34 Mr. II. 11. M. Bosanquet on the Tlworij of Sound. 



that the fluid should behave as if incompressible (which I 

 assume ttiroughout as a consequence of the analysis), and that 

 the motion should at the instant of examination be the same 

 as if a steady flow were going on. This hap])ens when a de- 

 finite value is ascribed to t, the velocity anywhere being then 



V S 



— ^ sin kvt. This being the case, the equation of continuity 



is satisfied, nnd the rest of the reasoning runs as before. The 

 iotal enei'gy at any instant is found by multiplying the prece- 

 ding result by sin''^ kvt (coefficient of V"^), and is 



^irp^^^lmi^ kvt. 



The observations made on the former result are true also of 

 tliis ; none of it lies at an infinite distance. Where, then, does 

 it go to for the values of t for which sin kvt approximates to 

 zero? Tlie answer is, that this sort of motion can only exist 

 in combination with some other form capable of receiving the 

 (>nergy and converting it either into another form of kinetic, 

 or into potential energy, as it disappears from the motion we 

 are considering. The simplest case is that of a resonator. If 

 we imagine a large vessel full of air to be connected with the 

 outer air by a system of spherical divergence of this kind, the 

 total energy in use is exchanged between the moving air and 

 the store expressed by the compression or rarefaction in the 

 vessel. The analysis of the energy into two streams corre^ 

 spending to those employed in the case of constant flow has 

 been already exhibited. 



To turn to the analytical side of the question. 



It is not in general sufficient, for a form of motion to be a 

 solution of a problem, that it should satisfy the diiferential equa- 

 tion of the second order to Avhich the problem is subject. It 

 must also satisfy the energy-conditions, and also in any parti- 

 cular case the conditions imposed by the initial circumstances. 

 Examples of this are found in the limitations on tlie superpo- 

 sition of plane-wave systems trammelling in the same direction, 

 which are discussed in Props. III. et sajq. of the last note. 



Now, if we turn to our integrals of the differential equation 

 of synnnetrical spherical divergence, we might say, if we over- 

 looked the above considerations, that (i) alone was an integral 

 of the equation and a possible form of motion, and that the 

 form to which it reduces when kr is Aery small is also a pos- 

 sible form, i. e. that a possible motion is capable of being ex- 

 pressed by a velocity 



v>sin/(r/ — ?'). 



7*" 



