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Full text of "Scientific Papers - Vi"

1915]              RESONATORS  EXPOSED TO  PRIMARY PLANE  WAVES                 283
R being the distance of any other resonator from the first, while (as before)
dr
We have now to distinguish two cases. In the first, which is the more important, the tuning of the resonators is such that each singly would respond as much as possible to the primary waves. The ratio of (16) to (17) must then, as 'we have seen, be equal to — ?\, when rą is indefinitely diminished. Accordingly
which, of course, includes (10).    If we write a= Aeia, then
•             jr    7 >   1 n  '       *'*
^ sin kR -
y"T1           T~     J. ~r ** ..... i~Tri
JcR  J      L            kR  }
The other case arises when the resonators are so timed that the aggregate responds as much as possible to the primary waves. We may then proceed as in the investigation for a single resonator. In order that no work may be done at the disturbing centres, (<Ģ + ^) and d(<p + ty)/dr must be in the same phase, and this requires that
11                  g-ifcR
_ _) --- ih + 'S, — =5- = real, a    TI                R
1         .     .,     ' ..., smkR                             ,__N
or                                   - = real + ifc + iS — p— ......................... (20)
CL                                                             Jiii
The condition of maximum resonance is that the real part in (20) shall vanish, so that
kR
or                                  A= - l/k  ... ............................... (22)
i j_ V sm /g^
1 + ^"~M"
The present value of A2 is greater than that in (19), as was of course to be expected. In either, case the disturbance is given by (15) with the value of a determined by (18), or (21).
The simplest example is when there are only two resonators and the sign of summation may be omitted in (18). In order to reckon the energy dispersed, we may proceed by either of two methods. In the first we consider the value of ty and its modulus at a great distance r from the resonators. It is evident that \/r is symmetrical with respect to the line R joining the resonators, and if 6 be the angle between r and R, r^ — rz = R cos 0. Thus
r2 . Mod2 ijr = A2 {2 + 2 cos (kR cos 0)} ;nces of the point where \Jr is measured from 1 various resonators, and a is a coefficient to be determined. The wh potential is <Ģ + ^, and it suffices to consider the state of things at the fi resonator. With sufficient approximation