A propagating sound wave causes periodic temperature...

A propagating sound wave causes periodic temperature variations in a gas. Thermal conductivity acts to remove these variations but it is generally claimed that the waves are adiabatic, that is, thermal conductivity is too slow. The coefficient of thermal conductivity for an ideal gas from kinetic theory is $k \approx 1.23 C_v \bar{v} l$ where $C_v$ is the heat capacity per unit volume, $\bar{v}$ is the mean thermal speed, and $l$ is the mean free path. What fraction of the temperature variation $\Delta T$ will be conducted away vs $\lambda$ and what is the condition on $\lambda$ for thermal conductivity to be ineffective? FIGURE CAN'T COPY.

A propagating sound wave causes periodic temperature variations in a gas. Thermal conductivity acts to remove these variations but it is generally claimed that the waves are adiabatic, that is, thermal conductivity is too slow.

The coefficient of thermal conductivity for an ideal gas from kinetic theory is $k \approx 1.23 C_v \bar{v} l$ where $C_v$ is the heat capacity per unit volume, $\bar{v}$ is the mean thermal speed, and $l$ is the mean free path.

What fraction of the temperature variation $\Delta T$ will be conducted away vs $\lambda$ and what is the condition on $\lambda$ for thermal conductivity to be ineffective?
FIGURE CAN'T COPY.

Key Concepts

Mean Free Path and Kinetic Theory in Thermal Conduction

In an ideal gas the kinetic theory gives the thermal conductivity as approximately k ? 1.23 C? v? l, where v? is the mean thermal speed and l is the mean free path. The mean free path sets a microscopic length scale for molecular transport, and when combined with the thermal speed it determines the thermal diffusivity (? = k/C?). This connection tells us how microscopic properties control the macroscopic effectiveness of heat conduction in the wave’s temperature variations.

Thermal Diffusivity

Thermal diffusivity (often denoted by ?) is the ratio of thermal conductivity to the heat capacity per unit volume. It quantifies the rate at which temperature inhomogeneities are evened out by conduction. In problems involving waves, it determines how fast heat can be spread out over the spatial scale of the wave and thereby compete with the time scale of the temperature oscillations.

Wave Propagation and Time Scales

A sound wave has a characteristic wavelength ? and propagates at the speed of sound v?, so the time for one wavelength to pass is approximately ?/v?. This period sets the time window during which thermal conduction can act on any temperature perturbations. Comparing the conductive time scale to this propagation time is central to determining whether the process will be nearly adiabatic or not.

Conductive Damping of Temperature Variations

For a sinusoidal temperature perturbation with wavelength ?, the damping rate due to conduction is proportional to the thermal diffusivity times the square of the wavenumber (2?/?)². Over one period, the fractional reduction in temperature amplitude is roughly f ? (2?)² ?/(v? ?). This fraction quantifies how much of the temperature variation ?T is removed by conduction as the wave advances one wavelength.

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