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The speed of sound is the propagation speed of a small mechanical pressure disturbance through a medium. Compression creates a restoring pressure gradient, while the medium's mass density supplies inertia. For an inviscid fluid undergoing locally isentropic disturbances,A stiffer pressure response increases the speed. For a perfect gas, this becomes $c_s^2=\gamma p/\rho$, with $\gamma$ the heat capacity ratio.

The disturbance travels while individual fluid parcels undergo much smaller oscillatory motions; acoustic particle velocity and propagation speed are different quantities. Timing a short sound pulse between two sensors a known distance apart gives a direct measurement after accounting for electronic delays. If heat exchange maintains a different thermodynamic response during compression, the relevant derivative changes, as in the isothermal sound speed. The formula also presumes a wavelength and timescale for which the continuum model is appropriate.

An isothermal sphere with constant positive isothermal sound speed $c$ obeys the title hydrostatic equilibrium equation. Between two positive finite radii with bounded enclosed mass, its right-hand side has a finite integral. A positive starting density therefore remains positive at any proposed finite outer radius. It cannot meet a vacuum boundary with $p=c^2\rho =0$ there without an additional confining structure or a change of equation of state. The singular isothermal sphere makes the obstruction explicit: $\rho =c^2/(2\pi G{r}^2)$ and $p=c^4/(2\pi G{r}^2)$ never vanish at finite radius, and its mass grows as $2c^{2}r/G$ without bound at infinity.