2.1.11.10.6-2 Last |
Book 2.1.11.10 |
Wireless Telegraphy on the Spherical Earth
We will assume that Earth is conducting perfectly (for example, is covered by sea water). We will deal with a vertical antenna near its surface. Its axis is to form a polar co-ordinate system ; its distance from Earth's centre is r_{0}; Earth's radius is a < r_{0}. The field has the components and does not depend on j. We will derive it from a scalar solution u of the wave equation.
Hertz's vector P would not be suitable for this purpose;it does not satisfy the simple wave equation DP + k²P = 0, but would satisfy the for curvilinear co-ordinates valid, yet more complicated form (31.3b). It is more convenient to start from the magnetic component
We use the table
to compute in terms of H
of then from the second equation (31.5) , by the same table above,
Thus, H satisfies the equation
It can be transformed into the wave equation Du + k²u = 0 by making H proportional to ; in particular, we will set most conveniently
In fact, we can then write instead of (2)
The first two terms of {} are, according to the same table above, the same as Du. Thus, if we select u as solution of
then , by (1) and (2a), our electro-magnetic field is described completely by
Hence, the boundary condition at Earth's fully conducting surface will be fulfilled by setting
To this is added the condition that u is to behave at like a unit source, which means for the the existence of a radially directed dipole.
This problem has already been solved by the method of (5.28) by the equation for G(P, Q) there, the only difference being that there was prescribed the boundary condition u = 0 instead of (3a) above. However, this only demands a change of the constant A there. While we drew a conclusion from (28.18) and the condition u = 0 regarding A's earlier value (18a), we obtain now from the present condition (5a)
where z_{n }here and in the sequel takes the place of z ^{1}_{n}.
First of all, we derive from the thus modified solution (28.22) the simplified formula for the limiting case r_{0} = a, where also the antenna is located directly on Earth's surface. By (6) and (28.18), one then has
The numerator of this fraction simplifies, by Exercise 4.8, Equation (II), (to i/(ka)², and we denote the denominator by x_{n}(ka), whence
Substituted into the upper row of (28.22), one finds for G there, i.e., the present u,
which is valid for all values naturally, the range of validity of the lower row of (28.22) has now shrunk to 0. This result (7) agrees with the earlier treatment of this case in Frank-Mises [Vol. II, Chapter XXII, Section 4, (10)] (apart from a factor which originates from our present definition of the unit source). Also, the results there for arbitrary soil could be readily added here by a corresponding continuation of 5.28 ( continuous extension of the sphere's inside instead of a boundary condition on the surface).
If we were to process also the horizontal antenna on a spherical Earth, we would have to introduce beside the here defined function u a function v, arising from an interchange of , and would find for it a similar to (7) but more complicated representation.
However, the convergence of the series (7), like that of our general representation of Green's function in Chapter V, is very bad. In order to see this in the present case, we need only note that ka and kr are due to the order of magnitude of the ratio Earth's radius/wave length > 1000. In fact, as long as n is of a moderate size, the asymptotic values of Hankel apply to z, which show that the fraction z_{n}/x_{n} in (7) is almost independent of n. One would have to include more than 1000 terms of the series until the asymptotic approximations of Debye (21.32) come into action, which only then would affect a real convergence of the series.
In order to arrive at an effective computation of u, we employ a method, which was applied first successfully to our problem by G.N.Watson [after the model of P. Debye in his thesis Munich 1908 or Ann. Physik 30 (1909) 67; cf. also Frank-Mises, Chapt. XX, Section 4, Proc. Roy. Soc. London 05 (1918)]; as we will see, it is connected with the in Appendix 5.2 developed method. In fact, we convert the sum (7) into a complex integral.
With this aim in mind, we write to start with the series (7) in the form
on the basis the following, for integer n (and only for integer n) valid relation,
Moreover, however, we replace n by a complex variable n and draw in the n-plane of Fig. 32 a loop which encircles in a clockwise direction all the points
We guide over this loop the integral
which arises out of the general term of (8) by interchange of n and n, but with suppression of the factor (-1)^{n} and addition of the denominator sin np. Naturally, as in Appendix 5.2, P_{n} is not Legendre's polynomial, but the (only for integer n with it identical) hypergeometric function
The integrand of (9) now has first order poles at all the zeroes of sin np ; as far as they lie inside the loop these are the locations (8b) and in the neighbourhood of the location n = n one has, in first approximation,
Hence, the residue of the first fraction in (9) is
and the integral (9), computed as product of -2p i and the sum of all residues, is
i.e., it is apart from the sign identical with (8).
The next step involves a transformation of the integration path and we note to start with: The hypergeometric series in (9a) is a symmetric function of its first two arguments, whence we have, valid for everyone (also complex) n :
On introducing the notation
Equation (11) becomes
Hence, P_{s-½} is an even function of s.
The same result applies to the last factor of the integrand in (9). In order to see this, we start from the initial, for arbitrary subscripts valid, integral representation (19.22) for H^{1}; calling the subscript s, it becomes
If we interchange here w and -w and simultaneously s and -s, we obtain after inversion of the direction of running along W_{1}:
If we multiply this equation by and thereby, thanks to (21.15), step over from H to z, we find
However, the same relation holds also for the quantity x(ka), defined in (6a), namely
Division of the two relations yields that also the quotient
is an even function of s.
Finally, as far as the first fraction in the integrand of (9) is concerned, it becomes, written in terms of s,
i.e., an odd function in s.
We now guide the loop over a parallel straight line , parallel to the imaginary axis of the n-plane, which passes through the point s = 0, i.e., n = -½ and along two paths , which at a large distance of the real axis, so to say, link the ends of with those of . We will discuss later on the poles of the integrand, which must here be taken into consideration. We will at first show that the integrals along the paths and vanish.
That this is true for the path follows after the preceding work directly from the odd character of the integrand of (9), written in the variable s. In order to show that the same applies to the linking paths , we examine the factor z_{n}/x_{n} of the integrand for large values of n. We start from the series (19.34)
in which all terms not written down can be neglected when |n| > r. By Stirling's formula,
whence
similarly for general complex n
Hence,
Its absolute value tends to zero as |n| in any manner tends to infinity with a positive real part. In the representation (19.31) of H^{1}, we can then neglect I_{n} compared with I_{-n} . Then, by (14), if we step immediately over from H_{n }to
and from z_{n}(kr) to the quotient of two z-functions
Since a/r is a true fraction, (15) vanishes, if n + 1 tends to infinity with a positive real part, which is the case along both link paths . The same is also true for the quotient z_{n}(kr)/z_{n}(ka), the denominator of which, by (6a) and (14), can be written as
We see from this that the third factor of the integrand of (9) vanishes; likewise, the first factor vanishes due to its denominator sin np ; for the second factor, the same results follows from (24.17) which, as noted there, applies also to arbitrary complex subscripts of the spherical function. Indeed, our initial path can thus be drawn to infinity of the positive real n-plane.
However, in the process, it gets stuck at the poles of the integrand, i.e., at
the locations of which we will now examine more closely. We set for the neighbourhood of the m-th of these roots
Then (9) yields through the formation of residues
Since our integral (9) was identical, apart from a sign, with the series (10) and this series, apart from a factor, with the solution (7) of our sphere problem, this solution is also represented by the series (16) so that we can write with suppression of an unimportant factor
As we have seen, the transition from the series (7), advancing with integers n, to the series (17), advancing with complex n, is mediated by the twofold formation of residues of a complex integral.
Before we continue the discussion of (17), we take a look back at Appendix II of Chapter V. There too, one is concerned with a series, advancing with integer n and complex non-integer n , i.e., the series (1) and (5) there. We will show that the identity of the two series also there is ensured by double formation of the residues for a complex integral. Written after the example (9), this is
It is seen, that one is here concerned with the zeroes of the denominator
which are common to the two functions u_{n}(kr) and u_{n}(k r_{0}), respectively. The corresponding residues are
with the meaning
differing from (15b). We also employ as the initial integration path of the two integrals (18) the path of Fig. 32. Like in that figure, it can be shifted across into the sum of the by-passes around n=n_{1},n_{2}, ··· , since the paths and also here do not contribute. Thus, one finds the for the two integrals (18) common representation in series
It agrees exactly with the series in A5.2, if one adds to it (II) of Exercise 4.8
according to which one has for r = ka and z_{n}(ka) = 0 that y_{n}(ka) is inversely proportional to z'_{n}(ka).
Thus, the new method, developed in A5.2, can indeed, although in a somewhat complicated manner, be derived by complex integration from the old hitherto used method in a series with advancing n. In particular, this derivation displays the mathematical reason for the there derived and emphasized strange fact that the two n-series, differing for , become one and the same n series (19).
In order to complete the discussion of our spherical Earth problem, we must show first of all that the roots (15a), just as it was the case with the roots (18a), lie in the first quadrant of the n-plane. In the case of (15a), one is dealing with the transcendental equation
We must use here again for z_{n} the special trigonometric form (11a) (both saddle-points at the same level) because in the case of the general exponential form of z_{n }there cannot appear roots in (20) whence we can extract dz_{n}/dr from (11c). Thus,
Since here the second term in the bracket exceeds, due to the magnitude of r = ka, the roots of z_{n}=0 (in contrast to the earlier ones of z_{n} = 0) are sufficiently exactly given by
Hence, similarly as in (21.40),
i.e., there exists indeed for m = 1, 2, ··· always one root n_{m }in the first quadrant of the n-plane.
Since the absolute values of these n_{m }are large numbers (due to ka > 1), we can compute in (17) from the asymptotic series , which with suppression of exponentially small components can be given the form
With the same accuracy, one has
sin np = e^{-i}^{np}/2i.
Hence, we can substitute in (17)
As far as the factor z /h in (17) is concerned, we want to specialize it immediately for the vicinity of Earth's surface, i.e., set there r = a. Then, by (11a) with sin z = (-1)^{m}, we have
and, by (15b) with restriction to the main term of (20a),
As in Appendix A52, we have Thus,
Finally, substituting from (22) and (23) into (17), we find
Here, the last factor a^{ -2} depends on the subscript m of the summation; in fact, as has already been expressed in (21), one has
In contrast, we can replace in the first factor under the sum sign of (24) v by the first, independent of m term of (21) n = ka, whence (24) is simplified into
We have indicated here all factors depending on which are not of interest to us by ··· . However, while in the initial sum (7), taken over n, as we have told, more than 1000 terms would have had to be taken into consideration, our present sum in m, due to the exponential dependence of , converges and, due to the increase of n_{m}, rises so rapidly, that one can already truncate it at the first or second term. This increase, due to the positive imaginary component of n, implies an exponential weakening of the wireless signal with a growing path along Earth's surface; the error suggests increasing intensity at the anti-podal point of the transmitter.
While the apparent becoming infinite of (22) for contradicts our general continuity demands, it is non-committal, because Equation (24.17) employed in (22) loses its validity for . A more accurate investigation of the location leads to a kind of Poisson refraction phenomenon with finite intensity (J. Gratiatos, Diss. Munich, Ann Physik 86 (1928).
We were able here to discuss briefly and forego all numerical details, because our present formulae for wireless telegraphy, due to the demanding role of the ionosphere, are unimportant. In contrast, they are of principal interest for the general mathematical treatment of Green's function in 5A.2 and display their importance by application to a special problem.
I should not omit to note that during a friendly visit, the English physicist Whipple in the Summer of 1945 has pointed out that Watson's results can be obtained directly without complex integrals. I suspect now that his physical considerations for this special case might be contained in the general method of 5A.2.