ATMOSPHERE AND THE ACOUSTIC EFFICIENCY OF FOG-SIGNAL MACHINERY. 215 
a simple harmonic type, small amplitude and negligible atmospheric losses. If we 
suppose the waves to be confined in a conical surface of solid angle w, the velocity 
potential which determines the motion at a distance r from the source is 
0 = (A /wr) cos k (at—r ). (ll) 
If \ is the wave-length and n the frequency we have 
k = 27r/x and n = Ka/(2ir) . (12) 
The rate of introduction of air at the source is A cos k at, and it is easily proved* that 
the pressure amplitude is given by 
I $P I = Po I 30/3^ | = p^aAj(oor) . (13) 
If the source be situated close to a rigid plane, we have w — 2tt, and obtain 
| Sp | = p 0 nA/r and a | s | = nA/(ar) .(14) 
The total rate of emission of energy as sound is given by the relation 
[j cTW/dt J = p 0 K 2 a,A 2 /(±Tr) = 7rii 2 p 0 A 2 /ci . (15) 
We have here assumed that the propagation takes place in a homogeneous medium 
so that the intensity of the sound as measured by | Sp \ 2 falls. off inversely as the 
square of the distance, and the wave-surfaces are spheres expanding outwards with 
the velocity of sound. In practice this condition is very far from being realized ; the 
atmosphere, even on an apparently calm day, is the seat of innumerable discontinuities 
of density due to the presence of eddies and convection currents arising from unequal 
heating. The amplitude of sound from a fog-signal falls off with distance in a manner 
which varies very greatly from day to day, the manner of propagation depending on 
the state of wind and weather to a remarkable degree, as an inspection of the charts 
of the acoustic survey described in Appendix II. clearly indicates. Under these 
conditions one must imagine the wave-surface to be undergoing severe distortions as 
it is being propagated outwards, with the formation at times of so-called “ silent 
zones ” and zones of abnormal intensity. Several illustrations of these are to be seen 
in the charts just referred to. 
§ 4. On the Efficiency of Sound-producing Instruments. 
The changes of pressure and density which occur in a sound-wave near the limits 
of audibility are of extremely small magnitude. The question appears to have 
* Rayleigh, 1 Phil. Mag./ vol. 6, pp. 289-305, 1903 ; ‘Scientific Papers,’ vol. v., p. 126. 
