2 PV = (P + P) (V – V) = PV – PV + VP – PV As P and V are very small, so PV term can be neglected. Hence, PV = PV – PV + VP Or, PV = VP P V P Or, P = = = Bulk modulus B V V V P v= For air, at NTP, P0 = 76 cm Hg = 1.01325 105 N/m2 and 0 = 1.293 Kg/m3 1.01325 10 5 So v0 = = 280 m/s. 1.293 Experimentally it is found that the velocity of sound in air at NTP is 332 m/s. Since theoretical and experimental values differ quite a lot, the formula needs correction. It was corrected by Laplace. Laplace’s correction According to Pierre Laplace (1817), a French scientist, propagation of sound in gas is an adiabatic process. Because, compressions and rarefactions occur very rapidly; the gas being poor conductor of heat, the heat developed in compression zone does not get time to flow to rarefaction zone, i.e. there is hardly any time for heat to be exchanged between a compression and rarefaction. As a result, the adiabatic condition is maintained instead of isothermal condition. Let due to arrival of a compression, pressure changes from P to P+P and volume from V to V – V. C Sp. heat of gas at constant pressure For adiabatic process, PV = constant , where, = P = CV Sp. heat of gas at constant v olume Applying adiabatic equation of state P V PV = (P + P) (V −V) = P (1 + ) V 1 P V V V V Since is very small, so 1 1 - V V V P V Hence, PV = PV 1 + 1 - P V V P PV or 1 = 1 − + – V P PV PV Neglecting the small term , we get PV V P 1 = 1− + V P V P or = V P P V P or P = = = Bulk modulus B V V V So velocity of sound, v = P
3 At NTP, for air P0 = 1.01325 105 N/m2 1.42 1.01325 10 5 = 1.42 280 = 333.53 m/s. 1.293 This value agrees quite well with the experimental value. 0 = 1.293 kg/m3 and = 1.42 v = Factors affecting velocity of sound (i) Effect of density: - We know, v = At a constant pressure, v P 1 v1 2 M2 = = 1 M1 v2 Velocity of sound varies inversely as the square root of the density of medium at constant pressure or molecular weight of the gas. O2 velocity of sound in H 2 16 4 E.g. = = = velocity of sound in O 2 H 2 1 1 Velocity of sound in hydrogen is 4 times that in oxygen. (ii) Effect of pressure P M v= , but = V v = PV M = RT M ............................................. (1) So velocity of sound is independent of pressure of the gas provided temperature remains constant.. (iii) Effect of humidity (moisture) As water vapours are lighter than dry air, density of moist air is less than that of dry air. 1 v But, m < d d v Now m = 1 vd m Hence vm > vd Sound travels faster in humid air than in dry air. This is why the sirens of the mills and the whistles of trains etc. are heard upto longer distances in rainy season as compared to summer season. (iv) Effect of temperature RT According to standard gas equation, PV = RT P = , T is in Kelvin scale. V RT RT P = So, v = = v T V M v1 T = 1 v2 T2 Velocity of sound varies directly as the square root of absolute temperature. Temperature co-efficient of velocity of sound It is defined as the change in velocity of sound for each 10 C rise of temperature. Or,
4 We know, velocity of sound at NTP = 332 m/s. T0 = 273K, v0 = 332 m/s. Let at T = (273 + t) K, velocity be v 1/ 2 v t 273 + t T Then, = = = 1 + 273 v0 273 T0 If t is very small, then t is also very small, hence 273 1/ 2 v t 1 t t = 1 + =1+ =1+ 2 273 546 v 0 273 t or v = v0 1 + 546 v0t 546 v t or v = 0 546 0 For t = 1 C, v is called temperature coefficient of velocity of sound (). v 1 332 = So = 0 = 0.608 m s-1 oC-1. 546 546 v = (332 + 0.608t )ms−1 Hence the velocity of sound in air increases approximately by 0.61m for every 1 oC rise in temperature. or v – v0 = (v) Effect of wind If wind blows making an angle with the direction of propagation of sound, then resultant velocity of sound vR = v + vw cos [Fig.13.8] Thus, sound will travel faster if is acute and slower, if is obtuse. So, sound velocity is maximum in the direction of wind and minimum against the direction of wind. The wind will have no effect on the velocity of sound if =90o. (vi) Effect of amplitude: - Velocity of sound is independent of amplitude and wavelength, provided amplitude is small. At higher amplitude, loud sounds move faster than ordinary sounds in the immediate neighbourhood of the source. (vii) Effect of frequency or pitch: -There is no effect of frequency or pitch on the velocity of sound in a medium. Sound waves of different frequencies travel with the same speed in air although their wavelengths in air are different. If the speed of sound were dependent on frequency, then we could not have enjoyed orchestra. Determination of speed of sound in air Resonance column method
5 [Fig.13.9] It is the simplest method used in the laboratory to measure speed of sound in air. The resonance tube consists of a long cylindrical tube supported on a vertical board. The tube is connected to a reservoir containing water. The height of water in the tube can be altered by changing the height of the reservoir. By this way the length of air column in the tube can be changed. A vibrating tuning fork is held near the open end so that the prongs vibrate parallel to the length of the tube. Standing waves are set up in it. Water level in the tube is adjusted so that at a certain length, resonance takes place and we hear a loud sound. Again the length of the tube is adjusted and length for second resonance is obtained. For first resonance, the length of tube is given by L1 + e = … … (1) 4 where e is called end correction for the tube. End correction is due to the fact that the antinode is obtained a little distance above the open end of tube. It is found that e = 0.3d, where d is the diameter of the tube. For second resonance, the length of tube is given by L2 + e = 3 … … (2) 4 Subtracting equation (1) from equation (2) L2 – L1 = 2 or = 2 (L2 – L1) Let N = frequency of the tuning fork, then v = N = 2N (L2 – L1) Hence velocity of sound in air can be calculated using above formula. Sound is produced whenever a body vibrates in a medium. Sound waves are longitudinal, threedimensional mechanical (or elastic) waves in air with frequencies between 20 Hz to 20 kHz (range of audibility- sonic sound). Above 20 kHz, it is called ultrasonic and below 20 Hz, it is called infrasonic. Being a mechanical wave, sound waves require a medium for their propagation. It is produced by a vibrating source such as a guitar string, the human vocal cords, the prong of a tuning fork or the diaphragm of a loudspeaker. Sound is generally divided into 2 classes: musical sound and noise. [Fig.14.1. Pure note and musical note] Music is a smooth, pleasant, continuous and uniform sound produced by the regular and periodic vibration of a source without any sudden change in amplitude and agreeable to the ear. Musical sound having a single frequency is called a tone and that having a number of frequencies is called note. Thus a note is a combination of tones. In a musical sound, the tone having the lowest frequency is called the fundamental or prime tone and higher ones are called overtones. The sounds produced by a vibrating tuning fork, vibrating air column, violin, piano, sitar and flute etc. are some examples of musical sounds. Sound level of music is usually between 10 dB and 30 dB.