1913]
PROPAGATION OF ELECTRIC WAVES ALONG CONDUCTORS
139
It may be remarked by the way that the above analogy throws Light upon the question under what circumstances, electric waves are guided by conductors. Some high authorities, it would seem, regard such guidance as ensuing in all cases as a consequence of the boundary condition fixing the direction of the electric force. But in Acoustics, though a similar condition holds good, there is no guidance of aerial waves round convex surfaces, and it follows that there is none in the two-dimensional electric vibrations under consideration. Near the concave surface of walls there is in both cases a whispering gallery effect *. The peculiar guidance of electric waves by wires depends upon the conductor being encircled by the magnetic force. No such circulation, for example, could ensue from the incidence of .plane waves upon a wire Avhich lies entirely in the plane containing the direction of propagation and that of the magnetic force.
Our "first special application "is"fcb the extreme form of Hertz's problem (as modified) which occurs when all the radii of the cylindrical surfaces concerned become infinite, while the differences CA, AB remain finite and indeed small in comparison with A. In fig. 2, A, B, 0 then represent
Fig. 2.
planes perpendicular to' the plane of the paper and the problem is in two dimensions. The two halves, corresponding to plus and minus values of x, are isolated, and we need only consider one of them. Availing ourselves of the acoustical analogy, we may at once transfer the solution given (after Poisson) in Theory of Sound, § 264. If the incident wave in GA be represented by fGA and that therein reflected by F, while the waves propagated along CB, AB be denoted by/^,/^, we have
„/ „, 2(7.A /,/ L/A „, ,Q\
>f CA ..............(°)
CA
CB
and
.(9)
* Phil. Mag. 1910, Vol. xx. p. 1001; Scientific Papers, Vol. v. p. 617.fixed walls, b then denoti velocity-potential. When. 6 is known, the remaining functions follow once. . . •