The Net Advance of Physics: SPECIAL REPORTS, No. 1


by David Reiss

First Edition, 1996 June 15.
Copyright © 1996 by David Reiss: no unauthorized reproduction is permitted.


The analysis and use of surface waves (see note on terminology) for remote sensing (of the sea surface, targets, ...) have a long history dating back to the beginning of the century. In essence, though, any discussion of the surface wave phenomenon is a discussion of the mathematical problem of diffraction of electromagnetic radiation around a spherical earth. Furthermore, since diffraction effects are more pronounced as the frequency of the radiation gets smaller (i.e. the wavelength gets larger), this discussion will ultimately become centered on lower frequencies (the HF region,3->30 MHz, and lower). In this note we will give a thumbnail sketch of a few of the high points in the history of surface waves.


J. Zenneck [1], in 1907, was the first to analyze a solution of Maxwell's equations that had a "surface wave" property. This so-called Zenneck wave is simply a vertically polarized plane wave solution to Maxwell's equations in the presence of a planar boundary that separates free space from a half space with a finite conductivity. For large conductivity -- this depends on the frequency and dielectric constant, too -- such a wave has a Poynting vector that is approximately parallel to the planar boundary. The amplitude of this wave decays exponentially in the directions both parallel and perpendicular to the boundary (with differing decay constants).

It is worth emphasizing at the outset that the term "surface wave" is often a misnomer. We should attach no more significance to the phenomenon than the equations that describe it imply. The term surface wave conjures up an image of energy flow that is confined to a region that is localized at or near the surface. Whereas this is true for acoustic surface waves or certain classes of seismic waves, for example, it is generally not true for the phenomenon that we are discussing here. In our case, the boundary generally does not serve to localize the energy associated with the phenomenon; rather it just serves to guide the wave. Most of the energy of the wave is not near the surface. More accurately, for the case of vertical polarization, the presence of the conducting boundary allows the energy of the wave to extend down to the boundary in a significant manner (in contrast to the horizontally polarized case where the boundary condition mostly excludes the wave from the region near the surface). When we allow the boundary to be curved, as in the case of propagation around a sphere, the curvature of the surface leads to diffraction effects, yielding propagation of the wave beyond the geometrical horizon. The electromagnetic surface waves that we are discussing are no more magical than these phenomena. Their analysis is, however, somewhat complicated.


The plane Zenneck wave, of course, is only of technical interest since such a wave requires a source that is infinite in extent for its creation. The same is true if we consider such a wave with cylindrical symmetry: such a wave requires an infinite line source for its generation. (Much work has been done on the question of whether a pure Zenneck wave can be excited by an emitter with a finite aperture: see for example [3, 4].) What is tantalizing about such a cylindrical Zenneck wave is the fact that (aside from the exponential decay factors) its field is inversely proportional to the square root of r, rather than to r itself as would be true for usual free space propagation with a source of finite extent. Again, this is not unexpected; it is simply due to the approximate two-dimensional nature of the propagation problem.

However, another question arises when considering a point source, such as an oscillating vertical electric dipole, located over this conducting plane. Far from the source, does the field from such a vertical dipole behave like the cylindrical Zenneck wave? Certainly the answer to this question depends upon which direction one looks in; but, near the boundary, there exists the possibility that the field might be approximated by that of the cylindrical Zenneck wave. This problem was analyzed in detail by Sommerfeld [2] who first concluded that the field of this dipole does go over to that of the cylindrical Zenneck wave near the surface as the distance is increased. However, Sommerfeld made a famous sign error that he later corrected [5] and which was rediscovered later by others [6, 7]. The corrected conclusions were that, in an intermediate region and near the surface, the field is approximated by that of the cylindrical Zenneck wave; but then, as the distance increases further, its decay goes over to the 1/r dipole form (modified slightly by the presence of the conducting surface): the direct wave overcomes the Zenneck mode at large distances over a planar boundary. (These issues are discussed in great detail in the book by Baños [8].) Even though the wave does not have the Zenneck form in general, there is still the important contrast between the vertical and horizontal polarizations (the latter being the case for a horizontal electric dipole). In the case of vertical polarization the field extends down to the surface.

When we consider the problem of a vertical electric dipole radiator over a conducting sphere we realize that, beyond the geometrical horizon, the only part of the field that exists is that which arrives there via diffraction: there is no direct ray. (We are, of course, ignoring ionospheric reflections here and we are not considering ``anomalous propagation'' via atmospheric ducting which may, however, occur frequently in the ocean environment.) Furthermore, since diffraction becomes more efficient at routing the electromagnetic energy around the curve of the earth as the frequency is decreased, the HF frequency region turns out to be the frequency range of choice. In fact, it can be shown that the field in this region is just a sum of modes that are a solution to the problem of the cylindrical Zenneck wave over a conducting sphere (in contrast to the planar boundary case just mentioned). This problem of a vertical electric dipole radiator over a conducting sphere was originally analyzed by Sommerfeld, whose motivation was the analysis of the propagation of radio waves around the earth; it was not realized at that time that the main mode of such propagation for large distances is via ionospheric reflections. This work is reviewed in [9], along with a discussion of the planar boundary case. Sommerfeld's solution was reexpressed in terms of Airy functions by Fock [10], (also discussed in Wait's book [11]).

NOTE ON THE TERM "SURFACE WAVE": There is no agreement on the name for this subject. The term ``ground wave'' is also used, as are other terms. Furthermore, the meanings of these terms vary from one author to another. We also note that the phenomenon of surface waves is closely related to that of creeping waves and traveling waves in electromagnetic scattering theory.

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BIBLIOGRAPHY (includes items not cited in the text):
  1. J. Zenneck, Ann. der Physik, 23, 846--866 (1907).
  2. A. Sommerfeld, Ann. der Physik, 28, 665--736 (1909).
  3. D. A. Hill and J. R. Wait, ``Excitation of the Zenneck surface wave by a vertical aperture,'' Radio Sci. 13, 969--977 (1978).
  4. J. R. Wait, Wave Propagation Theory, second edition, New York: Pergamon Press (1981).
  5. A. Sommerfeld, ``The propagation of waves in wireless telegraphy,'' Ann. der Physik, 81, 1135--1153 (1926).
  6. K. A. Norton, ``Propagation of radio waves over a plane earth ,'' Nature, 135, 954--955 (1935).
  7. K. A. Norton, ``The propagation of radio waves over the surface of the earth and in the upper atmosphere,'' Proc. IRE, 24, 1367--1387 (1936) and 1203--1236 (1937).
  8. A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space, Oxford: Pergamon Press (1966).
  9. A. Sommerfeld, Partial Differential Equations in Physics, New York: Academic Press (1964).
  10. V. A. Fock, Electromagnetic Diffraction and Propagation Problems, Oxford: Pergamon Press (1965).
  11. J. R. Wait, Electromagnetic waves in Stratified Media, second edition, Oxford: Pergamon Press (1970).
  12. M. Born and E. Wolf, Principles of Optics, sixth (corrected) edition, Oxford: Pergamon Press (1987).
  13. J. D. Jackson, Classical Electrodynamics, second edition, New York: John Wiley & Sons (1975).
  14. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, edited by M. Abramowitz and I. A. Stegun, New York: John Wiley & Sons (1972).
  15. L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Oxford: Pergamon Press (1977).
  16. D. A. Hill and J. R. Wait, ``Ground wave attenuation function for a spherical earth with arbitrary surface impedance,'' Radio Sci. 15, 637--643 (1980).