FORMULAS FOR CALCULATING THE SPEED OF SOUND …

1 1 FORMULAS FOR CALCULATING THE SPEED OF SOUNDR evision GBy Tom IrvineEmail: 13, 2000 IntroductionA SOUND wave is a longitudinal wave, which alternately pushes and pulls the materialthrough which it propagates. The amplitude disturbance is thus parallel to the directionof waves can propagate through the air, water, Earth, wood, metal rods, stretchedstrings, and any other physical purpose of this tutorial is to give FORMULAS for CALCULATING the SPEED of FORMULAS are derived for a gas, liquid, and Formula for Fluids and GasesThe SPEED of SOUND c is given byoBc = (1)whereB is the adiabatic bulk modulus,o is the equilibrium mass (1)

2 Is taken from equation ( ) in Reference 1. The characteristics of thesubstance determine the appropriate formula for the bulk or FluidThe bulk modulus is essentially a measure of stress divided by strain. The adiabatic bulkmodulus B is defined in terms of hydrostatic pressure P and volume V as V/VPB = (2)Equation (2) is taken from Table in Reference adiabatic process is one in which no energy transfer as heat occurs across theboundaries of the alternate adiabatic bulk modulus equation is given in equation ( ) in Reference = (3)

3 Note that = PP (4)where is the ratio of specific ratio of specific heats is explained in Appendix SPEED of SOUND can thus be represented as ooPc = (5)Equation (5) is the same as equation ( ) in Reference GasAn alternate formula for the SPEED of SOUND in a perfect gas iskTMRc = (6a)where is the ratio of specific heats, M is the molecular mass, R is the universal gas constant, Tk is the absolute temperature in mass is explained in Appendix A.

4 The SPEED of SOUND in the atmosphere isgiven in Appendix (6a) is taken from equations ( ) and ( ) in Reference SPEED of SOUND in a gas is directly proportional to absolute ,k1,k21 TTcc= (6b)LiquidA special formula for the SPEED of SOUND in a liquid isoTBc = (7)where is the ratio of specific heats,BT is the isothermal bulk modulus,o is the equilibrium mass (7) is taken from equation ( ) in Reference isothermal bulk modulus is related to the adiabatic bulk = BT (8)SolidThe SPEED of SOUND in a solid material with a large cross-section is given bycBG=+ 43 (9)whereG is the shear modulus, is the mass per unit (9) is taken from equation ( ) in Reference 1.

5 The c term is referred to as thebulk or plate SPEED of longitudinal shear modulus can be expressed as)1(2EG += (10)whereE is the modulus of elasticity, is Poisson s (10) is taken from Table in Reference equation (10) into (9).() + +=12E34Bc (11a) + +=1E32Bc (11b)The bulk modulus for an isotropic solid is)21(3EB = (12)whereE is the modulus of elasticity, is Poisson s modulus of elasticity is also called Young s (12) is taken from the Definition Chapter in Reference 3.

6 It is also given inTable of Reference equation (12) into equation (11b).() + + =1E32213Ec (13)The next steps simplify the algebra.() + + =11322131Ec (14)5()()()() + + + =12132121Ec (15)()() + =121333Ec (16)()() + =1211Ec (17)Equation (17) is also given in Chapter 2 of Reference Poisson terms in equation (17)

7 Account for a lateral effect, which can be neglected ifthe cross-section dimension is small, compared to the wavelength. In this case, equation(17) simplifies to =Ec (18)StringConsider a string with uniform mass per length L . The string is stretched with a tensionforce T. The phase SPEED c is given byLTc = (19)This SPEED is the phase SPEED of transverse traveling (19) is taken from equation ( ) in Reference a membrane with uniform mass per area a.

8 The membrane is assumed to bethin, with negligible membrane is stretched with a tension force per length LT. The tension is assumedto be uniform throughout the transverse phase SPEED c is given byaTcL = (20)This SPEED is the phase SPEED of transverse traveling (20) is the same as equation ( ) in Reference TopicsAppendix B gives the variation of the SPEED of SOUND in the atmosphere with C gives the SPEED of SOUND in properties of solids, liquids, and gases are given in Tables 1a, 1b, 2, and 3, 1a.

9 SolidsSpeed of SOUND (m/sec)SolidDensity(kg/m3)ElasticMo dulus(Pa)ShearModulus(Pa)Poisson (1010) (1010) (1010) (1010) (1010) (1010) (1010) (1010) (Pyrex) (1010) (1010) (Extreme Values from Reference 7)SolidSpeed of SOUND (m/sec)Granite6000 Vulcanized Rubber at 0 C54 Table 2. LiquidsLiquidsTemperature( C)Density(kg/m3)AdiabaticBulkModulus(Pa) Ratio ofSpecificHeatsSpeed ofSound(m/sec)Water (fresh) (109) (sea) (109) , (109) 3. Gases at a pressure of 1 atmosphereGasesMolecularMass(kg/kgmole)T emperature( C)Density(kg/m3)Ratio ofSpecificHeatsSpeed ofSound(m/sec) ( O2 ) ( H2 ) : 1 (kg/kgmole) = 1 (lbm/lbmole)ExamplesAirCalculate the SPEED of SOUND in air for a temperature of 70 degrees F ( K).

10 The properties for air are = = kg/kgmoleThe universal gas constant is8R = J/(kgmole K)The specific heat ratio is taken from Appendix 10 of Reference 1. The molecular massand gas constant values are taken from Reference formula for the SPEED of SOUND iskTMRc =() =c = 344 m/secc =1130 ft/secSolid, Aluminum RodCalculate the SPEED of SOUND in an aluminum rod. Assume that the diameter is muchsmaller than the material properties for aluminum are:E = 70(109) Pa = 2700 kg/m3 These properties are taken from Reference 6. The SPEED of SOUND is =Ec39m/kg2700Pa)10(70c=c = 5100 m/secc = 16,700 ft/sec9 References1.