The speed of sound in a gas is calculated as v=√( adiabatic bulk modulus/density ) . (a) Show th

step 1 For exercise 32, for part A, we have that V equals the root negative V over rho, D over DV, K ti

The speed of sound in a gas is calculated as v=√( adiabatic...

The speed of sound in a gas is calculated as $$ v=\sqrt{\text { adiabatic bulk modulus/density }} . $$ (a) Show that this is a dimensionally-correct equation. (b) This formula implies that the propagation of sound through air is a quasi-static process. On the other hand, the speed of sound for air is about $340 \mathrm{~m} / \mathrm{sec}$ at a temperature for which the rms speed of an air molecule is about $500 \mathrm{~m} / \mathrm{sec}$. How then can the process be quasi-static?

The speed of sound in a gas is calculated as
$$
v=\sqrt{\text { adiabatic bulk modulus/density }} .
$$
(a) Show that this is a dimensionally-correct equation.
(b) This formula implies that the propagation of sound through air is a quasi-static process. On the other hand, the speed of sound for air is about $340 \mathrm{~m} / \mathrm{sec}$ at a temperature for which the rms speed of an air molecule is about $500 \mathrm{~m} / \mathrm{sec}$. How then can the process be quasi-static?

Key Concepts

Dimensional Analysis

Dimensional analysis is a method used to check the consistency of physical equations by ensuring that both sides of an equation have the same base units. In the context of the speed of sound formula, it helps verify that the square root of a ratio of an adiabatic bulk modulus (which has units of pressure) to density yields a result with the dimensions of speed, ensuring the formula is physically plausible and mathematically correct.

Adiabatic Process

An adiabatic process is one in which there is no heat transfer into or out of the system, meaning that any compression or expansion occurs without exchange of heat. For sound propagation in gases, the rapid compression and rarefaction of the medium occur so quickly that there is no time for significant heat exchange, justifying the use of the adiabatic bulk modulus in the speed of sound equation.

Quasi-static Process

A quasi-static process is one that occurs slowly enough to assume that the system remains in a near-equilibrium state throughout. In the context of sound waves, even though the propagation speed is finite, the changes in pressure and density are very small in each oscillation, allowing the use of equilibrium thermodynamic relations. This approximation is valid because the wave disturbances are infinitesimal, despite the finite speed of the sound wave compared to the random motion of the gas molecules.

Separation of Macroscopic and Microscopic Speeds

There is an essential distinction between the macroscopic speed of sound, which is a collective property of wave propagation through a medium, and the microscopic or rms speed of individual gas molecules, which is related to thermal energy. Despite the fact that individual molecules have high speeds, the slight, coordinated fluctuations that constitute a sound wave travel much more slowly, and it is these small deviations from equilibrium that are responsible for the propagation of sound.

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