ON WAVE MOTION. 413 



In the period of revolution, T=_ , the wave advances L, and so with 



n 



velocity c=-,= — =nl, thus 



(4) c'^=gh cos^o, reducing to c^=gh, 



when the rope is taut, and a=0, as in the usual elementary theory. 



If the stiff helical wire is passed through a corkscrew hole, like a 

 screw through a fixed nut, the helix advances as it turns, and passes 

 through its own shape, so as to appear stationary as a fixed curve. 



In the flexible rope of the same shape, the velocity of advance is c, 

 and the velocity of the helix through the nut is v—c sec a, so that the 

 relation above becomes v^=gh. This is in accordance with the general 

 result for any flexible rope or chain, of any invariable shape through 

 which it is passing with tangential velocity v=\/(gf/!.) ; seen sometimes 

 in heaving the lead, or in the helical curve of the life-line drawn out 

 by the life-saving rocket of the coastguard. 



With the axis of the helix horizontal and passing perpendicularly 

 through a nut in the wall, or else with the helix revolving about the axis, 

 the projection shadow thrown on the wall by a horizontal ray aslant is a 

 trochoid, fixed or moving, visual realisation of Rankine's trochoidal wave, 

 stationary or advancing. 



The shadow thrown by rays at right angles to the axis will be a 

 sinusoid ; but this cannot represent a wave motion of the rope unless the 

 curve is very flat. 



6. To give an elementary demonstration of the condition of a stationary 

 wave, revolving bodily, as on a musical cord fixed at the ends, the method 

 must be restricted to the taut chain, of length a=|L, displaced into a 

 flat sine curve, taken as given by 



(1) y=b sinmx, mL = 27r, 



with a slope mb at each end ; and the 2'Pmb balances the C . F. 



The average value of y in the half wave is ^, so that the C . F. is 



^■^ 



(2) w'''iL^^=2Vmb, '""'-p™ 



g i^ gm-^'"'' 



(3) gh=g = =_ = c-, 



w m^ T* 



as before ; and the number of revolutions, cycles, or double vibrations 

 per second, is 



^ ' T 27r '^ L2 V 4^2* 



For a revolving slack chain, such as a skipping rope, the curve would 

 be given by the elliptic function 



(5) g=bsnmx. 



To realise a plane sine curve of finite amplitude, the rope would 

 require to be of variable linear density, such that the axial distribution 

 of density was uniform, 



