ACOUSTICS. 81 
g. On the Beats of Tones. 
If two tuning-forks of very nearly the same pitch, or two strings or 
pipes of almost precisely the same tone, be sounded simultaneously, we 
shall hear a variation of the tone, consisting in an alternate increase or 
diminution of its intensity. This is caused by the fact, that both sounds 
are produced by undulations of very nearly, but not quite the same 
rapidity, so that at one time these will come together in the same phase of 
vibration, and at another time in opposite phases. In the first case the 
intensity will be double that of a single sound; in the latter, no sound 
whatever would be perceived but for the momentary persistence in the ear 
of the sound of the instant previous. The tone will consist then in a 
gradual increase or diminution between these extremes. The greater the 
difference in the rapidity of undulation, the more frequent will be these 
beats; when the two instruments are in unison they cease entirely. Any 
number of strings may thus be brought into unison by tuning until the beats 
are found to have disappeared altogether. When two sounds are heard, of 
which the vibrations stand in a simple ratio to each other, as of two to 
three, three to four, or four to five, and in which the coincidence of two 
impulses or undulations recurs with sufficient frequency, a third sound is 
produced by this coincidence, always deeper than the primary notes, and 
generally the fifth or the octave below the lower of the two. These are 
called tones of combination, or the accessory sounds of Tartini, and must 
not be confounded with harmonic notes. 
h. Sound in various Media. 
Sound diffuses itself through all ponderable matter, although with various 
velocities. Newton gave an expression for the motion of sound in the air, 
which was much too small, being but about five sixths of the actually 
observed velocity; Laplace explained the difference by showing that a 
motion of sound cannot take place but by compression of the molecules of 
the air, during which, in all cases, there must be a development of heat; 
and that then the heat, now become sensible, must influence the law of 
elasticity in such a manner as to bring about an acceleration in the 
transmission of sound. Consequently, temperature would influence the 
motion of sound, as we find to be actually the case. Laplace has given a 
formula for the rapidity of this motion in vapors and gases; according to 
him, v= Womb (1 + al) k, where v is the. velocity in a second; g, the 
accelerating force of gravity, 386.29 inches; m, the ratio of the density of 
mercury to that of atmospheric air, found by experiment to be 10.466, at a 
temperature of 32° F., and a barometric pressure of 29.927; b, the standard 
height of mercury in the barometer; a, the constant co-efficient of 
expansion, ascertained by experiment to be .00208; and k, the square-root 
of the quotient, which is found by dividing the number which expresses the 
specific heat of the air (or other gas) under a constant pressure, by that 
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