34 



Prof. G. H. Darwin. 



[Nov. 22, 



I am disposed to regard this as the transitional state from the mode 

 of oscillation, which produces the octave with small height of ripple- 

 crest, to the fundamental with considerable height. 



In gentle oscillation over high ripple-mark the tree vortices are, 

 in the first instance, seen to be started by the mushroom vortices, and 

 the same is probably true of the condition we are considering. 



If the suggested view as to the mode of transition be correct, 

 then we must suppose that at first every alternate tree vortex is 

 started by its correlated mushroom vortex. If there be no tree vor- 

 tices, or if there be only every alternate one, the vortices can pack 

 twice as close as if the trees are symmetrical ; but the existence of a 

 half tree vortex tends to generate its other half, and this half cannot 

 exist normally unless e^ery alternate ripple-mark is removed. The 

 degradation of the alternate ripple-mark must arise then from the 

 existence of the second half of the ink tree. In these early stages the 

 phenomenon is not highly regular, and therefore, besides the smali- 

 ness of the scale and the rapidity of the motion, we have the difficulty 

 of irregularity to contend with. 



Other observers have endeavoured to determine the laws con- 

 necting the wave-length in the ultimate formation with the various 

 concomitant circumstances, and I shall leave this subject to the 

 following section, where some account of their work will be given. 



We may summarise the results of these observations as follows : — 



The formation of irregular ripple-marks or dunes by a current is 

 due to the vortex which exists on the lee of any superficial inequality 

 of the bottom ; the direct current carries the sand up the weather 

 slope and the vortex up the lee slope. Thus any existing inequalities 

 are increased, and the surface of sand becomes mottled over with 

 irregular dunes. The velocity of the water must be greater than one 

 limit and less than another, the limiting velocities being dependent on 

 the average size and density of the particles. Existing regular 

 ripple-mark is maintained by a current passing over it perpendicular 

 to the ridges. A slight change in form ensues, the weather slope 

 becoming less steep and the lee slope steeper. The ridges are also 

 slowly displaced to leeward. The regular ripple-mark may also thus 

 be somewhat prolonged, so that although a uniform current cannot, 

 as I believe, form regular ripple-mark, yet it may increase the area 

 over which it is to be found. 



Regular ripple-mark is formed by water which oscillates relatively 

 to the bottom. A pair of vortices, or in some cases four vortices, are 

 established in the water ; each set of vortices corresponds to a single 

 ripple- crest and the vortices oscillate about a mean position, changing 

 their shapes and intensities periodically, but not with a simple 

 harmonic motion. 



The successive changes in the vortex motion, whilst ripple-mark 



