70 Prof. E. H. Barton on Spherical 
This point is here illustrated graphically, the reason for the 
shifts which occur and their relative amounts being made 
clear in a single diagram. 
Fundamental Assumptions. — In attacking the problem of 
the spherical radiation of sound in air we simplify the analysis 
by making at the outset the following assumptions : — 
(1) That the action of gravity is negligible. 
(2) That the effect of viscosity is negligible. 
(3) That the motion is vibratory and devoid of rotation. 
(4) That the vibrations are small, so that writing the 
density 
the " condensation " s is to be regarded as a small 
quantity often negligible in comparison with unity. 
(5) That the velocities and accelerations of the air are 
small quantities whose squares and products are 
negligible. 
It is easily seen that the above assumptions simply narrow 
down the discussion to the case in question, and do not 
involve the loss of any generality we wish to retain. 
In estimating the acceleration of the air two methods are 
open to us : (a) We may follow in thought an individual 
particle and note how much its velocity is increased per 
second ; or : (b) We may fix attention on a spot in space and 
note how the speed changes of that particle (whichever it is) 
which is found there at the time in question. In other words, 
we may note the increase of speed of an individual in the 
procession, or the increase of speed of the procession as it 
passes a fixed point on the route. The relation between the 
two accelerations is given in hydrodynamical treatises. In 
our use of acceleration the first form should in strictness be 
taken, but, with the limitation (5). the distinction drops, as 
the difference is only of the second order of small quantities. 
Thus the second form, which is simpler, may always be used. 
We have now to derive the differential equation for aerial 
vibrations in space of three dimensions, solve it and simplify 
to the case of spherical radiation, then apply the solution to 
the various cases of conical pipes. 
The differential equation is based upon (i.) the so-called 
equation of continuity, and (ii.) the equations of motion. 
These we now take in the above order. 
. Equation of Continuity, — Consider an infinitesimal parallele- 
piped of edges dx, dy, and dz, and let the velocities of the 
air parallel to the axes of as, y, and z be denoted by u, v, and w 
respectively. 
