A SELECTIVE HOT-WIRE MICROPHONE. 
411 
and substitutingUsinytf for U, we find that the total resistance change can now be regarded 
as being made up of five parts, namely :— 
* 
m, = p' u a (i+i^u>), 
£Ro — a'U ( 1 +f U 2 ) sin pt, 
■a ) 
m 3 = -p'Wi + y u 2 j cos 2 P t, 
= —^-c'U 3 sin 3 pt, 
^R 5 = ^d'XJ* cos 4 pt. 
The interpretation of the various terms is obvious, and it remains only to estimate 
their relative importance. To do this, we can take as an example the grid examined 
in the experiment described above and use the numerical values of a', b ', d and d' 
already given. It will also be supposed that U = 2 -5 cms. per second, which, as 
previously shown (§ 5), would be the maximum velocity produced in the neck of the 
resonator if its natural frequency were 240 vibrations per second, and the amplitude in 
the primary wave were 200 times as great as the minimum amplitude audible. We 
find then that 
rlRj = — 0'23U 2 (l — 0'0072U 2 ) = — 1 ’ 37, 
SB,= + 1 ‘ 19U (l — 0*014U 2 ) = +1-09 sin pt, 
£R 3 = + 0'23U 2 (l—0‘0096U 2 ) cos 2pt = +1*35 cos 2 pt, 
dR 4 = + 0'0055U 3 sin 3 pt = + 0'086 sin 3 pt, 
£R 5 = + 0’00055U 4 cos Apt = +0'021 cos Apt. 
So that, even with a comparatively loud sound, the notes of pitch three and four 
times the fundamental are quite unimportant. 
One other point remains to be noted. From the expressions just given it can be seen 
that the simple rule, that oik is proportional to the intensity of the sound stimulating 
the resonator, does not hold for very loud sounds. Similarly, the amplitude of the 
fundamental oscillatory effect is not proportional to the amplitude in the primary wave 
when very intense sounds are used. In both cases the effect with very loud sounds 
falls short of what it would be if the simple relations continued to hold. 
§ 7. Experiments on the Measuremeyit of Sound. 
Two experiments will now be described which were undertaken with the object of 
testing the correctness of two of the conclusions arrived at in the previous section, 
viz. 
'i 
