412 REPORTS ON THE STATE OF SCIENCE. — 1919. 



The motion is visible in a taut cord or chain, as a rope, hawser, or 

 cable, and it is realised experimentally with ease in this way, for theo- 

 retical illustration, not requiring the complication of a tank of water. 



An elementary and exact treatment can be submitted, where the 

 motion is not restricted to be small, if the wave motion in the rope is 

 helical, and appears progressive, each particle whirling round the axis 

 in a circle with the same velocity, linear and angular. 



A stiff wire, wound into some convolutions of a uniform helix, is 

 useful in elementary explanation. Suspended vertically from a point in 

 its axis and revolved, the advance of the waves is .seen in the helix, as 

 well as in the shadow on a wall or the floor. 



Take a half length, ^L, measured axially, of a helical wave of the 

 equivalent flexible rope, wound on a cylinder of radius b, so that if a is 

 the angle of the helix with the axis, 27r6=L tan a. 



Suppose the rope weighs w, lb /ft, and is stretched to a tension or 

 pull P, lb, so that /z=-^ > called the tension length, is the length of rope 

 hanging vertically to produce tension P. 



Projected on a plane perpendicular to the axis, the half wave appears 



as a semicircle, of line density ~. , lb/ft ; and it is in equilibrium 



sm a 



under the transverse component tension P sin a at each end, and the 

 centrifugal force (C F.) of the rotation n, radians/ second, equivalent 



to an internal pressure - — . — =.__._, lb/ft, acting radially over 



sin a g sm a i 



the semicircle, putting g=zln-, so that I is the height of the equivalent 

 conical pendulum. 



The resultant C.F. thrust over the semicircle is the same as 

 the thrust over the diameter, 2b, due to the same pressure, and 



so is -. — — 2b ; so that for equilibrium 

 sin a g 



(1) 2Psina=^^^'— 2&, 



sm a g 



li sin' a= = — I -- 1 tan"' a, 



g g \27rJ 



(2) ^- cos^ a=lh COS^ a= 1 ^r- ) . 

 n^ 21? J 



Denoting by A. the length of rope for one radian angle of turn, 

 - =A COS o, the relation reduces to the simple form 



Ztt 



(3) Zfe=A2. 



