A sound wave in a fluid is a travelling pattern of small compressions and expansions. Its propagation speed is set by a competition between the fluid's resistance to compression and its inertia.

Let an equilibrium fluid have mass density $\rho$ and an appropriate small-signal bulk modulus $K$. Consider a longitudinal displacement field $u(x,t)$ in a narrow fluid column.

Compression produced by displacement

Take a thin fluid element of equilibrium length $dx$. If the displacement varies with position, its fractional change in length, and therefore its fractional volume change in this one-dimensional picture, is approximately

$$\frac{\Delta V}{V}=\frac{\partial u}{\partial x}.$$

The bulk-modulus relation gives the pressure perturbation

$$\boxed{p'=-K\frac{\partial u}{\partial x}}.$$

A region with negative $\partial u/\partial x$ is compressed and therefore has positive pressure perturbation.

Pressure-gradient force

The pressure force on the left and right faces of a small element do not exactly cancel when $p'$ varies with $x$. For cross-sectional area $A$, the net force is approximately

$$dF=-A\frac{\partial p'}{\partial x}dx.$$

The element mass is

$$dm=\rho A,dx.$$

Newton's second law gives

$$\rho A,dx\frac{\partial^2u}{\partial t^2} =-A\frac{\partial p'}{\partial x}dx.$$

Canceling $A,dx$ and substituting

$$p'=-K\frac{\partial u}{\partial x}$$

gives

$$\rho\frac{\partial^2u}{\partial t^2} =K\frac{\partial^2u}{\partial x^2}.$$

Therefore

$$\boxed{\frac{\partial^2u}{\partial t^2} =c^2\frac{\partial^2u}{\partial x^2}},$$

with

$$\boxed{c=\sqrt{\frac{K}{\rho}}}.$$

Stiffness versus inertia

The formula has the same structure as other mechanical-wave speeds:

• larger $K$ means stronger restoring pressure for a given compression, so the wave travels faster;
• larger $\rho$ means greater inertia, so the wave travels more slowly if stiffness is unchanged.

The relevant bulk modulus must describe the material response on the timescale of the acoustic disturbance. In gases, for example, rapid acoustic compression and expansion can differ from a slow isothermal compression, so the appropriate acoustic bulk modulus must match the thermodynamic process.

Worked example: sound speed in water

Take approximately

$$K=2.2\times10^9,\mathrm{Pa}$$

and

$$\rho=1000,\mathrm{kg/m^3}.$$

Then

$$c=\sqrt{\frac{2.2\times10^9}{1000}} =\sqrt{2.2\times10^6} \approx1.48\times10^3,\mathrm{m/s}.$$

Thus

$$\boxed{c\approx1480,\mathrm{m/s}}.$$

A pressure disturbance therefore crosses a $100,\mathrm m$ water path in roughly

$$t=\frac{100}{1480}\approx0.068,\mathrm s.$$

The relation $c=\sqrt{K/\rho}$ is the simplest continuum model for longitudinal acoustic propagation in a fluid. It makes explicit what the generic mechanical-wave picture only states qualitatively: a wave travels because elastic compression supplies a restoring force while the medium's mass supplies inertia.