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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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 Spring '08
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 Fundamental physics concepts

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