the Conservation of Force. 115 



Suppose, for instance, that we have moving in opposite 

 directions, so as to encounter each other, two condensed waves 

 of the same length, having each a single position of maximum 

 condensation — the condensation at the maximum, and at equal 

 distances from the maximum, being the same in both. Under 

 these circumstances — the particle-velocity throughout the one 

 wave being necessarily in the opposite direction to that 

 throughout the other* — as the waves meet and gradually 

 overlap, a constant destruction of velocity, and therefore of 

 vis viva, will take place in the overlapping portions, till, when 

 the position of complete occultation is attained (that is, when 

 the centre of one wave is coincident with the centre of the 

 other), the vis viva of the system of two waves will have 

 entirely disappeared. After passing the position of complete 

 occultation the velocity, and consequently the vis viva, will 

 revive, and will go on increasing till the waves are entirely 

 separated, when the vis viva of the system of two waves will 

 be exactly what it was at first. 



To ascertain the effect of the encounter of the two waves on 

 "the sum of the tensions," the equation of motion applicable 

 to this case being 



= D^ + ^, (1) 



dt 2 dx v ' 



we shall have for the equation of vis viva 



= V(dxf\\(dt£d^ d f; ... (2) 

 2 J dt\ J J dt dx v 7 



where x and y are the ordinates of a particle in the position of 

 rest and at the time t respectively, p the pressure at the time 

 t, and D the density of equilibrium. The last term of (2) 

 represents " the sum of the tensions ; " the integration with 

 respect to x being supposed to extend over the entire dis- 

 turbance existing at a given time, that with regard to t being 

 taken between any epochs we may fix upon. 



Before interference we shall have S dx -Q -^ (which call a) 



} dt dx v ; 



equal to zero, and the sum of the tensions will vanish. But 

 during the interference the value of a will vary with t; in 

 other words, a will be a function of t, and therefore " the 



* It should always be borne in mind that when waves are propagated 

 unalterably the particle- velocity in a wave of condensation will through- 

 out be in the direction of propagation; that in a rarefied wave will' be 

 throughout in a direction contrary to that of propagation, — conclusions 

 which can be readily verified, either with or without the aid of analysis. 

 The velocity at each point will also be always in the same fixed ratio to 

 the condensation, positive or negative, at that point. 



12 



