying action of the orthogonality condition (21.9a) becomes effective, which we rewrite once more in the present notation and specialize to n = 1:
It shows that in (7) the last integral vanishes for all values of l except for l = l and that the execution of the last integral yields f2(l,z), whence (7) reduces to the simple integral
A further simplification results if one, as we have announced already above, lets the two planes z=h±e move very close to the position of the dipole antenna, i.e., lets
One has then instead of (5) and (6)
Hence, the product f*1(l)f2(l) would consist of four summands. However, if we go to the difference S+ - S-, only two of them remain, namely those which correspond to the double sign in f*1(l), whence ,with the definition (1a),
where we could, due to , cancel in the exponential function of the last integral m*e against 2mh. The integral ahead is readily obtained. In fact, for l > k, m and, of course, also m+m* are real, whence the re vanishes the real part of the by -i multiplied partial integral from . Thus, there only remains the partial integral from 0 to k in which one can obviously go to the limit e = 0. With the now convenient integration variable m instead of l, one has
Due to the sign of m, the continuations over the permissible leaf of the Riemann surface in Fig. 28 must be taken into consideration.
As far as the second part of (12) is concerned, we compute the first summand in (32.8), which does not vanish for , i.e., F(l) = l/m, i.e.,
Due to m being real for l > k, one is again only concerned with the partial integral from l=0 to l=k. Written in the variable m with the abbreviation z = 2kh, one finds instead of (14)
Hence, by evaluation of the real part, one finds
Combination of (12), (13) and (15) yields
where K is the contribution due to of F(l) from (32.8), which had not yet been taken into account in (14), i.e.,
Regarding (16), we yet note that the first two terms on the right hand side, since they do not depend on the condition of the soil, can already be derived without our analytic apparatus from the ideas in 6.31. However, the evaluation of the correction term K is only possible on the basis of our complete theory. The discussion of these formulae will be deferred to C.
B. Due to the combined action of Px and Pz, the formulae for the horizontal antenna become more complicated; however, thanks to the orthogonality relation (8), they lead eventually to similar simplifications as in the case of the vertical antenna. Instead of (2), one has
and instead of (3) and (4) for the present
Since, by (34.1) and (33.15), are proportional to cos j, in contrast, by (33.12), Px does not depend on j, we draw from (3a) and (4a) the conclusion that Er and Hj contain the factor cos j, Ej and Hr contain the factor sin j. The integration with respect to j in (2a) can then be performed and one finds instead of (7) a rather complicated triple integral with respect to l, l and r. However, if one forms the difference W = S+ - S-, it becomes simpler, because then only the term arising from the primary excitation of Px and contained in has a double sign. Moreover, if one employs for the elimination of the differential quotient of I0 Bessel's differential equation, one finds with equal meaning of e as before
As a result of the orthogonality condition (8), this yields now the simple integral
The integration in the first term of (6a), which does not depend on kE, can again be executed as in (13) and (14), where, due to the real part sign, only the interval 0 < l < k need be considered. Thus, one finds instead of (16) with equal meaning of z as there
with the abbreviation
The expression (16a) just as (16) (cf. start of section) does not contain Bessel functions. One obtains the same expressions, as F. Renner has pointed out to me, by a method which is perhaps closer to the thinking of a practical person and which we will encounter in Exercise 6.3. However, this approach only yields the performance W = S+ - S- and not S+ and S- separately, which are also of considerable practical interest. Hence, for their computation, the preceding approach is unavoidable.
C. Discussion: To start with, consider the main terms in (16) and (16a), neglecting for the time being the corrections K and L:
As , both yield the common value 2/3. According to the meaning of z = 2kh, z = is the same as h = . Naturally, for h = , Earth does not affect the radiation of the antenna, i.e., the vertical and the horizontal antenna must behave in the same way. For both, the entire performance becomes radiation. Correspondingly, both Equation (16) and (16a) yield the same limiting value
This is identical with a formula, which Hertz already has given for the radiation of a dipole (oscillating in infinite space). Note, however, that the factor k4 corresponds to the reciprocal fourth power of the wave length in Rayleigh's law of the blue of the sky, which truly arises from the superposition of the radiation of infinitely, far distant dipoles distributed in the air space, which are excited into oscillations by the Sun's rays.
In Hertz's already mentioned work, this formula is on page 160 of the collected works, Vol. II. When comparing (18a) with Hertz's formula, one must take into account the dimension factor to be obtained in (22).
On the other hand, if we go to the limit h = 0, the expressions in (18), expanded in rising powers of z and truncated with z 0, yield
From here, we understand the just found factors 2 and 0 in (18a) by means of Fig. 27: Reflection in the infinitely well conducting earth doubles the radiation of the vertical antenna for h = 0, that of the horizontal antenna is annihilated by its mirror image. In this context, we must note that we have neglected the correction terms K and L in (18). This neglect means that we have performed simultaneously the limiting processes
Fig. 31 presents generally the expressions (18). Above the abscissa, one has the values of z , below it those of h. The figure shows that for the vertical as well as for the horizontal antenna the transition into the final value 2/3 occurs with continuing oscillations about the latter. The difference of the abscissae of two successive extreme values is for both curves in the scale of the h about equal to half the wave length, which corresponds to the interference between the entering and from the ideally conducting soil reflected radiation.
Moreover, there has been entered for both curves a first correction (broken line) as given by the terms with K and L in (17) and (17a). The value k/|kE|=1/100 corresponds to the conditions of sea water for a wave length of 40 m. It is seen that the ordinates of the broken curve in both cases increase more steeply with decreasing h; the difference in the ordinates becomes of course, for finite h compared to the limiting case the stronger the more kE deviates from the limiting value . We are here obviously dealing with a rather complicated double limiting transition, in which connection we recall the double limiting process in Gibbs' phenomenon in 1.2: If we let first of all and then h = 0, we end up with the ordinates 4/3 and 0. On the other hand, if we stop at a finite value of kE and go first of all to , we end up at an infinite ordinate; naturally, this happens also then when we afterwards go to .
What is the physical meaning of this infinite growing of W? It does not benefit the useful radiation S+, but gets lost as heat -S- in Earth. Indeed, the Joule heat generated in the soil per unit volume * increases for a fixed antenna current with growing kE more and more, while the useful radiation remains finite. In order to prove this,we would have to discuss the formulae for S+ separately and also compute the corrections K and L, which would take us here too far.**
* Since the volume in which Joule heat is generated shrinks with growing |kE| more and more (skin effect), one understands that, without prejudice of the statements in the text, there does not occur a heat loss in the limiting case .
** Regarding this, we refer to A.Sommerfeld and F.Renner, Radiation Energy and Earth's absorption for dipole antennae. Ann. Phys. 41 (1942), where there are also statements regarding the in Industry customary concept of radiation resistance and the form factor in the case of finite antenna length.
D. Normalization for given Antenna Current: We have developed the entire theory of this chapter without consideration of the physical dimensions of the introduced quantities, which we must now do.
In fact, we set the factor in (31.1) of Hertz's dipole equal to 1. In reality, its is a defined number. Its dimension follows from the connection between P and in (31.4). Accordingly, P has the dimension Since, on the other hand, according to our start (31.1), P would have the dimension 1/r, i.e., M-1, it follows from the factor of P, set by us equal to 1, as dimension We compare it it with Maxwell's dielectric displacement which has the dimension charge per unit area, i.e., Q/M² with Q as dimensional symbol for charge. Thus, especially written for the vacuum, we have the dimensional equations:
QM is an electric moment. We set it equal to el. In the image, on which Hertz based his statements, e is the charge of a particle, which towards a charge at rest - e and beyond it oscillates.
What does now take the place of this moment in the concept of our earlier described short, with end capacities loaded antenna? The current jt, flowing in it, is, by assumption, to be treated along the entire antenna at each instant as equal phased and constant. We write it in the form
Let the corresponding charge of the end capacities be
According to the general relation
one must have e = j/w. At time t = 0, when the current is 0, the charges of the end-capacities are ±e. Since they are located at the distance l of the length of the antenna, they represent an electric moment of magnitude
We must set this product el instead of our moment QM in (19). Moreover, we have to add in (19) the factor 1/4p, which follows from a comparison of the field (31.4) in the neighbourhood of the dipole with that of the antenna current. Thus, we obtain for the dimension factor of our P:
The radiation S as well as the performance W must be multiplied by its square. Thus, one finds , using yet the often quoted relation
instead of (16)
This formula yields dimensionally faultlessly the performance in Watt. In fact, has, as has been shown in Volume III, Section 6, the dimension of a resistance and the numerical value 120p=377W. The quantity j must be measured in our measuring system, which is based on the electric unit Q = 1 Coulomb, measured in Ampère. Since lk has the dimension 0, W must be expressed directly in performance units WA² = Watt.
In the same manner, (16a) for the horizontal antenna is made dimensionally correct by multiplication by the same factor: