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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)?

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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 at 2:22
    
Somewhat off-topic, but super interesting about sounds at low temperatures is the "third sound". –  Patrick M Feb 11 at 16:42
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3 Answers

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: speed of sound graphs, actual (for ideal gas) and linearly approximated

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 $473\space\mathrm{K}$.

Plot of error of approximation

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.

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

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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).

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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 at 2:56
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