# Speed of sound at temperatures below 0 °C

How can the speed of sound be calculated for temperatures below 0 °C (down to -40 °C)?

Does the calculation $v=331\ \frac{m}{s} + 0.6 \frac{m}{s°C} \times T$ still hold (where T's unit is °C)?

• Not a bad question, but an even better one (I think) would be to ask what the range of validity of that equation is. – David Z Feb 11 '14 at 2:22
• Somewhat off-topic, but super interesting about sounds at low temperatures is the "third sound". – Patrick M Feb 11 '14 at 16:42

The speed of sound in an ideal gas is given by

$$a = \sqrt{\gamma R T}$$

Where $\gamma = \frac{C_p}{C_v}$, $R$ is the specific ideal gas constant and $T$ is the absolute temperature.

Taking standard values for air, this makes a graph like this: The linear approximation is plotted by your formula, $a = 331\ \frac{m}{s}\ +\ 0.6 \frac{m}{sK} (T - 273\ K)$, with the 273 K to convert it to the Kelvin scale.

As you can see, the linear approximation is nearly equal to the actual value in the range marked by the two black lines, from $T \approx 240\space\mathrm{K}$ to $T \approx 350\space\mathrm{K}$.

If you don't care about accuracy so much, you could even stretch your definition to $T\ \epsilon\ [200\space\mathrm{K},375\space\mathrm{K}]$, as shown by the green lines.

The error is:

• $\approx +1.3\%$ at $T=200\space\mathrm{K}$
• $\approx +1.0\%$ at $T=375\space\mathrm{K}$

As seen in the following graph of the percentage error of your approximation between $173\space\mathrm{K}$ and $473\space\mathrm{K}$. Of course, at low temperatures air doesn't behave like an ideal gas, so it all breaks down, but for the purposes of this question, I believe it's a fair assumption.

• the speed of sound approaches 0 as you approach absolute zero? – Michael Feb 11 '14 at 18:23
• As you approach absolute zero air is no longer approximable to an ideal gas, so the relation breaks down anyway – Pranav Hosangadi Feb 11 '14 at 18:29

Wikipedia gives the formula $c_{air}=331.3\sqrt{1+\frac {T(^\circ C)}{273.15}}$, valid anywhere the ideal gas law is valid. The expression you quote is given at the first two Taylor series terms.

I don't know about your formula, but the speed of sound is proportional to the square root of the absolute temperature (for ideal gases, and approximately so in air).

• We have measured the speed of sound at temperatures ranging from 15 F to -42 F and wanted to see how close our measurements are to calculated values. They aren't that close, but we read that the equation above (v=331m/sec + (.06m/s/C) X T) is only valid for the range 0 to 100 C. We also read that there are other equations for calculating the speed of sound in air outside of that range, but can't seem to find them. – user40343 Feb 11 '14 at 2:56

In absolute zero, the molecular vibration is in its least possible extend. Therefor it is almost impossible to fluctuate under the influence of sound wave. Imposing any form of energy including sound energy will cause to increase in temperature . in supposed situation(Absolute Zero ) and a source that keeps the situation stable, sound would not be transmitted!!!

• Note: absolute zero $\neq\ \mathrm 0^\circ\mathrm C$. – AccidentalFourierTransform Dec 30 '17 at 20:36