The corresponding motion is expressed by the complex potential (<j> ential, ^r stream-function)
sin \ir(Zo — z rr W
(\ 0 sin (TT (z0 + z }/l}' ......... ' .......
y "h ": W . •
V 7 ...... ^ v
which f denotes the strength of a vortex, 2 = # + i'2/, z0 = $,l + ih. The ,xis is drawn midway between the two lines of vortices and the y-axis .ves the distance -between neighbouring vortices with 'Opposite rotation. ,rman gives a drawing of the stream-lines thus defined. The constant velocity of the processions is given by
is velocity is relative to the fluid at a distance.
The observers who have experimented upon water seem all to have used stacles not susceptible of vibration. For many years I have had it in my nd to repeat the aeolian harp effect with water*, but only recently have raght the matter to a test. The water was contained in a basin, about era. in diameter, which stood upon a sort of turn-table. The upper part, wever, was not properly a table, but was formed of two horizontal beams >ssing one another at right angles, so that the whole apparatus resembled ;her a turn-sfo'fe, with four spokes. It had been intended to drive from a tall water-engine, but ultimately it was found that all that was needed lid more conveniently be done by hand after a little practice. A metro-me beat approximate half seconds, and the spokes (which projected beyond e basin) were pushed gently by one or both hands until the rotation was iform with passage of one or- two spokes in correspondence with an assigned .mber of beats. It was necessary to allow several minutes in order to
* Prom an old note-book. "Bath, Jan. 1884. I find in the baths here that if the spread gers be drawn pretty quickly through the water (palm foremost was best), they are thrown into nsverse vibration and strike one another. This seems like zeolian string..,. The blade of a sh-brush about 1£ inch broad seemed to vibrate transversely in its own plane when moved •ough water broadways forward. It is pretty certain that with proper apparatus these vibration s ght be developed and observed."teady motion is possible in two arrangements (a) and (6), fig. 1, of which (a) is symmetrical. Karman shows that (a) is always unstable, whatever may be the ratio of h to I; and further that (6) is usually ( unstable also. The single exception occurs when cosh (rrh/l) = ^2, or h/l = 0'283. With this ratio of hjl, (b) is stable for every kind of displacement except one, for which there is neutrality. The only procession which can possess a practical permanence is thus defined.