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Full text of "Scientific Papers - Vi"

1919]                                   FROM A PERFORATED WALL                                       667
representing the two members of (27), regarding 7^ and k2 as abscissas and taking as ordinates
y = k1 sin 2k1}    y'  kg. sinh 2&.2......................(30)
If k-i and kz be both small,
so that at the origin both curves touch the line of abscissae and start with the same curvature. Subsequently y1 > y and increases with great rapidity. On the other hand, y vanishes whenever ^ is a multiple of |TT, although the successive loops increase in amplitude in virtue of the factor klf The solutions of (27) correspond, of course, to the equality of the ordinates y and y'. It is evident that there are no solutions when y is negative. The most important occur when &2 is small and 27^ just short of TT. But to the same small values of k2 correspond also values of 27^ which fall just short of STT, 5?r, etc., or which just exceed 27r, 4?r, etc. More approximately these are
,........................(32)
TOTT where TO = 1, 2, 3, etc.
In order to examine whether these solutions are really available, we must calculate S.    By (25)
k8 - kz  l -    k             +                 '   *} + fr tan ( +   CS m" 
3     / \ 2              ??i7r        /                \ 2              m-TT
If TO is odd, we have approximately
/^ = ^|(1+/^);  ........................... (33)
and if m is even,
Since 7c is approximately ^mir, we see that when m is odd, $ is large, and the condition of no reflexion can be satisfied, as when m = 1. On the other hand, when m is even, S is small, and here also tfee condition of no reflexion can be satisfied, at any rate at high angles of incidence.
It should be remarked that high values of TO, leading to high values of k, correspond with overtones of the resonating channels.
A glance at Fig. 1 shows that there is no limitation upon the values of the positive quantities &a and &2. And since Arx is always greater than &2, k, as derived from 7^ and kz, is always real and positive.
So far we have supposed that the values of 7ca, corresponding with small values of &a, are finite, as when m = 1, 2, 3, etc. But the figure shows thatof (0).