456 SURFACE WAVES. [CHAP. IX 



Indeed, either of these methods the latter especially gives very 

 beautiful wave-patterns*.&quot; 



The solution of the problem here stated is to be derived from 

 the results of the last Art. in the manner explained in Art. 228. 



For a line of pressure making an angle JTT with the 

 direction of the stream, the distances (p) of the successive wave- 

 ridges from the origin are given by 



kp = (2m - %) TT, 

 where m is an integer, and the values of k are determined by 



l&T -&c 2 cos 2 + # = (1). 



If we put c m = (40r)* (2), 



and cosa = c m /c, a = (m - J) 7rc 2 /# (3), 



this gives _- 2^cos*0+ cos 4 a = (4), 



whence p/a = cos 2 9 (cos 4 6 cos 4 a)* (5). 



The greater of these two values of p corresponds to the down 

 stream and the smaller to the up-stream side of the seat of 

 disturbance. 



The general form of the wave-ridges due to a pressure-point at 

 the origin is then given, on Huyghens principle, by (5), considered 

 as a tangential-polar equation between p and 6. The four lines 

 for which 6 = a. are asymptotes. The values of ^TT a for several 

 values of c/c m have been tabulated in Art. 246. 



The figure opposite shews the wave-system thus obtained, 

 in the particular case where the ratio of the wave-lengths in the 

 line of symmetry is 4 : 1. This corresponds to a= 26 34 *h 



In the outlying parts of the wave-pattern, where the ridges 

 are nearly straight, the wave-lengths of the two systems are 

 nearly equal, and we have then the abnormal amplitude indicated 

 by equation (16) of the preceding Art. 



When the ratio c/c m is at all considerable, a is nearly equal to |TT, and 

 the asymptotes make a very acute angle with the axis. The wave-envelope 



* Lord Eayleigh, I. c. 



t The figure may be compared with the drawing, from observation, given by 

 Scott Russell, I c. 



