104 Mr. Sudhansukumar Banerji on Aerial 



We can now assume for y]r the following expression 



d f(ct—r) . 



* = -dr' r • C0S<9; .... (5) 



and we can easily determine the arbitrary function involved 

 in this expression by a method first given by Prof. Love * 

 so as to satisfy all the conditions enumerated above. 

 The method consists in assuming 



f(ct—r) = AeM ct - r+0 » and x = Be* ct , 



and then on substitution in the boundary conditions (1) and 

 (2), we notice that X satisfies a biquadratic equation, two of 

 whose roots are zero. The constants A's and B's are then 

 determined with the help of the remaining conditions. 



If we assume that the ratio of the mass of the air 

 displaced by the sphere to its own mass is a very small 

 quantity, we see that the expression for yjr can be written 

 in the simple form 



( ct+a-r \ 



, a d A\ a \/ 2 Ua 3 d r« " /ct + a-r 1 \T - 



+= A M«» e -*- brl — ~ °° S (~ a J »)J«»*. 



.... (6) 



where A is an indeterminate constant. 



The first term in this expression is a degenerated function 

 wnich does not satisfy the usual differential equation for 

 wave-propagation, and consequently does not represent a 

 wave-disturbance. This term arises from the subsequent 

 motion of the sphere with a nearly constant velocity which 

 involves only a local reciprocating motion of the neigh- 

 bouring air. 



The wave-motion produced is therefore given by the 

 expression 



/ct + a— r\ 



^/2LV dp" V ~""~ ' (ct + a-r 1 \-| ' .„ 



f = r~ &l -J-~ cos (— i*)J«»«- (7) 



Thus we see that the wave-motion generated by an 

 instantaneous change in velocity of a single sphere is of 

 the damped harmonic type which is practically confined to 

 a small region near the front of the advancing wave. 



* Love, "Some Illustrations of the Modes of Decav of Vibratory 

 Motions," Proc. Lond. Math. Soc. (2) vol. ii. p. 88 (1904). [See also 

 Lamb's ' Hydrodynamics/ Art. 295.] 



