ON WAVES. 357 



No great number of experiments has been made on channels of other forms 

 of variable depth, such as have been made coinciding witli those in the tri- 

 angular channel, so far as to shovi^ that we may take the simple arithmetical 

 mean depth as the depth of the rectangular channel of a wave of equal velo- 

 city, and so in general reckon the mean depth as 



h 





The form of transverse section does not therefore affect the velocity of the 

 wave otherwise than as it becomes necessary to use the mean depth as the 

 argument in calculating it, and not the maximum depth. 



The Form of Channel affects the Form of the Wave as loell as its Velocity 



When the channel is very broad the wave ceases to have a velocity, it 

 loses unity of character, and each part of it moves along the channel 

 independent of the velocity of the other, and Avith the velocity due to the 

 local depth of the channel. Where the water is shallow the wave becomes 

 sensibly higher and shorter, and when the difference of depth is not consi- 

 derable, the wave is found to increase in height so as to give in the shallow 

 part a velocity equal to that in the narrow part. When the channel is narrow 

 in proportion to its depth, this unity of propagation exists Avithout sensible 

 difference of velocity toAvards the side, and without very great difference in 

 height at the sides. In a channel of the form of a right-angled and isosceles 

 triangle, Avith the hypothenuse upwards and horizontal, it is visible to the 

 eye that the Avave is somewhat longer and loAver in the middle, but higher 

 and shorter at the sides, but that it retains most perfect unity of form and 

 velocity, and moves along unbroken with the velocity due to the mean depth. 

 The same figure with the angle at the bottom increased so that each side has 

 a slope of one in four, still contains a single wave propagated with a single 

 velocity, being that due to half the depth, but breaks at the shallow side, be- 

 coming disentegrated in form though not in velocity. 



In a channel 12 inches Avide, 5 inches deep on one side, and 1 inch deep 

 on the other, the following observations were made : — 



Height of the Wave. 



On the Incidence and Reflexion of the Wave of the First Order. — When 

 a Avave of the first order encounters a solid plane at right angles to the direc- 

 tion of its propagation, it is wholly reflected and is thrown back in the oppo- 

 site direction Avith a velocity equal to that in Avhich it Avas moving before 

 impact, remaining in every respect unchanged, excepting in direction of 



