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Lecture 6

# Lecture 6 - adiabatic To derive the wave velocity in a...

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EEL 4930 Audio Engineering Lecture 6 Wave Equation for Sound Propagation in a Fluid Sound waves are traveling pressure waves in a fluid (gas or a liquid). P ( x , t ) = P 0 + p ( x ! ct ) Corresponding to the pressure wave is a displacement wave ! ( x " ct ) of atoms from where they would be if there had been no sound propagating. The relationship between the displacement wave and the deviation from average pressure is a characteristic property of the fluid: p = ! ( " P 0 ) d # dx Note that d ! / dx rather than ! appears on the right hand side of the equation because d ! / dx measures the deformation of the fluid from equilibrium: A constant ! would correspond to an overall displacement of the fluid. The negative sign is there because a contraction d ! / dx < 0 gives an increase in pressure and thus a positive p . ! = 7/5 is the ratio of the specific heat at constant pressure to the specific heat at constant volume. It comes from thermodynamics and reflects the fact that compression of air causes not only a pressure rise but also a temperature rise. (The process is

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Unformatted text preview: adiabatic .) To derive the wave velocity in a fluid we write Newton's Second Law for a cylindrical slice of thickness " x and area A . The force on this slice is F = A { P ( x ) ! P ( x + " x )} = ! A " x dp dx = A " x ( P ) \$ 2 % \$ x 2 We also calculate ma = A " x # 2 \$ # t 2 Equating F and ma gives us the wave equation F = ma ! A ! x ( P ) # 2 # x 2 = A ! x # 2 # t 2 ! 2 ! x 2 = P % & ' ( ) * ! 2 ! t 2 = 1 c 2 ! 2 ! t 2 From which we conclude that the speed of sound is c = P Putting in numbers we get for air at ambient pressure ( P = 10 5 w N/m 2 ,), and for a typical density of air ( # = 1.3 kg/m 2 ,) we get c = (1.4)(10 5 Nt / m 2 ) (1.3 kg / m 2 ) = 330 m/s which is a typical measured value. Note that at constant pressure the velocity of sound is slower in a denser gas. Adapted from http://www.pha.jhu.edu/~broholm/l28/node4.html...
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Lecture 6 - adiabatic To derive the wave velocity in a...

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