RADIATION OF HEAT ON THE PROPAGATION OF SOUND.
experiments by which M. Biot proved that the velocity of propagation of sound in air is independent of the pitch.
Let the origin be situated at the vibrating plane, and let us consider the motion of the fluid situated at the side of x positive. Let m be what m' becomes when ri is replaced by J — In. The equation (9) furnishes two values of m, corresponding to two series of waves, which travel, one in the positive, and the other in the negative direction. Of course we are only concerned with the former. We get from (9)
m* = --fjf(coa2Tlr-J^I sin 2^)............(10),
where
.(11)
r = tan"^J -—
afinq
......(12).
Choosing that root of m^2 which corresponds to waves travelling in the positive direction, we get from (10)
cos i/r — J- 1 sin i|r).
Substituting in (8), introducing another function got by changing the sign of */— 1 and taking a new arbitrary constant, changing the arbitrary constants so as to get rid of the imaginary quantities, and altering the origin of the time so as to get rid of one of the circular functions, we get
_s _= ^-Min*. * _Cos (nt - /A COS i/r . #)............(13).
It will be easily seen that the expressions for 0, u, and p are of the same form, that is to say, that they involve the same exponential multiplied by a sine or cosine of the same angle. Had the actual expressions been required, it would have been shorter to defer the substitution of real for imaginary quantities until after the imaginary expressions for 6, u, and p had been obtained.
Now the formula (13) shows, that unless sin ty be insensible, sound cannot be propagated to a distance, but must be stifled in the neighbourhood of the vibrating body by which it is excited. Since we know very well that this is not the case, we are taught that sin ty is insensible, and therefore ^ itself, since ^ denotes an angle lying between 0 and 7r/4. The formula (13) shows, that if V be the velocity of propagation, V=npT^l sec >/r, which, when s. in.                                                                          10