60 JOURNAL OF THE WASHINGTON ACADEMY OF SCIENCES’ VOL. 13, No. 4 
Let two sounds have the same amplitude, but slightly different 
frequencies, a and b, a>b. Let them start together with maximum 
displacement, or at the top, P, of the phase circle, fig. 5. Let the 
phase position of the quicker, after a few intervals, and at the instant 
when the resultant amplitude is at a maximum, be at A (having 
described the angle 2n7+ a, n being a whole number) and that of the 
slower at B (having described the angle 2n7 — 8); let s be the velocity 
along the circumference of the phase circle of the slower, and s+és 
the correspnding velocity of the quicker. Then, since the combined 
amplitude is at a maximum, 
ssin 6 = (s + és) sina,and 8B >a. 
For the next maximum 
ssin (8 + 68) = (s + és) sin (a + 6a), da > 48, 
and so on for succeeding maxima. 
Furthermore, since the rate of increase of an angle is greater than 
that of its sine, the successive values of 68 will slowly increase so 
long as 6 and @ are greater than 0, and less than 7/2, and then increase 
while they are greater than 7/2 and less than z. In short, the com- 
bination tone will have a varying pitch intermediate between those of 
its constituents. And this is also true of any number of constituent 
sounds—whatever their initial and subsequent phases their resultant 
always has a quasi-average pitch. Hence, as is well known, the hum 
of a swarm of bees is pitched to that of the average bee, and the 
concert of a million mosquitoes is only the megaphoned whine of the 
type. 
This problem may of course be concisely treated analytically, but 
the above simple method is sufficient for the present purpose. 
The final law of sound essential to the explanation of the roar of 
the mountain, namely, the intensity of a blend in terms of that of its 
constituents, has been found by Lord Rayleigh* in substantially the 
following manner: 
Let the number of individual sounds be 7, all of unit amplitude and 
the same pitch, but arbitrary phase—conditions that approach the 
aeolian blend of a forest, or even a single tree. Clearly, if all the 
individual sounds had the same phase, and unit amplitude, at any 
given point, their combined intensity at that point would be n*. If, 
however, half had one phase, and half the exactly opposite phase, 
the intensity would be zero. Consider then the average intensity 
? Encyclopaedia Britannica, 9th ed., ““Wave Theory.’”’ Scientific Papers 3: 52. 
