Theory of Short and Long Waves 1 9 



Table 1 and Fig. 9 show, according to Airy, for different values of /i/A listed in the first column, 

 the ratio of the horizontal motion at the bottom to that at the surface listed in the second column 

 and shown in curve one; the third column and curve 2 gives the ratio of the vertical to the horizontal 



0-8 



0-6 



0-4 



02 



3 / 2 A V 



0? 



0-4 



0-6 



0-8 



1-0 



Long 

 wave ^ 



Surface -or short waves 



FiG. 9. Relation between orbital motion and velocity of propagation and the ratio h\X. 



vertical 



bottom 

 (1) sech /;: horizontal motion , (2) orbit: 



(3) 



surface 

 velocity of waves 



,(4) 



horizontal 

 velocity of waves 



diameter at the surface, 



velocity of short waves velocity of long waves 



axis of the elliptic orbit of a surface-particle; the fourth and fifth columns (curves 3 and 4) give 

 the ratios of the wave velocity to that of waves of the same length on water of infinite depth, and 

 to that of long waves on water of the actual depth respectively. This presentation shows clearly that 

 deep water waves start with h = JA, whereas the long waves extend up to a maximum where h = 

 = OT A; the interval between those two belongs to a transition zone with more complicated 

 conditions. 



Lord Rayleigh (1876) has given another elegant derivation of the wave theory when the 

 disturbance is small compared with the wave length. If waves travel in a certain direction with 

 a velocity c and if we give the water-mass a velocity equal but opposite to the direction of propaga- 

 tion, the motion becomes steady, while the forces acting on the individual particles remain the 

 same. Prandtl (1942) uses this method in a brief derivation of the velocity of propagation of surface 

 or short waves. A "reference system" travelling with the waves has a horizontal velocity in the 

 wave crest u 1 = c— InrjT, r being the radius of the orbit and T the time of a revolution (period of 

 the waves). Then 2nr\T is the velocity on the circular orbit. The horizontal velocity in the wave 



2* 



