Question
The speed of longitudinal waves of small amplitude in an ideal.gas is$$C=\sqrt{\frac{d p}{d \rho}}$$where $p$ is the ambient gas pressure and $\rho$ is the corresponding gas density. Obtain expressions for(a) The speed of sound in a gas for which the compressions and rarefactions are isothermal.(b) The speed of sound in a gas for which the compressions and rarefactions are adiabatic.
Step 1
Step 1: To find the speed of sound in a gas, we start with the given formula for the speed of longitudinal waves: $$ C = \sqrt{\frac{d p}{d \rho}} $$ where \( p \) is the pressure and \( \rho \) is the density of the gas. Show more…
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Review. As a sound wave passes through a gas, the compressions are either so rapid or so far apart that thermal conduction is prevented by a negligible time interval or by effective thickness of insulation. The compressions and rarefactions are adiabatic. (a) Show that the speed of sound in an ideal gas is $$v=\sqrt{\frac{\gamma R T}{M}}$$ where $M$ is the molar mass. The speed of sound in a gas is given by Equation $16.35 ;$ use that equation and the definition of the bulk modulus from Section 12.4 . (b) Compute the theoretical speed of sound in air at $20.0^{\circ} \mathrm{C}$ and state how it compares with the value in Table $16.1 .$ Take $M=28.9 \mathrm{~g} / \mathrm{mol}$. (c) Show that the speed of sound in an ideal gas is $$v=\sqrt{\frac{\gamma k_{\mathrm{B}} T}{m_{0}}}$$ where $m_{0}$ is the mass of one molecule. (d) State how the result in part (c) compares with the most probable, average, and rms molecular speeds.
Review Problem. (a) Show that the speed of sound in an ideal gas is $$ v=\sqrt{\frac{\gamma R T}{M}} $$ where $M$ is the molar mass. Use the general expression for the speed of sound in a fluid from Section $17.1 ;$ the definition of the bulk modulus from Section $12.4 ;$ and the result of Problem 57 in this chapter. As a sound wave passes through a gas, the compressions are either so rapid or so far apart that energy flow by heat is prevented by lack of time or by effective thickness of insulation. The compressions and rarefactions are adiabatic. (b) Compute the theoretical speed of sound in air at $20^{\circ} \mathrm{C}$ and compare it with the value given in Table $17.1 .$ Take $M=28.9 \mathrm{~g} / \mathrm{mol} .$ (c) Show that the speed of sound in an ideal gas is $$ v=\sqrt{\frac{\gamma k_{\mathrm{B}} T}{m}} $$ where $m$ is the mass of one molecule. Compare your result with the most probable, the average, and the rms molecular speeds.
The speed of sound in a gas, $c,$ is a function of the gas pressure, $p,$ and density, $\rho .$ Determine, with the aid of dimensional analysis, how the velocity is related to the pressure and density. Be careful when you decide on how many reference dimensions are required.
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