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Full text of "Scientific Papers - Vi"

In Hertz's problem, fig. 2, the method is similar. In the region of regular waves on the left in CA we may retain (13), (14), and for the regular waves on the right in CB we retain (15), (16). But now in addition for the regular waves on the left in AB, we have
............... (20)
............. (21)
Three conditions are now required to determine Kl} Hz, K3 in terms of We shall denote the radii tak-en in order, viz. \BB, $AA, %CC, by ^2. fa respectively.    As in (17), the electric forces give
1og    + ^3log    = tfBlog    ................ (22)
The magnetic forces yield two equations, which may be regarded as expressing that the currents are the same on the two sides along SB, and that, since the section is at a negligible distance from the insulated end, there is no current in A A. Thus
H.-K^-K^H* ........................... (23)
From (22) and (23)
^ - % rt - log n
tfriogr.-logV ........................... (    }
H --K _l<>Sr*-logr9                                 (25}
•LJ-Z — — Jl-3 — T - i ------ ............................ V^"/
log ra - log n
If rz exceeds rx but little, K^ tends to vanish, while Hz and — K9 approach unity. Again, if the radii are all great, (24), (25) reduce to
1 — rg ~ TI         "ff —.     TT — TS ~ TZ
~     —                                   •*jta -- •"•% --
as already found in (8), (9).
The same method applies with but little variation to the more general problem where waves between one wire and sheath (rl5 r/) divide so as to pass along several wires and sheaths (r&, r2'), (rs, r/), etc., always under the condition that the whole region of irregularity is negligible in comparison with the wave-length*. The various wires and sheaths are, of course, supposed to be continuous. With a similar notation the direct and reflected waves along the first wire are denoted by IIl, Kls and those propagated
* This condition will usually suffice. But extreme cases may be proposed where, in spite of the smallness of the intermediate region, its shape is such as to entail natural resonances of frequency agreeing with that of the principal waves. The method would then fail.