Waves generated by Impact. 103 



changes of velocity U during the short interval of time 

 known as the duration of impact, in consequence of which 

 an impulsive pressure is communicated to the surrounding 

 medium and a train o£ sound-waves is started travelling 

 forward with a definite wave-front. As a simplification we 

 shall assume that the change in velocity is instantaneously 

 acquired by the balls. As a matter of fact, if we examine 

 the curve for the relative velocities of the centres of mass 

 for the period the balls are in actual contact, we notice that 

 the most rapid changes in velocity occur only at the epoch 

 of greatest compression, and as the duration of impact itself 

 is a very small quantity (usually less than the 2000th part of 

 a second), we see that the effect of the duration of the impact 

 on the sound-waves is generally not of very great importance. 

 At any rate, we are not wide of the mark in taking the 

 change in velocity as practically instantaneous. 



(1) The case of a single sphere. 



We shall first consider the case when a single sphere 

 suffers an instantaneous change in velocity U. 



If M be the mass of the sphere, its equation of motion can 

 be written in the form 



M'|f = -§p cos 0. a* dco, . . . . (1) 



where a is the radius of the sphere, p is the pressure at a 

 point on the surface of the sphere, and dco an elementary 

 solid angle. 



Also if i/r denote the velocity potential of the wave-motion 

 started, the condition of continuity of normal motion on the 

 surface of the sphere gives 



— jl _ . ^cos#, when r = a. ... (2) 



or at 



The initial circumstances at time £ = give 



w^0 and ^=U (3) 



Further, the condition of discontinuity at the spherical 

 boundary of the advancing wave gives 



to be satisfied for r = ct-\-a, c being the velocity of sound. 



