41 
•will result. Take the case of an elastic wire or rod, natural 
length l, 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 MY, and according to the usual way 
of looking at the subject of impact, the liability to rupture 
should be independent of l and proportional to MV. But 
in reality the mass MY 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 <x 3 =— (ju 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 l this wave is reflected at the fixed 
point, and returns to the point of attachment of the mass 
M, and the effects 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 l, and on M and V independently, not 
only on the product MV. 
First take the case of an infinite wire; let x be the 
unstretclied distance of any point from the initial position 
of the extremity which is fast to M, x -f £ the distance of 
the same point from this origin at time t. The equation of 
motion is 
cpz_ 
df~ a dx 1 
and we have the condition 
( 2 ) M S= E § whena: = a 
The general solution of (1) is %=f(at — x). 
