June 1, 1800.] 



KNOWLEDGE 



137 



amount of water passes per second across a shallow section 

 (under the troughs) and across a deep section (under the 

 crests). 



If the stone be tugged up stream the length from crest 

 to crest is greater than if the stone be at rest, for the 

 velocity of the current relatively to the stone is increased. 

 As the speed of the stone up stream is increased the wave- 

 length continually increases, but the train of waves becomes 

 shorter, as in the ease of a canal boat, until, when the 

 velocity of the stone relatively to the current is equal to 

 the velocity of a free long wave in a channel of the given 

 depth, there is no longer a train of waves but a single 

 hilloclv accompanying the motion of the stone. When this 

 state of things has been reached the vertical oscillation of 



when the current rolls a stone down stream, we should 

 find the wave-length shorter than in the case of a stationary 

 stone, because the relative velocity of stone and current is 

 diminished. 



In all cases where a [iroup of waves is formed, one wave 

 making many, the energy travels slower than the wave ; 

 it falls behind the wave and makes another, so that the 

 group lengthens, the tail of the group moving more slowly 

 (relatively to the current) than the head, just as in the 

 track of ships {vi<te KxcvN-LEDaE, March and April, 1896). 



It wiU often be noticed that the first wave crest to lee- 

 ward of a stone in a shallow stream (the first wave being 

 that which has greatest amplitude) is a cusped or 

 breaking wave, the cusp being, of course, on the up-stream 



Fio. 1. — Rliossilby Bay. From a Photojrapli by Mr. F. II. Worsley-Beuison. 



the water particle is insignificant compared with the hori- 

 zontal component of the motion. For shorter waves from 

 stationary obstacles in ordinary shallow streams the motion 

 is presumably elliptical or approximately so, the ellipse 

 becoming more nearly a circle for shorter waves in deeper 

 water. If we were to push our experiment of dragging a 

 stone up stream beyond the critical velocity, the effect would 

 be that the stone would leave the " solitary" wave behind 

 it, the velocity of a free solitary wave being strictly limited 

 by the depth of the stream. In this way a number of 

 independent solitary waves might be produced on the lee 

 side of the stone. 



If, on the other hand, we wero to watch what takes place 



side. This happens when the downward rash of the 

 water on the lee side of the obstacle carries the water 

 surface far below the proper level of the stream, so that 

 under the trough the water is very shallow, and under the 

 crest relatively deep. This is a similar condition to that 

 of the breaker of the seashore : the wave velocity in the 

 crest is greater than that in the trough ; the crest therefore 

 gains upon the preceding trough and overhangs it with a 

 trembling cusp. But the cusped wave to leeward of a 

 stationary stone in a shallow stream docs not usually fall 

 bodily forward, as finally happens with the wave breaking 

 on the beach, for it remains in the same depth of water 

 , Occasionally, liowcver, when the current happens to be. 



