ON WAVES. 345 



placed ; and this happens successively in every point of the fluid. The hori- 

 zontal range of motion is thus equal to the volume of water employed to form 

 the wave. 



While, therefore, the front of the wave is transmitted from A' to d, the 

 water particle A' is transferred to A. The same particle is also raised and de- 

 pressed through the height of the wave. These motions in the vertical and 

 horizontal plane are simultaneous. It is required to represent accurately 

 these motions : take c «f = the height of the wave, A A' = the range of trans- 

 lation : describe an ellipse whose major axis is the range of translation, and 

 whose semi-minor axis is the height of the wave : describe the wave circle 

 rf 1 , 2, 3, 4, c, and having divided as formerly its circumference into equal 

 parts, draw the horizontal ordinates 11, 22, 33, 44, &c., as in fig. 3, and let 

 the curve of versed sines A' C d be drawn as in fig. 3, then will the curve 

 A' 8, 7, 6, C 4, 3, 2, 1, d, represent the wave curve, the vertical motion only 

 being considered. But at the same time that the particle rises and falls through 

 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 on the diameter c d, and in the curve of versed 

 sines, the particle A' will advance to A, through A' 1, 2, 3, 4, 5, 6, 7, 8, 9, A. 

 Thus every point in the curve will have to be advanced forward in the direc- 

 tion of translation in order to represent the actual form of the wave. This is 

 done in fig. 4, and also for a larger wave in fig. 5. While the wave rises to 

 1, 2, 3, 4, C, &c., it also advances simultaneously at each point by the quan- 

 tity A' 1, A' 2, A' 3, A' 4, A' 5, A' 6, A' 7, &c., and thus the wave A' C d be- 

 comes transformed in both figures into K C d. This curve represents the 

 form of the wave as corrected for the horizontal translation. Thus are re- 

 conciled to each other the apparently diverse motions of the particle, by one 

 of which it describes the observed sinuous wave surface, and by the other 

 the semiellipse of its path of translation. 



Finally, as the motions of translation are equal and simultaneous through- 

 out all particles situated in the same vertical line, the path of translation of 

 each particle is an ellipse having the same major axis with that of the particle 

 on the surface, but having its minor axis less in proportion to its distance 

 from the surface of the liquid in repose. (See Plate XLVII. fig. 5.) 



Hence, when the wave is not large, the amplitude of the particle path or 

 range of translation is 3*1416 times the height of the wave ; this quantity 

 gradually diminishes as the height increases, and becomes nearly 2* when the 

 height approaches the limit of equality with the height of the wave. But 

 near this limit it is not capable of accurate observation. 



Mechanism of the Wave. — The study of the phsenomena of the translation 

 of water particles during the transit of a wave is peculiarly valuable, as 

 affording us the means of correctly conceiving the real nature of wave trans- 

 mission of the first order ; it therefore deserves great attention. 



We perceive, in the first place, that the vertical arrangement of the water 

 particles is not deranged by wave transmission ; that is, if we conceive the 

 whole fluid in repose to be intersected by transverse vertical planes, thin, and 

 of the specific gravity of water, these planes will retain their parallelism 

 during transmission and will not affect that transmission. 



We may therefore accurately conceive the whole volume of water as re- 

 posing in rectangular vessels, each of them formed between two successive 

 vertical thin moveable planes, and bounded by the two sides and bottom of 

 the channel, and above by the plane of repose. The water in each of these 

 elementary vessels undergoes in successive instances the same change as each 

 of the others preceding it, and therefore we may direct our attention to one 

 individual among them. 



