3.2.5.1. Speed of Sound

Table of Contents

What the speed of sound means

The speed of sound is the rate at which a sound disturbance travels through a material. When a source vibrates, it creates regions of compression and rarefaction in the medium, and these disturbances move outward. The particles of the medium do not travel with the sound over long distances, instead they oscillate around their equilibrium positions while the wave itself carries energy forward.

For sound, the symbol $v$ is commonly used for wave speed. If a sound wave has frequency $f$ and wavelength $\lambda$, then its speed is

$$
v = f\lambda
$$

This relation is true for all periodic waves. For sound, it tells us that if the medium changes, the wave speed changes, and the wavelength changes accordingly if the frequency remains fixed by the source.

Important relation for sound waves:
$$
v = f\lambda
$$
The frequency is set by the source, but the speed depends on the medium.

Why sound has a finite speed

Sound needs a material medium to travel. Its speed is determined by two competing properties of that medium. A stiffer medium transmits disturbances more quickly, while a denser medium tends to slow them down because the particles have more inertia.

In a simple qualitative form,

$$
\text{speed of sound} \propto \sqrt{\frac{\text{stiffness}}{\text{density}}}
$$

This explains why sound travels faster in solids than in liquids, and faster in liquids than in gases. Solids are usually much stiffer, so even though they may be dense, the stiffness effect often dominates.

Speed of sound in different media

In fluids, especially gases and liquids, the relevant measure of stiffness is the bulk modulus $B$. The speed of sound is

$$
v = \sqrt{\frac{B}{\rho}}
$$

where $\rho$ is the density of the medium.

For a gas, a more specific expression can be written as

$$
v = \sqrt{\frac{\gamma P}{\rho}}
$$

where $\gamma$ is the ratio of specific heats, and $P$ is the pressure of the gas.

Using the ideal gas law, this can also be written as

$$
v = \sqrt{\frac{\gamma RT}{M}}
$$

where $R$ is the gas constant, $T$ is the absolute temperature, and $M$ is the molar mass.

This form shows an important result, in an ideal gas the speed of sound depends mainly on temperature and the type of gas, not directly on pressure when temperature is fixed.

For sound in a fluid:
$$
v = \sqrt{\frac{B}{\rho}}
$$
For an ideal gas:
$$
v = \sqrt{\frac{\gamma RT}{M}}
$$
In gases, increasing temperature increases the speed of sound.

Sound in air

In dry air at about room temperature, the speed of sound is approximately

$$
v \approx 343\ \text{m/s}
$$

at $20^\circ\text{C}$.

A useful approximation in air is

$$
v \approx 331 + 0.6\,T
$$

where $T$ is the temperature in degrees Celsius and $v$ is in $\text{m/s}$.

This shows that sound travels faster in warmer air. Warmer air has molecules moving more rapidly, so pressure disturbances are passed along more quickly.

Comparison across materials

The speed of sound differs greatly from one medium to another.

Medium	Approximate speed of sound
Air at $20^\circ\text{C}$	$343\ \text{m/s}$
Water	$1480\ \text{m/s}$
Steel	$5000\ \text{m/s}$

These values are approximate, but they clearly show the trend that sound generally travels slowest in gases and fastest in solids.

What changes when sound enters a new medium

When sound passes from one medium into another, its speed usually changes. The frequency does not change because it is determined by the source. Since $v = f\lambda$, a change in speed causes a change in wavelength.

If sound goes from air into water, the speed increases greatly. Because the frequency stays the same, the wavelength must become larger in water.

At a boundary between media:
Frequency stays the same, speed may change, wavelength changes.
$$
v = f\lambda
$$

Measuring the speed of sound

The speed of sound can be measured by timing how long a sound takes to travel a known distance. If a sound travels a distance $d$ in time $t$, then

$$
v = \frac{d}{t}
$$

A common everyday example is an echo. If you clap near a wall and hear the echo after a time interval $\Delta t$, the sound has traveled to the wall and back, so the distance to the wall is

$$
d = \frac{v\Delta t}{2}
$$

This is useful because the sound covers twice the one way distance.

A simple picture of sound propagation

The following sketch shows compressions moving through air from a vibrating source.

Rendering scientific sketch…

Sound wave moving through air

In this picture, the compressed regions repeat at regular intervals. The distance between two successive compressions is the wavelength $\lambda$.

Key ideas to remember

The speed of sound tells us how fast a sound disturbance moves through a medium. It is not a universal constant, because it depends on the material. In general, sound travels faster in stiffer media and slower in denser media. In air, the speed increases with temperature. The basic wave relation

$$
v = f\lambda
$$

connects speed, frequency, and wavelength, and helps explain what happens when sound moves from one medium to another.

Key facts:
$$
v = f\lambda
$$
$$
v_{\text{fluid}} = \sqrt{\frac{B}{\rho}}
$$
Sound requires a medium and travels faster in warmer air.

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