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Note for Applied Physics-I - AP1 by Sudipta Dash

• Applied Physics-I - AP1
• Note
• Engineering Institute - KIIT
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1 14. SOUND WAVES [Learning objective: - 1. Standing waves in organ pipes, fundamental mode and harmonics, 2. Beats. 3. Doppler’s effect] Sound is a form of energy that produces the sensation of hearing. Sound requires a material medium for its propagation. Sound waves are longitudinal waves. The elastic and inertial properties of a medium determine the speed of sound in that medium. Sound can propagate in solids, liquids and gases. Human ear is sensitive to sound waves whose frequencies lie between 20 – 20,000 Hz. Sound waves of frequencies less than 20 Hz are called infrasonic. Sound waves of frequencies more than 20 Hz are called ultrasonic. The branch of physics that deals with the generation, propagation and reception of sound is called ‘acoustics’. The waves produced by vibrating bodies in air are longitudinal. This is because air has only one elastic co-efficient, namely the bulk modulus. Longitudinal waves are a sequence of alternate compressions and rarefactions. Velocity of longitudinal wave (sound waves) in an elastic medium In case of propagation of a longitudinal wave, the particles of the medium vibrate in the direction of propagation of wave. So its velocity must depend upon the density  of the medium. Greater the density, greater is the opposition and hence smaller is the velocity. Again, vibrations are sustained due to the property of elasticity, so velocity must depend upon the co-efficient of elasticity (E) of the medium. Let v  E a  b v  E a  b  v = kEa  b , where, k is a dimensional constant. Taking dimensions on both sides [M 0 L1T −1 ] = [ML−1T −2 ]a [ML−3 ]b = [M a + b L− a −3bT −2a ] Applying principle of homogeneity, a+b =0 − a − 3b = 1 − 2a = −1 1 1 a= , b= 2 2 From experiment k=1 E v = .................................. (1)  This formula is known as Newton’s formula for velocity of sound. Y (i) Velocity of sound in solids= v = , where, Y= Young’s modulus of elasticity.  (ii) (iii) Velocity of sound in liquids= v = B  , where, B= Bulk modulus of elasticity. Velocity of sound in gases Newton showed that the elasticity of gas is equal to its pressure. According to him, when sound waves travel through a gas, compressions and rarefactions are formed so slow that the temperature of the medium remains the same. In the regions of compression, heat is produced and is readily lost to the surroundings. Likewise, in the region of rarefaction, cooling is produced; heat is gained from the surroundings. Thus according to Newton, the propagation of sound waves in gas takes place under isothermal conditions and appropriate modulus of elasticity is the isothermal bulk modulus. Consider a small section of the medium having pressure P and volume V. Due to arrival of compression, the pressure there increases to P+P while volume decreases to V– V. According to Boyle’s law,

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2 PV = (P + P) (V – V) = PV – PV + VP – PV As P and V are very small, so PV term can be neglected. Hence, PV = PV – PV + VP Or, PV = VP 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 PV or 1 = 1 −  + – V P PV PV 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 

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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,

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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