October 31, 19 12] 



NATURE 



207 



The problem of the propagation of electric waves 

 over the earth's surface involves, therefore, three im- 

 portant factors which greatly influence the result, 

 first the imperfect conductivity and rather high di- 

 electric constant of the earth, making it a semi- 

 dielectric. Secondly, the effects of atmospheric ionisa- 

 tion, natural electrilicalion, and sunlight, and, thirdly, 

 the earth's curvature. The German mathematical 

 physicists have of late years considered the first of 

 these factors very carefully, and arrived at some in- 

 teresting results, which I will endeavour to epitomise. 

 Prof. A. Sommerfeld published in 1909 a very able 

 paper on the propagation of the waves in wireless 

 telegraphy over the earth's surface.' 



He supposes that a small Hertzian oscillator is 

 placed with axis vertical at the flat boundary surface 

 of two media, each having conductivity ir, dielectric 

 constant K. and permeabilitv /i, and that the bounding 

 surface is plane and indefinite. Taking E as the 

 electric force and H as magnetic force, we have at 

 any point in space the two circuital or Maxwellian 

 equations fulfilled, viz. : — 



KE +47r(rE = c curl H 



- \i\\ =c curl E. 



where £=3x10'°. The quantities E, a and K are 

 measured in electrostatic units and i^ and H in electro- 

 magnetic. Hence, if E and H are both simple har- 

 monic quantities of frequency n varying as the real 

 part of 6-JPt, where p = 2-irn, and if we write 



!^MK^'-+.;^4^ovi 



we have /■'-H =curl- H, 



A-'E = ciu'l- E. 



If we then take the magnetic force to be the curl of 

 a vector potential n and bear in mind that for vector 

 fields with no divergence the operator 



\S.r-' + fiy^8sV' 

 we see that n satisfies the differential equation 



A2n + /--n = o (1) 



and ih.-it H = ciirln, (2) 



4770- - JpK ^^' 



If r- = x' + y- + z-, then a particular solution of (i) is 



r 



To obtain a solution applicable to the case in ques- 

 tion, we have to satisfy the boundary conditions. 



These conditions are that the horizontal component 

 of the electric force and the vertical component of the 

 magnetic flux or induction must be continuous across 

 the boundary. Taking suffixes i and 2 for the air and 

 earth regions, these boundary conditions are 



n =n i^ 8rr,_^, m, 



Sommerfeld then shows that a solution of the 

 equation fi) is 



where r- = x- + y- + z- and x is an arbitrary parameter, 

 and Jo(^r) is a Bessel's function of zeroth degree. 



By a series of difficult transformations, the validity 

 of which must be tested by our pure mathematicians. 



■ '■ Ueber die Ausbreitung der Wellen in der drahllosen Telegraphic," 

 :al£n der Pliyiik^ vol. xxviil., p. 665, 1909. 



NO. 2244, VOL. go] 



Sommerfeld then proves that v can be expressed as- 

 the sum of three quantities P and Q such that 



n = P + Q, + Q„ 



where the quantities Q, and Q^ correspond to space 

 waves (Raumwellen), and P to a surface wave (Ober- 

 tlachenwellen). 



It is, of course, not new to suggest that the waves 

 involved in radio-telegraphy resemble electric waves on 

 wires or are surface waves. It was long ago surmised 

 that the sending antenna, the earth, and the receiving 

 antenna might be regarded as one single oscillator in 

 which oscillations were set up. This view has been 

 held, amongst others, by A. Blondel, E Lecher, and 

 F. G. Baily> 



Prof. Bailv pointed out in 1903 that the energy of 

 surface waves would decrease only inversely as the 

 distance, and therefore at large distances survive when, 

 space waves would have vanished. The strict mathe- 

 matical proof of their possibility has, however, only 

 been lately given. 



The space waves are subject to diffraction, and are 

 hindered by obstacles. On the other hand, the surface 

 waves pass round, and are unhindered, apart from 

 damping, by the curvature of the surface. Also, owing, 

 to the surface waves decreasing in amplitude less fast 

 with distance, the surface waves survive when the 

 space waves are extinguished. If, then, Sommer- 

 feld's investigation is valid, we need no longer seek 

 for an explanation of such achievements as the detec- 

 tion of electromagnetic waves one-quarter of the way 

 round the earth in any abnormal diffraction. If Som- 

 merfeld is right, diffraction has nothing to do with 

 the matter. The effect at such distances is entirely 

 due to these " Oberflachenwellen," or surface waves, 

 which, like electric waves or w-ires, are propagated 

 along the surface, no matter what the curvature may 

 be. There is a certain analogy between these space 

 and surface electric waves, and corresponding effects, 

 in the case of earthquakes. From the time of Poisson 

 it has been known that a shock communicated to an 

 elastic solid created in it two waves, one of dilatation- 

 and one of distorsion, travelling at different speeds 

 through the mass. 



In 1885, Lord Rayleigh showed that, in addition, 

 there was a surface wave dependent on the fact that 

 the surface can be distorted and resists distorsion with 

 a different elasticity to that of the interior of the mass. 



These effects are recognised by seismologists as re- 

 presented in the preliminary tremors and main shock 

 in an earthquake. 



In this case, two kinds of disturbance are found to 

 be propagated through the earth, with velocities of 

 10 and S km. per second respectively. Also another 

 main shock arrives later, which moves with a speed of 

 about 3 km. per second. The latter is a surface wave 

 travelling along the surface crust of the earth, and 

 the two former are space waves travelling through 

 the mass. 



In the same manner we can say that in wireless 

 telegraphv we are concerned with three waves — one 

 travelling' throun-h the air above the earth, the second 

 through the crust of the earth, and the third a surface- 

 or cylindrical wave, which is confined to a limited 

 region at the boundarv of the two dielectrics. 



It is suggested, however, that long-distance radio- 

 telegraphv is chieflv effected by means of the surface 

 waves or " Oberfl.achenwellen " of Sommerfeld, which 

 fall off in amplitude inversely as the square root of 

 the distance and are not limited bv diffraction as they 

 follow- round the surface. The earth's curvature. 



« See A. Blondel, Com/>tes Rcmlus du Cotigris dc Nantes, i8q8 : al^o 

 K. T echer, Physik. Zeitschr., vol. iii., n. 2-3. igoi. and Prof. F. G. Baily, 

 Trans. Royal Sco'tish Society of Arts, February 9, 1903. 



