I am doing a presentation on sound waves and I need to know if they are adiabatic or isothermal. I know that they can generate heat, but is the amount of heat created so small that it can still be considered adiabatic?
[Physics] Are sound waves adiabatic or isothermal
adiabaticenergythermodynamics
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Yes quite! The ideal gas law (and in general any state equation) holds only on equillibrium while the 1st law (and all the rest) hold in general. So, your anslysis mr James Hoyland is inacurrate.
Mr Steven, the post includes the word "expansion" so volume changes by assumption.
And mr or ms PhysC, the first two cases are correct. About the third, expansion would occur by pulling the piston thus removing energy from the system and causing decrease on temperature
For starters, to quote Allan Pierce in Acoustics,
The often stated explanation, that oscillations in a sound wave are too rapid to allow appreciable conduction of heat, is wrong.
That one surprised me when I learned it myself.
In fact, sound is not an adiabatic process for all frequencies. For any medium there is a thermal conduction frequency, $$ f_{\mathrm{TC}} =\frac{\rho c_{p} c^{2}}{2 \pi \kappa}. $$ Frequencies much lower than this value will be well-approximated as adiabatic. However, increasing the frequency through and above this point will transition the process from adiabatic to isothermal. For air, this frequency is $\sim 10^{9} \, \mathrm{Hz}$, well above the range of human hearing, so we almost always treat sound as adiabatic.
The physical reason this occurs is that heat transfer due to conduction is proportional to the temperature gradient. This is just a statement of Fourier's law for heat conduction. Consider what happens as the frequency of a harmonic wave decreases: The wavelength increases, and the slope of the oscillating waveform decreases as it is "stretched out." Assuming equal amplitudes, lower frequency waves will therefore set up smaller temperature gradients, which will conduct heat less effectively. If the heat conduction is negligible, then the entropy is conserved by the process.
So, in summary, the thermal gradients set up by sound waves for typical frequencies of interest are small enough to be neglected, hence sound is a (very nearly) adiabatic process. However, as Thomas pointed out below, in reality frequencies that cross into the potentially-isothermal regime are almost always affected by attenuation first, and the principal effects from conduction and viscosity are actually to damp out the sound wave.
In case you decide you do want to see some math, the energy equation is $$ \rho T \frac{d s}{d t} = \kappa \nabla^{2} T.$$ The previous arguments can be seen mathematically by linearizing about a quiescent base state and assuming harmonic wave solutions for $s$ and $T$. The equation can be rewritten as $$ - i \omega \rho_{0} T_{0} \hat{s} = - \kappa \frac{\omega^{2}}{c^{2}} \hat{T}, $$ $$ \hat{s} = - i \frac{\kappa \omega}{\rho_{0} T_{0} c^{2}} \hat{T}. $$ As the angular frequency $\omega = 2 \pi f \rightarrow 0,$ so must the amplitude of the entropy oscillation, $\hat{s}$. As with the quote at the beginning, much of my answer draws from Acoustics by Allan Pierce.
Best Answer
Sound waves are approximately adiabatic, and the sound speed is determined by the adiabatic compressibility. The reason is that long wave length disturbances are approximately solutions to the Euler equation (which conserves entropy), and the Navier-Stokes terms (which generate entropy) are small corrections. Sound is not isothermal, the pressure disturbances induce temperature oscillations.
The amount of heat generated is proportional to the amplitude squared, the frequency squared, and the dissipative coefficients (shear viscosity, bulk viscosity, and thermal conductivity). The sound absorption coefficient (the inverse sound absorption length) is $$ \gamma = (\langle\dot E\rangle/\langle E\rangle)/(2c) $$ with $$ \gamma = \frac{\omega^2}{2\rho c^3} \left[\frac{4}{3}\eta+\zeta + \kappa\left(\frac{1}{c_v}-\frac{1}{c_p}\right) \right] $$ where $\eta$ is shear viscosity, $\zeta$ bulk visosity, and $\kappa$ thermal conductivity.
Under typical conditions the sound absorption length is quite long, and not that much heat is produced. A more efficient mechanism for producing heat is sound absorption by a solid body. This is because the surface of the solid is isothermal, so large temperature gradients can occur at the surface boundary layer.