1883.] 



On the Formation of Ripple-mark. 



41 



length is a function of the height of the existing undulations ; that is 

 to say, not only of the amplitude of oscillation of the upper part of the 

 vortices, but also of their intensity. On the parts of the plate where 

 the sand is thick, a continual rearrangement of ripple-mark goes on ; 

 the wave-length extends by the excision of short patches of inter- 

 calated ripple-mark, and by general rearrangement. Finally the 

 sand reaches an ultimate condition as regards wave-length, although 

 rearrangement of ripple-mark still appears to go on for a long time. 

 Then we find in this final condition most of the sand arranged with a 

 certain fundamental wave-length, but where the sand is thin, patches 

 remain with the octave or half wave-length. 



It is not easy to understand precisely the mode in which the 

 oscillation of the water over the undulating bottom gives rise to 

 vortices, but there are familiar instances in which nearly the same 

 kind of fluid motion must occur. 



In the mode of boat propulsion called sculling, the sailor places an 

 oar with a flat blade through a rowlock in the stern of the boat, and, 

 keeping the handle high above the rowlock, waves the oar back- 

 wards and forwards with an alternate inclination of the blade in one 

 direction and the other. This action generates a stream of water 

 sternwards. The manner in which the blade meets the water is 

 closely similar to that in which the slopes of two ripple-marks 

 alternately meet the oscillating water ; the sternward current in one 

 case, and the upward current in the other are due to similar causes. 

 We may feel confident that in sculling, a pair of vortices are formed 

 with axes vertical, and that the dividing line between them is 

 sinuous. The motion of a fish's tail gives rise to a similar rearward 

 current in almost the same way. These instances may help us to 

 realise the formation of the ripple-making vortices. 



Lord Rayleigh has considered the problem involved in the oscilla- 

 tions of a layer of vortically moving fluid separating two uniform 

 streams.* At the meeting of the British Association at Swansea in 

 1880, Sir William Thomson read a paper discussing Lord Rayleigh's 

 problem. f He showed that, in a certain case in which the analytical 

 solution leads to an infinite value, there are waves in the continuous 

 streams in diametrically opposite phases, and that the vortical 

 stratum consists of a series of oval vortices. Fig. 13 illustrates this 

 mode of motion. The uniform current flowing over existing ripple- 

 mark exhibits almost a realisation of this mode of motion, one of the 

 streams of fluid being replaced by the sandy undulations. The same 

 kind of motion must exist in air when a gust of wind blows a shallow 

 puddle into standing ripples. 



* " On the Stability or Instability of certain Fluid Motions." Proc. Lond. 

 Math. Soc. (Feb. 12, 1880), vol. xi, p. 57. 



f Nature, Nov. 11, 1880, pp. 45-G, and see correction on p. 70, 



