326 REPORT— 1844. 



coincides with the velocity assigned in Table II. when the height of the wave 

 is inconsiderable. 



1 have found that this deviation is to be reconciled, without at all destroying 

 the simplicity of the formula, by a very simple means. In order to obtain 

 perfect accuracy, we have only to reckon the effective depth for calculation, 

 from the ridge or crest of the wave instead of from the level of the water at 

 rest ; and having thus added to the depth of the water in repose, the height 

 of the wave crest above the plane of repose, if we take the velocity which a 

 heavy body would acquire in falling through a space equal to half the depth 

 of the fluid (reckoning from the ridge of the wave to the bottom of the chan- 

 nel), that number accurately represents the velocity of transmission of the 

 •wave of the first order. 



We have, therefore, for the velocity of the wave of the first order, 



approximately «;= ^ gh, A. 



accurately i;=V'^(A+^), B. 



where v is the velocity of transmission, 



g is the force of gravity as measured by the velocity which it will com- 

 municate in a second to a body falling freely =32, 



h is the depth of the fluid in repose, 



h is the height of the crest of the wave above the plane of repose. 



The velocities of waves of the first order in channels of different depths are, 

 therefore, as the square roots of the depth of these channels. 



Nevertheless, when the height of one of the waves is considerable compared 

 with the depth of the channel, a high wave in the shallower channel may move 

 faster than a lower wave in a deeper channel ; provided only the excess in 

 height of the higher wave be greater than the difference of depth of the 

 channels ; in short, that wave will move fastest in a given channel whose 

 crest is highest above the bottom of the channel, and in channels of different 

 depths waves may be propagated with equal velocities, provided only the sum 

 of the height of wave and standing depth of channel amount to the same 

 quantity. 



Table II. 



History of a Solitary Wave of the First Order, from observcUion. 



Depth of fluid in repose in the channel 5'1 inches. 



Breadth of the channel 12 inches ; the form rectangular. 



Volume of generating column 445 cubic inches. 



Column A is the observed height of the crest of the wave in inches above 

 the bottom of the channel. 



Column B is the observed height of the crest of the wave in inches above 

 the surface of the water in repose. 



Column C is the time in seconds occupied in traversing the distances in 

 column D. 



Column D is the spaces traversed by the wave in feet previous to each 

 observation of time. 



Column E is the velocity of the wave through each length of 40 feet de- 

 duced from observation. 



Column F is the velocity deduced from the formula Vg{h+k)=v. 



