1919] FROM A PERFORATED WALL 669
and in like manner (by changing the sign of S},
Moda Denominator = k-(8 + $0)2 11 + tan2 &i tann2 ^2} ;
and hence Mod2 B = ( ^— | ?V, .............................. (39)
where 8 = cos 6> (<r + <r')/<r ...................... (21)
If cr, the perforated area, is relatively great, it makes little difference what its actual value may be, but if cr is relatively small, as in the case of strong resonance, it is otherwise.
It would be preferable to suppose S fixed at S0 and to calculate the effect of a variation of k with h given. The resulting expressions are, however, rather complicated, and it is evident without calculation that the reflexion will be very sensitive to changes of wave-length when there is high resonance as a consequence of small dissipation and accurate tuning. The spectrum of the reflected light [in the corresponding optical circumstances] would then show a narrow black band.
{f some importance to consider whether when <r, or', and $, determining S, are given, the reflexion can always be annulled by a suitable choice of A?i and A;2. It appears that the answer is in the affinnative. Let. us consider the various loops of Fig. 1 which give" possible valutas of A-a. The, ranges for 2Aa are from 0 to TT, from 2?r to STT, from 4?r to .r>7r, and HO on. As we have seen, the intermediate ranges are excluded. In the first range, between 0 and TT we found that S may be made as great as we please by n sufficiently close approach to TT. At the other end where A.1! = 0, the value. of S was V3, or 1-7321. This is the smallest value which occurs. When 2&j = -|TT, it appears that 7c2= '5656, k = -5449, and S ~ 1776*. And again, when 2^ = ITT, &8='5797, (8=1-964. We conclude that within this range some value of 7^ with its accompanying kz can be found which Hhall annul the reflexion, provided S exceed 1-7321, but not otherwise.