ON WAVES. 355 



to the mean depth, that is, with the same velocity as in a rectangular channel 

 of a depth equal to the mean depth of the channel ; or we might expect that 

 each portion of the wave would move with a velocity due to the depth of that 

 part of the channel immediately below each part of the wave, and so each 

 part passing forward with a velocity of its own, have a series of waves, each 

 propagating itself with an independent velocity, and speedily becoming dif- 

 fused, and so a continued propagation of a wave in such circumstances would 

 become impossible from disintegration ; and instead of a single large wave 

 we should have a great many little ones. Or, finally, we might have a perfect 

 wave moving with a velocity, the mean of the velocities which each of these 

 elementary waves might be supposed to possess. 



I soon found that the propagation of a single wave, i. e, one of which all 

 the parts should have a given common velocity, was possible in a channel 

 whose depth at different breadths is variable ; that the wave does not neces- 

 sarily become disintegrated ; that its parts do not move with the different 

 velocities due to the different depths of the different parts of the channel, but 

 that the entire wave does (with certain limits) move with such velocity as if 

 propagated in a channel of a rectangular form, but of a less depth than the 

 greatest depth of the channel of variable channel. 



It became necessary therefore to determine the depth of a rectangular 

 channel equivalent to the depth of a channel of variable transverse section ; 

 to determine, for example, in a channel of triangular section v, the depth of 

 rectangular channel in Avhich a wave would be propagated with equal velocity. 

 In this case the simple arithmetical mean depth of the channel is half of the 

 depth in the middle. But on the other hand, if we calculate the velocity due 

 to each point of variable depth, and take the mean of these velocities, we shall 

 find a mean velocity such as would be due to a wave in a rectangular channel 

 two-thirds of the greatest depth. 



In the first series of experiments I made on this subject, I conceived that the 

 results coincided sufficiently well with the latter supposition ; but they were on 

 so small a scale, that the errors of observation exceeded in amount the diffe- 

 rences between the quantities to be determined, and the results did not esta- 

 blish either. Mr, Kelland arrived at the opposite conclusion, his theoretical 

 investigations indicating the former result. I examined tlie matter afresh, and 

 after an extensive series of experiments, have established beyond all question 

 the fact, that the velocity in a triangular channel is that due by gravity to 

 one-fourth of the maximum depth. Although therefore the absolute velocity 

 assigned by Mr.Kelland's investigations deviates widely from the true velocity, 

 yet he has assigned the true relation between the velocities in the triangular 

 and the rectangular channel ; and if therefore we take the absolute velocity 

 which I have determined for the rectangular channel, and deduce from it the 

 relative velocity which Mr. Kelland has assigned to the triangular form, we 

 obtain a number which is the true velocity of the wave in a y'" channel. 



Table XV. 

 Observations on the Wave of the First Order in triangular Channels. 



The sides of the channels are planes, and slope at an angle with the ho- 

 rizon = 45°. 



Col. A is the observed depth of the channel in the middle, reckoned from 

 the crest of the wave. 



Col. B is the height of the wave taken as the mean between the observa- 

 tions at the beginning and end of the experiment. 



Col. C is the observed time in seconds occupied by the wave in describing 

 the distance in column D. 



2 a2 



