340 REPORT — 1844. 



depending like that velocity only on the depth of the fluid and the height of 

 the wave crest. 



That this wave-form has its surface wholly raised above the level of repose 

 of the fluid. This is what I mean to express by calling this wave wholly po- 

 sitive. I apply the word negative to another kind of wave whose surface ex- 

 hibits a depression below the surface of repose. The wave-proper of the first 

 order is wholly positive. 



The simple elementary wave of the first order assumes a definite length 

 eqiial to about six times the depth of the fiuid beloiu the plane of repose. 

 When the height of the wave is small the length does not sensibly diff'er from 

 that of the circumference of a circle whose radius is the depth of the fluid ; 

 or h being the depth of the fluid in repose, the length of the Avave is repre- 

 sented by the quantity ^ith, v being the number 3'14159, we may use this 

 notation, 



A=2Tr/i E. 



The length, therefore, increases with the depth of the fluid directly, being 

 equal to about 6'28 times the depth. The length does not, like the velocity 

 of the wave, increase witli the height of the wave in a given depth of fluid. 

 On the contrary, the length appears to diminisli as the height of the wave is 

 increased, and the length of the wave when thus corrected is 



X=i2ich-a . . , . F. 



the value of a will be afterwards examined. 



The form of the wave surface when not large is a surface of single curva- 

 ture, the curvature being in the longitudinal and vertical planes alone, and 

 the curve is the curve of sines, or rather of versed sines, the horizontal ordi- 

 nates of which vary as the arc; and the vertical ordinates, as the versines of a 

 circle whose radius is the depth of the fluid in repose, 2irA being the length 



of the wave, and — an arc of that circle =6>. We have for the equation of 

 m 



the wave curve, x—lB 



y:=\k.vevs,m d . . . . G. 



the height of the wave being denoted by k, reckoned above the plane of re- 

 pose of the surface of the fluid. 



The height of the wave above the surface of the water in repose may in- 

 crease till it be equd to the depth of the fluid in repose. When it approaches 

 this height it becomes acuminate, finally cusped, and falls over breaking and 

 foaming with a white crest. The limits of the wave height are, therefore, 



k=0, and k=h . . . K. 



that is to say, the height of the wave may increase from to k, but can never 

 exceed a height above the level of repose equal to the depth of the fluid in 

 repose ; that is, the height total reckoned from the bottom is never greater than 

 twice the depth of the fluid in repose. 



The absolute Motions of each Water-Particle during Wave- Transmission. 

 — This is one of the subjects on which, prior to last Report, I had not made 

 a sufficient number of observations to enable me to make a full report. The 

 methods I had employed for such observations as I had then already made, 

 were the observation of the motions of small particles visible in the water of 

 the same, or nearly the same specific gravity with water, or small globules of 

 wax connected to veiy slender stems, so as to float at required depths. The 

 motions of these were observed from above, on a minutely divided surface on 

 the bottom of the channel, and from the side through glass windows, them- 



