41 



will result. Take the case of an elastic wire or rod, natural 

 length I, modulus E, fixed at one end, the other end is 

 supposed to become suddenly attached to a mass M moving 

 with velocity V, which the tension of the wire brings to 

 rest. The wire is thus submitted to an impulsive tension 

 due to the momentum MV, and according to the usual way 

 of looking at the subject of impact, the liability to rupture 

 should be independent of I and proportional to MY. But 

 in reality the mass MV is pulled up gradually, not instanta- 

 neously, and the wire is not at once uniformly stretched 

 throughout, but a wave of extension or of tension is trans- 



E 



mitted along the wire with velocity a when a^=— (^ being 



the mass of a unit of length of the wire) ; in an infinite 

 wire this wave would be most intense in front, as in the 

 figure in which the ordinates are proportional to the tension. 

 In the wire of length I this wave is reflected at the fixed 

 point, and returns to the point of attachment of the mass 

 M, and the efi'ects of the direct and reflected waves must be 

 added, and again we must add the wave as reflected from 

 M back towards the fixed point. The question then of the 

 breaking of the wire is very complicated, and may depend 

 not merely on the strength of the material to resist rupture, 

 but also on a, E, and I, and on M and V independently, not 

 only on the product MV. 



First take the case of an infinite wire; let x be the 

 unstretched distance of any point from the initial position 

 of the extremity which is fast to M, oj 4- ? the distance of 

 the same point from this origin at time t The equation of 

 motion is 



and we have the condition 



The general solution of (1) is ^—fipi — a?). 



