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Isothermal versus Isentropic

If air compression/expansion were isothermal (constant temperature $ T$), then, according to the ideal gas law $ PV=nRT$, the pressure $ P$ would simply be proportional to density $ \rho$. It turns out, however, that heat diffusion is much slower than audio acoustic vibrations. As a result, air compression/expansion is much closer to isentropic (constant entropy $ S$) in normal acoustic situations. (An isentropic process is also called a reversible adiabatic process.) This means that when air is compressed by shrinking its volume $ V$, for example, not only does the pressure $ P$ increase (§B.7.3), but the temperature $ T$ increases as well (as quantified in the next section). In a constant-entropy compression/expansion, temperature changes are not given time to diffuse away to thermal equilibrium. Instead, they remain largely frozen in place. Compressing air heats it up, and relaxing the compression cools it back down.

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Adiabatic Gas Constant
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Ideal Gas Law