In physics, the **speed of sound** is the distance traveled per unit of time by a sound wave through a medium. It is highest for stiff solids and lowest for gases. There is no sound or speed of sound in a vacuum because sound (unlike light) requires a medium in order to propogate.

### What Is the Speed of Sound?

Usually, conversations about the speed of sound refer to the speed of sound of dry air (humidity changes the value). The value depends on temperature.

- at 20
*°*C or 68*°*F:**343 m/s**or**1234.8 kph**or**1125ft/s**or**767 mph** - at 0
*°*C or 32*°*F:**331 m/s**or**1191.6 kph**or**1086 ft/s**or**740 mph**

### Mach Numher

The **Mach number** is the ratio of air speed to the speed of sound. So, an object at **Mach 1** is traveling at the speed of sound. Exceeding Mach 1 is breaking the sound barrier or is **supersonic**. At Mach 2, the object travels twice the speed of sound. Mach 3 is three times the speed of sound, and so on.

Remember that the speed of sound depends on temperature, so you break sound barrier at a lower speed when the temperature is colder. To put it another way, it gets colder as you get higher in the atmosphere, so an aircraft might break the sound barrier at a higher altitude even if it does not increase its speed.

### Solids, Liquids, and Gases

The speed of sound is greatest for solids, intermediate for liquids, and lowest for gases:

**v _{solid} > v_{liquid} >v_{gas}**

Particles in a gas undergo elastic collisions and the particles are widely separated. In contrast, particles in a solid are locked into place (rigid or stiff), so a vibration readily transmits through chemical bonds.

Here are examples of the difference between the speed of sound in different materials:

- Diamond (solid): 12000 m/s
- Copper (solid): 6420 m/s
- Iron (solid): 5120 m/s
- Water (liquid) 1481 m/s
- Helium (gas): 965 m/s
- Dry air (gas): 343 m/s

Sounds waves transfer energy to matter via a compression wave (in all phases) and also shear wave (in solids). The pressure disturbs a particle, which then impacts its neighbor, and continues traveling through the medium. The **speed** is how quickly the wave moves, while the **frequency** is the number of vibrations the particle makes per unit of time.

### The Hot Chocolate Effect

The hot chocolate effect describes the phenomenon where the pitch you hear from tapping a cup of hot liquid rises after adding a soluble powder (like cocoa powder into hot water). Stirring in the powder introduces gas bubbles that reduce the speed of sound of the liquid and lower the frequency (pitch) of the waves. Once the bubbles clear, the speed of sound and the frequency increase again.

### Speed of Sound Formulas

There are several formulas for calculating the speed of sound. Here are a few of the most common ones:

For gases these approximations work in most situations:

For this formula, use the Celsius temperature of the gas.

**v = 331 m/s + (0.6 m/s/C)•T**

Here is another common formula:

**v = (γRT) ^{1/2}**

- γ is the ratio of specific heat values or adiabatic index (1.4 for air at STP)
- R is a gas constant (282 m
^{2}/s^{2}/K for air) - T is the absolute temperature (Kelvin)

The Newton-Laplace formula works for both gases and liquids (fluids):

**v = (K _{s}/ρ)^{1/2}**

- K
_{s}is the coefficient of stiffness or bulk modulus of elasticity for gases - ρ is the density of the material

So solids, the situation is more complicated because shear waves play into the formula. There can be sound waves with different velocities, depending on the mode of deformation. The simplest formula is for one-dimensional solids, like a long rod of a material:

**v = (E/ρ) ^{1/2}**

- E is Young’s modulus
- ρ is the density of the material

Note that the speed of sound *decreases* with density! It increases according to the stiffness of a medium. This is not intuitively obvious, since often a dense material is also stiff. But, consider that the speed of sound in a diamond is much faster than the speed in iron. Diamond is less dense than iron and also stiffer.

### Factors That Affect the Speed of Sound

The primary factors affecting the speed of sound of a fluid (gas or liquid) are its temperature and its chemical composition. There is a weak dependence on frequency and atmospheric pressure that is omitted from the simplest equations.

While sound travels only as compression waves in a fluid, it also travels as shear waves in a solid. So, a solid’s stiffness, density, and compressibility also factor into the speed of sound.

### Speed of Sound on Mars

Thanks to the Perseverance rover, scientists know the speed of sound on Mars. The Martian atmosphere is much colder than Earth’s, its thin atmosphere has a much lower pressure, and it consists mainly of carbon dioxide rather than nitrogen. As expected, the speed of sound on Mars is slower than on Earth. It travels at around 240 m/s or about 30% slower than on Earth.

What scientists did *not* expect is that the speed of sound varies for different frequencies. A high pitched sound, like from the rover’s laser, travels faster at around 250 m/s. So, for example, if you listened to a symphony recording from a distance on Mars you’d hear the various instruments at different times. The explanation has to do with the vibrational modes of carbon dioxide, the primary component of the Martian atmosphere. Also, it’s worth noting that the atmospheric pressure is so low that there really isn’t any much sound at all from a source more than a few meters away.

### Speed of Sound Example Problems

#### Problem #1

Find the speed of sound on a cold day when the temperature is 2 *°*C.

The simplest formula for finding the answer is the approximation:

v = 331 m/s + (0.6 m/s/C) • T

Since the given temperature is already in Celsius, just plug in the value:

v = 331 m/s + (0.6 m/s/C) • 2 C = 331 m/s + 1.2 m/s = 332.2 m/s

#### Problem #2

You’re hiking in a canyon, yell “hello”, and hear an echo after 1.22 seconds. The air temperature is 20 *°*C. How far away is the canyon wall?

The first step is finding the speed of sound at the temperature:

v = 331 m/s + (0.6 m/s/C) • T

v = 331 m/s + (0.6 m/s/C) • 20 C = 343 m/s (which you might have memorized as the usual speed of sound)

Next, find the distance using the formula:

d = v• T

d = 343 m/s • 1.22 s = 418.46 m

But, this is the round-trip distance! The distance to the canyon wall is half of this or 209 meters.

#### Problem #3

If you double the frequency of sound, it double the speed of its waves. True or false?

This is (mostly) false. Doubling the frequency halves the wavelength, but the speed depends on the properties of the medium and not its frequency or wavelength. Frequency only affects the speed of sound for certain media (like the carbon dioxide atmosphere of Mars).

### References

- Everest, F. (2001).
*The Master Handbook of Acoustics*. New York: McGraw-Hill. ISBN 978-0-07-136097-5. - Kinsler, L.E.; Frey, A.R.; Coppens, A.B.; Sanders, J.V. (2000).
*Fundamentals of Acoustics*(4th ed.). New York: John Wiley & Sons. ISBN 0-471-84789-5. - Maurice, S.; et al. (2022). “In situ recording of Mars soundscape:. Nature. 605: 653-658. doi:10.1038/s41586-022-04679-0
- Wong, George S. K.; Zhu, Shi-ming (1995). “Speed of sound in seawater as a function of salinity, temperature, and pressure”.
*The Journal of the Acoustical Society of America*. 97 (3): 1732. doi:10.1121/1.413048