456 SURFACE WAVES. [CHAP. IX 



Indeed, either of these methods the latter especially gives very 

 beautiful wave-patterns*." 



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 = (2772 J) 7T, 



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



& 2 r'-&c 2 cos 2 + # = (1). 



If we put c ra = (VO* (2), 



and cosa = c m /c, a = (ra J) 7rc 2 /g (3), 



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



(.i a 



whence p/a = cos 2 6 (cos 4 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. 



