232 Reflexion of Sound from a Perforated Wall. 



Hence by (24) 



JfeS = k x tanh & 2 + k 2 tan ^ = 2£ 2 2 (1 + p 2 2 ), . (37) 

 and S=v/3.(l + p 2 2 ) (38) 



Here again the condition of no reflexion can be satisfied, 

 whatever the angle (6) of incidence, by a suitable choice of 

 (7 l /(T. But the damping is no longer small, in spite of the 

 smallness of k 2 , since k 2 is not now small in comparison with 

 k } and k. On the contrary, k x and k 2 are nearly equal, and B 

 is small in comparison with k 2 , so that this case stands apart. 



Not only is it always possible to find a series of values 

 of ki satisfying (27) with any assumed value of k 2 , but the 

 values so obtained make S positive. For in (25) k^ k 2 , tanh k 2 

 are positive, and so also is tan An, since 



tan &!= sin 2 kj 2 cos 2 k 1} 



and sin 2k l is positive. 



It is a question of some importance to consider whether 

 when a, o\ and 0, determining S, are given, the reflexion 

 can always be annulled by a suitable choice of k Y and k 2 . 

 It appears that the answer is in the affirmative. Let us 

 consider the various loops of fig. 1 which give possible 

 values of k 2 . The ranges for 2k\ are from to tt, from 27r 

 to 37r, from 4-7T to 57r, and so on. As we have seen, the 

 intermediate ranges are excluded. In the first range between 

 and 7r we found that S may be made as great as we please 

 by a sufficiently close approach to it. At the other end 

 where #i = 0, the value of S was \/%, or 1'7320. This is the 

 smallest value which occurs. When 2# 1 = ^7r, it appears 

 that k 2 — '5656, £ = '5449, and S = 1-947. And again, when 

 2A; 1 = f7r, # 2 = *5795, S = 1'964. We conclude that within 

 this range some value of k\ with its accompanying k 2 can be 

 found which shall annul the reflexion, provided S exceed 

 1*7320, but not otherwise. 



In each of the other admissible ranges, S takes all positive 

 values from to go . At the beginning of a range when 2& a 

 slightly exceeds 27r, 47t, &c, S starts from 0, as appears 

 from (34) ; and at the end of a range, as 37r, 57r, &c. are 

 approached, S is very great (33). Within each of these 

 ranges it is possible to annul the reflexion by a suitable 

 choice of k u k 2 , whatever cr, o\ and 6 may be. 



If the actual value of S differs from that calculated, the 

 reflexion is finite, and we may ask what it then becomes. 

 If we denote the value of S, as calculated from £ l5 k 2 , by S , 

 (24) gives 



Mod. 2 Numerator = P(S-S ) 2 {l + tan 2 h x tanh 2 k 2 ], 



