366 REPORT — 1844. 



pai'ticles of the stream pass through this area of the stream with a diminished 

 velocity, the area of transverse section must be increased at this point ; thus 

 the elevation of surface, enlargement of section, diminution of velocity above 

 the obstacle are its necessary consequences of that obstacle. Again, below 

 the obstacle the accumulation above generates an additional velocity due to 

 that height in addition to the mean motion of the stream ; the same volume 

 of water which passed through the large area, with its increased section and 

 diminished velocity, being now a higher velocity, is transferred through the 

 smaller area which allows its transmission. Thus the constant volume pass- 

 ing down the stream varies its velocity with the conservation of its forces 

 by means of a varying area of transference ; and thus we are enabled to con- 

 ceive how the observed form of the surface becomes at once possible and 

 necessary to the transmission of the fluid under the action of the disturbing 

 force. 



I am not aware that this species of standing wave in moving water has 

 ever before been made the subject of philosophical examination. But I con- 

 ceive that its study is highly important, especially in a theoretical view, as 

 the means of conveying sound elementary conceptions of wave motion, as 

 exhibiting the transition from the phaenomena of water currents to those of 

 water waves, as the intermediate link between motions of the first degree 

 and motions of the second degree, and as affording a basis from which we 

 may commence, with some prospect of success, the application of the known 

 principles and laws of motion to the investigation of the difficult theory of waves. 



M&viny Waves of the Second Order — Sea Waves. — It is not difficult to pass 

 from the conception of standing waves in running water to the conception of 

 running waves in standing water. Let us first conceive the waves in Plate LV. 

 to be formed in water running in the direction there indicated from right to 

 left, with a given mean motion, and a given motion of uniform circular oscil- 

 lation : and next let us conceive the whole water channel and waves to be 

 transferred uniformly in the opposite direction with a velocity equal to the 

 mean velocity of transference ; then the absolute motion of transference of 

 the water will become nothing : the waves formerly standing are now moved 

 in the opposite direction with a velocity equal to the former mean velocity 

 of the running stream, and the motion of oscillation remains. Thus, the 

 running water becoming still, the waves become moving waves, and if we 

 reverse the hypothesis once more, and conceive the waves which move with 

 a given velocity to exist in water which has a motion of transference with 

 equal velocity in the opposite direction, it is manifest that these waves run- 

 ning up the stream as fast as the waters run down, the wave-crests remain 

 fixed in place. Thus then the same oscillating phasnomenon which in stand- 

 ing water gives moving waves, will give in moving water standing waves ; 

 taking for granted always that the motions of oscillation are such as to be 

 possible, consistent with the nature of the fluid, and independent of the com- 

 mon mean motion of the fluid ; a condition equally essential to the possibility 

 of the wave motion and of our conceptions of it. 



I have been able accurately to observe the phenomena of wave motion in 

 still water, the waves being of the second order and gregarious, under the 

 following circumstances : — 



1. I have drawn a body through the water with a uniform motion, and 

 have observed the group of waves which follow in its wake. 



2. I have propagated the negative wave of the first order, and observed 

 the group of waves M-hich follow in its wake. 



I have not observed in the results of these two methods any distinction of 

 form, velocity, or other character. 



