Page:Elementary Text-book of Physics (Anthony, 1897).djvu/175

§ 135] the different pressures $$p_{a}$$ and $$p_{b}$$ on the moving cross-sections, since the interactions of the portion of matter between those cross-sections cannot change the Uiomentum of that portion. Hence we have $$M(v_{a} - v_{b}) = p_{a} - p_{b}.$$

If we for convenience assume $$v_{b} = 0$$, we have $$p_{b} = P$$, the pressure in the medium in its undisturbed cudition. If we further substitute for $$v_{a}$$ its value, we obtain $$MV = d_{a} \frac{p_{a} - P}{d_{a} - D}\cdot$$If the changes in pressure and density be small, the quantity $$d_{a} \frac{p_{a} - P}{d_{a} - D}$$ equals $$E$$, the modulus of elasticity of the medium. If we further substitute for Jf its value VD, we obtain finally

135. Velocity of Sound in Air.—In air at constant temperature the elasticity is numerically equal to the pressure (§ 105). The compressions and rarefactions in a sound-wave occur so rapidly that during the passage of a wave there is no time for the transfer of heat, and the elasticity to be considered, therefore, is the elasticity when no heat enters or escapes (§ 313).

If the ratio of the two elasticities be represented by $$\gamma$$ we have for the elasticity when no heat enters or escapes $$E = \gamma P$$, and the velocity of a sound-wave in air at zero temperature is given by $$V = \sqrt{\frac{\gamma P}{D}}\cdot$$ The coefficient $$\gamma$$ equals 1.41. $$P$$ is the pressure exerted by a column of mercury 76 centimetres high and with a cross-section of one square centimetre, or 76 x 13.59 x 981 = 1013373 dynes per square centimetre. $$D$$ equals 0.001293 gram at 0°, hence $$V = \sqrt{\frac{1.41 \times 1013373}{0.001293}} = 33240$$ or 332.4 metres per second.

Since the density of air changes with the temperature, the velocity of sound must also change. If $$d_{t}$$ represent the density at temperature $$t$$, and $$d_{0}$$ the density at zero, $$d_{t} = \frac{d_{0}}{1 + \alpha t}$$ from § 211.