226 DR. LOUIS VESSOT KING ON THE PROPAGATION OF SOUND IN THE FREE 
(48) holds at every instant of time during which A (t) varies, the rate of mass-flow 
through the ports of the siren is given by 
m = qpk. (t) .(49) 
If, now, we assume that the entire flow contributes to the velocity u in the 
sound-wave in the cylindrical resonator of area S in the neighbourhood of the siren 
ports {x — 0), the equation of continuity gives 
m — puS, .(50) 
so that from (48) (49) and (50) we obtain for the velocity in the wave the expression 
u = 
A (t) 
S 
2y P ( 
_(y— 1 ) Pi V 
/ 6 
: i-^- 
K 
(51) 
where we have written k = {pjp 0 ) y 1 and f = (p/p 0 )* <Y A Under the assumption just 
mentioned, writing U = 0 in (26), 
M =-{2«/(y-l)}(l-f) = -{2/(y-l)}( yft /ft)*(l-f). • • ■ (52) 
Identifying (51) and (52) we are enabled to solve for f from the quadratic 
+ = 0 ....... (53) 
where we have written 
*W = *(y-i)[A(0/S]>. 
The root of (53) which makes £ = 1 for k = 1 is 
(= .(54) 
Inserting this value of £ in equation (52) we obtain the initial velocity conditions, 
so that from the results of § 5 (ii.), we are enabled (theoretically) to trace the 
progress of the wave generated by any arrangement of siren ports for which the 
function A (t) may be expressed as a function of the time. 
It will be evident that in the problem under consideration any attempt to estimate 
the rate at which energy is transmitted along the resonator can have little meaning, 
as such an expression will include the energy of translation of the air in the resonator 
as a whole, in addition to what may be called the “ acoustic output.” The energy 
transmitted as sound at a distance will depend on the type of wave-motion which 
results after discontinuity has set in. The present theory, which takes no account of 
energy dissipation, is entirely inadequate to inform us on this subject and requires to 
be supplemented by data from suitable experiments which will be referred to in 
Part II. 
