December 2012 - URSI

Figure 1a. The closed contour of integration as implemented
by Sommerfeld in the lower complex plane.
Figure 1b. The actual location of the pole in the
lower complex plane.
of these are the necessary conditions for the convergence of
the integral at infinity fulfilled. According to Sommerfeld,
the path of integration can be resolved on this sheet into
three parts. The first part is a loop from infinity about the
branch point k 1the second part is a similar loop about
k 2 , and the third part can be any small circle about
the pole P
, as seen in Figure 1a. The contributions of
the loops about the branch points give the dominant terms
jk1R
jk
at e
2R
and e , which can be identified as the space
waves. R The residue R at the pole has a variation of the form
jPR
e , which had all the hallmarks of a true surface wave,
but Rnot a Zenneck wave. (We will discuss the properties of a
surface wave in the next section. However, suffice it to say
that it is a slow wave for a lossless dielectric medium and
there is practically no loss in the direction of propagation.
The fields in the transverse directions are evanescent, and
as the frequency increases, the wave is confined close to
the surface. However, it is not a Zenneck wave, as it is a
slow wave.) Here, R is the distance between the source and
the observation (field) point. The mathematical subtleties
that differentiate a surface wave from a Zenneck wave are
explained in the next section.
Sommerfeld thus obtained results that lent considerable
credence to Marconi’s view that the electromagnetic wave
was guided along the surface. As Collin noted in [3], as
early as 1902, Kennelly [19] and Heaviside [20] predicted
the existence of an ionized layer at a considerable height
above the surface of the Earth. It was thought that such
a layer could possibly reflect the electromagnetic waves
back to the Earth. This was experimentally verified by
Brett and Tuve in 1926 [21]. No serious challenge to the
surface-wave mechanism for long-distance propagation
thus occurred until a decade later, when Weyl published a
paper on the same subject. Weyl obtained a solution very
similar to that found by Sommerfeld [17], but without the
surface-wave term [23]. Weyl [23] obtained an asymptotic
series representing the diffracted field by applying a method
of steepest descent. Weyl’s solution also reduces to a form
that can be interpreted as the superposition of a space and
Figure 1c. The contribution of the pole generating
the surface wave is excluded when the branch-cut
contour is chosen vertically (Kahan and Eckart
[22], Baños [6, pp. 55]).
a surface wave, but the Weyl surface wave is not identical
to that of Sommerfeld [17] and Zenneck [15].
In 1926, Sommerfeld returned to the same problem.
That time, he solved it [22] using a different approach, and
confirmed the correctness of Weyl’s solution [23]. With a
better understanding of the ionospheric mode of propagation,
the concept of the surface wave being the important factor
for long-distance propagation lost favor. It is the presence
of the Kennelly-Heaviside layer of the ionosphere that
is responsible for the long-distance propagation of the
long-wavelength fields. In 1930, Van der Pol and Niessen
published a new solution to the old Sommerfeld problem,
using yet another method of solution [24]. Again, the
results of Weyl and the later results of Sommerfeld were
confirmed. This was followed by another paper of Van der
Pol on the same problem [25]. Each independent solution
of the old Sommerfeld problem agreed with Sommerfeld’s
1909 solution, except for the surface-wave term.
In the 1930s, Norton undertook the task of reducing
the formulas of Van der Pol and Niessen to practical form
for the radio engineer [26-28]. As a part of this undertaking,
8 The Radio Science Bulletin No 343 (December 2012)