668 ON RESONANT BEFLEXION OF SOUND " [440
solutions of (27) may exist when kl} as well as L, is small. In this case we obtain from (31)
making 7c2 = kf — k* — $/ca4............................w''/
Hence by (25)
and ^=V3.(l+fL2)............................(««)
Here again the condition of no reflexion can be satisfied, whatever the angle (0) of incidence, by a suitable choice of tr'/er. But the damping is no longer small, in spite of the smallness of A'a, since A-a is not now small -in comparison with A?! and k. On the contrary, /^ and k, are nearly equal, and tt is small in comparison with 7c2, so that this case stands apart.
Not only is it always possible to find a series of values of A*, satisfying (27) with any assumed value of kz, but the values so obtained make Ay positive. For in (25) k1} k2, tanhA;2 are positive, and so also is ban 7^, since
tan k-i — sin 2/^/2 cos2 7^, and sin 2/ca is positive.
It is a question of 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.
In each of the other admissible ranges, S takes all positive values from 0 to QO . At the beginning of a range when 27c: slightly exceeds 2?r, 4?r, etc., ;S' starts from 0, as appears from (34); and at the end of a range, as BTT, fa, el;c. are approached, 8 is very great (33). Within each of these ranges it is possible to annul the reflexion by a suitable choice of klt kz, whatever a-, a', and 6 may be.
If the actual value of 8 differs from that calculated, the reflexion is finite, and we may ask what it then becomes. If we denote the value of 8, as calculated from fa, k2, by $0, (24) gives
Mod2 Numerator = k2 (S - Sn}* (1 + tan2 7q tanh2 k,}, * [This result (1-776) is a correction of the value (1-947) given in the original. W. F. S.]is probably for the most part legitimate. In imagination we may suppose the film to be infinitely thin or, if it be of finite thickness, that the diffusion takes place strictly in one dimension.