VELOCITY OF SOUND It has already been shown that the velocity of sound-waves of small amplitude in an unlimited fluid medium is given by Al(KI p) where tc is the bulk modulus of elasticity and p the density of the medium. In the case of a solid in bulk, K must be replaced by (lc+ 4,u) where kt is the rigidity of the solid. In special cases where the volume of material is limited, e.g., in solid bars or columns of liquid in tubes, it is important that the appro priate elastic constant be chosen.
In a gas is compressed it is heated, and when rarefied it is cooled. If the heating or cooling effects are not neutralised by removal or supply of heat, e.g., by convection, then the temperature of the gas will rise or fall accordingly. In ordinary sound-waves the alternate condensations and rarefac tions take place so rapidly that there is no time for a transfer of heat between adjacent layers of gas. Consequently the temper ature rises and falls with the same frequency as the waves of pressure. Such conditions are described thermodynamically as adiabatic. In an adiabatic compression, the relation between pressure p and density p of the gas is p/p7= constant, that is 7 where Po and p p refer to initial and final states of pressure and density, and y is the ratio of specific heats of the gas. Thus for small values of the condensation s(=bp/ p). Now K =6p/s (see equation 9). Therefore Consequently the velocity of sound is given by C= V(k/ P)= AteiPci PO. (20) Now Boyle's law indicates that p/p is a constant at constant temperature, consequently the velocity is independent of the pressure of the gas, for the density p changes in proportion to the pressure p.
Taking dynes/cm.' and 0.00129, and assuming 7= 1.41 for air, we obtain a value for the velocity This is in good agreement with observation.
Change of Velocity with Temperature.—Since p0= (1.-Fa0) where a is the temperature coefficient of volume expansion at constant pressure, and 0 is the temperature in degrees centigrade, we have ce =1/ +a0)/ Now for any gas a = 1/273 per degree centigrade nearly, therefore where T and are the absolute temperatures. That is, the velocity varies directly as the square root of the absolute temperature. The velocity given by (20) is independent of frequency, but is dependent on the nature of the gas. The velocity varies inversely
as the square root of the density, provided y is the the velocity is four times as great in hydrogen as in oxygen.
Experimental Determinations of Velocity of Sound in Free Air.—The velocity of sound as measured in free air is affected by the wind, being greater with the wind than against it; in the first case the sum, and in the second the difference, of the ve locities of the sound and the wind. It is also affected by humidity. The wind-velocity may be eliminated from the result by the method of reciprocal sound being produced first at one end and then at the other of a measured base line with a receiver also at each end. Alternatively the wind-velocity may be measured directly and a correction applied to the observed velocity of the sound. The latter method has been employed by Esclangon (Comptes Rendus, Jan 20, 1919). The sound-waves from guns of various calibres were received by sensitive electrical sound-detectors at 1400 m. and 14000 m. along the same line. The time intervals were measured to ±0.002 second, the observations being made under various conditions of wind and humidity. The final result for the speed of sound in dry air at C. was 339.9 metres per second which compares well with Regnault's value 339.7 metres/sec. (1863). Measure ments of the velocity of sound at different levels (on mountain heights and plains) confirm the theory that there is no change of velocity with pressure. Observations at low temperatures in the Arctic give a value co= (333+0.60) metres/sec. at Other methods of measuring the velocity in free air (1) Echo methods—the time t between the transmission of a sound and arrival of its echo from a reflector at a distance d being measured whence velocity = 2007 (2) Coincidence Method—The sound is transmitted at regular, accurately known, time inter vals simultaneously from two points at a known variable dis tance apart. When close together, the sounds are heard together but when one is moved further away the sounds are separated by an interval, at first increasing and then decreasing, till coinci dence is again established. The difference in distance between two such positions of coincidence, combined with a knowledge of the time intervals, gives a value of the velocity.