414 REPORTS ON THE STATE OF SCIENCE.^1919. 



The velocity c=s/(gh) is that acquired in falling freely under gravity 

 through i 7^, half the tension length, analogous to long flat tidal waves 

 in water of depth h. But in surface waves, rolling over deep water, 



c = ^~- =s/gl, where I is the equivalent conical pendulum length. 



Other physical results may be stated in a similar manner. 



Thus the velocity of longitudinal waves in an elastic cylindrical rod 

 is due to half the elastic length k, where the length ek hanging vertically 

 will produce extension e, provided the tension is under the elastic limit, 

 so that e is small. 



And the bursting velocity of the rim of a flywheel, treated as a 

 circular wire filament, is due to half the tensile breaking length of the 

 material. 



A comparison is made between the waves of vibration of a taut hawser 

 or piano wire, and the slack helical waves, or the plane waves of finite 

 amplitude seen in laying a cloth or shaking a blanket, or heaving the 

 lead, and firing a life-line rocket. 



7. Tie a knot on the chain, to represent a weight attached, and 

 investigate in this way the reflected and transmitted wave, and the 

 mechanical illustration of the Pupin coil on a telephone wire. 



Suppose the knot or weight is the equivalent of a length a of the 

 chain, of line density a, stretched to a tension P, and represent the 

 incident, transmitted, and reflected waves by 



(1) y—b sin [nt + mx], y^=bi sin (nt + mx — e^), ^2=^2 sin (ni — mg — cg). 

 The geometrical condition at the weight, x=0, is 



(2) y + y;^z=y^, b sin nt — b^ sin {nt — e^) + b2 sin {nt—e.2)=0, 



for all values of ^ ; so equating to zero the coefficients of sin nt and 

 cos nt, 



(3) b — bi cos ei + &2 cos €2=0, b sin €, — 62 sin £2=0- 



The dynamical equation of the weight at cc = is, in gravitation 

 units, 



(4) P ^-^ - P '^p- + P ^p - .a ^^' = 0, with ^' =i'?, 



ax ax ax gaP m^ <j 



leading to 



(5) 6 cos Mi— 6] cos {nt—ei) — b2Coa {nt—e2) + inabi sin (ni— €,) = 0, 



(6) b — bi cos ci — &2 cos €.,—mabi sin €, = 0, 



(7) —6, sin ci— &2 sin eg + maS, cos c, =0. 

 Thence from these equations 



(8) ^raa:=tan€, = — cot €2, e2 = ^7r + e,, 



bi = b COB ei, 62 = 6 sin €1. 



(9, y\=t) cos e, sin (wi+mw— €,). 



(10) y+y.2 = 2b sin mx cos nt + b cos ci (cos nt—mx + e^). 



