Speed of a Longitudinal Wave (Speed of Sound)

The Speed of a Longitudinal Wave

We now turn to the speed of a longitudinal wave — a wave in which the particles of the medium oscillate parallel to the direction of wave propagation. Sound waves in air, compression waves in a spring, and seismic P-waves are all longitudinal. The key question is: what determines how fast such a disturbance travels?

The answer, as for transverse waves, depends on two properties of the medium: its elasticity (how strongly it resists compression) and its inertia (how massive the medium is). For a longitudinal wave, the relevant elastic property is the bulk modulus $B$, and the inertial property is the density $\rho$.

This is the fundamental expression for the speed of a longitudinal wave in a fluid (liquid or gas). Let us derive it carefully.

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Derivation of $v = \sqrt{B/\rho}$

Consider a long fluid column (say, a tube of air) of cross-sectional area $A$. Imagine a piston at one end that is pushed inwards with a constant speed $u$ for a short time $\Delta t$. This creates a compression pulse that travels down the tube at speed $v$.

Step 1: Geometry of the pulse.

In time $\Delta t$, the piston moves a distance $u \Delta t$ into the tube. Meanwhile, the front of the compression pulse has moved a distance $v \Delta t$ ahead. The compressed region therefore has length $v \Delta t$, and the extra volume of fluid that has been forced into this region is $A u \Delta t$.

Step 2: Volume strain.

The original volume of the compressed region was $V = A v \Delta t$. The change in volume is $\Delta V = - A u \Delta t$ (negative because the volume decreases). The volume strain is therefore

Step 3: Pressure change from bulk modulus.

The bulk modulus $B$ is defined by $B = - \frac{\Delta p}{\Delta V / V}$. Hence the excess pressure in the compressed region is

Step 4: Newton’s second law on the compressed slug.

Consider the slug of fluid of mass $\Delta m$ that lies in the compressed region. Its mass is $\rho \times \text{volume} = \rho A v \Delta t$. The net force on this slug comes from the pressure difference across it: the left face experiences the higher pressure $p + \Delta p$, the right face the undisturbed pressure $p$. The net force to the right is

This force acts on the slug for the time $\Delta t$ during which the compression front passes. The slug’s momentum changes from zero to $\Delta m \, u$ (the particles in the compressed region acquire the piston’s speed $u$). By the impulse–momentum theorem:

Substitute $F$ and $\Delta m$:

Cancel $A$, $u$, and $\Delta t$ (all non-zero):

Thus

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Speed of Sound in a Gas: The Role of Temperature

For sound waves in a gas, the bulk modulus $B$ is not a constant; it depends on how the compression and rarefaction occur. Sound propagates so rapidly that there is no time for heat to flow between adjacent compressed and rarefied regions. The process is adiabatic, not isothermal.

For an adiabatic process in an ideal gas, pressure and volume satisfy $p V^\gamma = \text{constant}$, where $\gamma = C_p / C_v$ is the ratio of specific heats. From this relation, one can show that the adiabatic bulk modulus is

Substituting into $v = \sqrt{B/\rho}$ gives

Now use the ideal gas law $p V = n R T$. Writing density $\rho = \frac{M}{V}$, where $M$ is the total mass, we have $p = \frac{\rho R T}{M_0}$, where $M_0$ is the molar mass. Then

This is the standard formula for the speed of sound in an ideal gas. It shows three important features:

1. Speed increases with temperature — as $T$ rises, molecules move faster and transmit disturbances more quickly.
2. Speed is independent of pressure — $p$ cancels out when expressed in terms of $T$ and $M_0$.
3. Speed depends on the gas — lighter gases (small $M_0$) have higher sound speeds; for example, sound travels faster in helium than in air.

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Speed of a Longitudinal Wave in a Solid Rod …