THE OBSERVATIONS. 



29 



ing, to the same scale, the acceleration of the 

 grain by gravity. Connect C and F; the line 

 CF represents in magnitude and direction the 

 resultant acceleration of the grain. These 

 relations are independent of the particular 

 directions of motion of the grain and the water. 

 Let us now introduce the assumptions, believed 

 to be practically true for the laboratory condi- 

 tions, that the water in the region of saltation 

 moves parallel to the bed and that its velocity 

 increases notably with distance from the bed. 

 In the ascending part of its path the grain en- 

 counters filaments of the current with higher 

 and higher velocity. This tends to increase 

 the relative velocity, but the grain is at the 

 same time gaining in horizontal velocity and 

 the gain tends to diminish the relative velocity. 

 Unless the leap is short in relation to the size 

 of the grain, the second of these tendencies is 

 the greater, and at the highest point of its 

 path the grain is moving nearly as fast as the 



FIGURE 8. Theoretic trajectory of a saltatory particle, the initial point 

 being at /. Arrows indicate acceleration. 



water. In the descending part of its path it en- 

 counters slower moving filaments of current, 

 and at some point (H, fig. 8) its horizontal mo- 

 tion may equal that of the adjacent water. 

 Then beyond H it passes through filaments 

 moving still more slowly, and its acceleration 

 from the reaction of the current becomes nega- 

 tive. The acceleration due to gravity is of 

 course uniform and downward, and its combina- 

 tion with that due to the current yields a system 

 of directions and magnitudes of the type indi- 

 cated in figure 8 by short arrows. In the 

 shorter and lower trajectories it is probable 

 that the critical point //is not reached. 



If the position of the grain before leaping 

 (fig. 6) is such that only a relatively short roll 

 suffices to free it, then its initial velocity is 

 small and the angle of ascent at which it is 

 freed is low. It has a short, flat trajectory, and 

 its velocity at the highest point is moderate. 

 If the original roll is longer there is time to 

 acquire speed before the leap; the initial ve- 

 locity is large and the angle of ascent is rela- 

 tively high. It has a long and high trajectory 

 and when at the crest has been accelerated to 

 high velocity. IF a grain at the end of a leap 



touches the bed at a favorable point it may leap 

 again without coming to rest, and the impetus 

 of the first flight will thus enhance the initial 

 velocity of the second. 



In the observations with the moving field 

 the grains seen most distinctly were those which 

 moved horizontally with the field and at the 

 same time had little vertical motion. So each 

 belt of distinctness contained grains at the tops 

 of their trajectories and was practically made 

 up of such grains. The grains producing the 

 curved lines in figure 3 were ascending or de- 

 scending obliquely, and their horizontal com- 

 ponents of motion coincided with the motion of 

 the field for an instant only. 



In general the observations seem to show 

 that the summit velocities of the leaping grains 

 increase systematically with the height of the 

 leap, and this generalization is in perfect accord 

 with the hypothesis that the paths of grains 

 are determined primarily by initial impulse. 



Under the hypothesis the series of velocities 

 observed by aid of the moving field are not 

 velocities of current, for the initial velocities of 

 grains, being caused by the current, require that 

 the water outspeed all the grains at the bottom 

 of the zone of saltation. At the top of the 

 zone there must be at least a slight advantage 

 with the current, provided the water velocities 

 increase upward. That the water velocities do 

 increase upward can hardly be doubted, for in 

 sweeping along the sand the stream expends 

 energy, and as its energy subsists in velocity, 

 the expenditure involves retardation. More- 

 over, the grains of sand are at the same time 

 most numerous and slowest near the bottom 

 of the zone, so that their effect is there greatest. 



In this connection it is to be observed that 

 the width of the belt of distinct vision in the 

 moving field (fig. 5) is greater for the upper 

 part of the zone of saltation than for the lower. 

 As distinct vision is limited to a certain (unde- 

 termined) range in horizontal velocities, this 

 fact implies that the increase in horizontal 

 speed of sand grains with distance from the 

 bed is less rapid in the upper part of the zone 

 than in the lower. 



The preceding discussion is subject to two 

 qualifications, the first of which is connected 

 with the retardation of the current at the side 

 of the trough. By reason of that retardation 

 the zone of saltation is shallower near the side 

 and does not include the longer and higher 



