398 
DR. W. S. TUCKER AND MR. E. T. PARIS ON 
twelve equidistant holes. This plate was rotated by an electric motor, the speed of 
which could be regulated by means of a rheostat in series with the armature. A stream 
of air, forced through a nozzle against the ring of holes, was supplied from a gas- 
compressor. The speed of the siren-plate was given by an EllioO. speed-indicator attached 
to the motor, the readings of this instrument being proportional to the frequency of 
the fundamental note of the siren. The end of the nozzle and the holes in the siren- 
plate were so shaped that the area through which air cculd escape from the nozzle was 
proportional to (1 — cospf), p/2?r being the number of holes passing the end of the nozzle 
per second.* It was found by experiment (using a Hot-Wire Microphone) that with 
this arrangement the note produced by the siren was remarkably free from harmonics. 
It will be assumed in what follows that, within the limited range of frequencies over 
which a resonance curve is plotted, the amplitude of the sound produced by the siren 
remained sensibly constant. 
On the understanding that the results may be subject to some revision on account 
of these assumptions, we can deduce the degree of damping of the Helmholtz resonator 
used to obtain the curve in fig. 5. 
If the equation of motion of the forced vibration in the neck of the resonator is 
written 
d 2 x f.-rr dx 2 x 
rr +211 ——l -n x — r cos pt, 
dt 2 dt J 1 
where — is the instantaneous current of air m the neck, and n 2 
dt 
forced vibration is 
V 2 c/S, then the 
x 
f 
{{n 2 — p 2 ) 2 + 4K 2 jo 2 } 
i cos (pt — 6), 
and the average energy of the vibration in the neck is proportional to the average 
dx \ a 
dt J 
r 
Vc 
alue of 
that is, to 
n 
5-2Y+4K- 
p n) 
It is found that the experimental curves can be fairly well represented by an 
expression of this type, provided that a suitable value is given to K. Thus, by choosing 
II = 38 -5, we obtain the dotted curve shown in fig. 5, which approximates closely 
enough to the experimental curve to show that this is about the proper value for the 
damping factor. In all cases so far examined the value of II required to fit the 
experimental curves has been found to lie between 20 and 40, and it has generally been 
found that II is less for the low-pitched resonators. For example, with a resonator 
whose natural pitch was 112 vibrations per second, the value of K was found to be 22 -2. 
* For the design of this siren-plate we are indebted to Messrs. R. H. Fowler and E. A. Milne, of 
Trinity College, Cambridge. 
