# How to relate speed of sound with relative humidity?

I am exploring the idea of measuring the humidity of a space using sound waves, however I am having trouble finding a mathematical relationship between the speed of sound and the humidity level.

$c_{air} = 331.3 \sqrt{1 + \frac{T}{273.15}}$ but this is for dry air (0%RH)

How can I factor the effects of humidity into this relationship?

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## 2 Answers

Speed of sound in a gas is given by the equation: $$c = \sqrt{\gamma R T}$$

where $\gamma = c_p/c_v$ ( $c_p$ and $c_v$ are specific heats), $R$ is the gas constant, and $T$ is temperature. The specific heat of a gas changes with humidity, so varying these will vary your calculated speed of sound.

This page has a calculator as well as a great explanation of how their formula works.

Hope this helps!

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The speed of sound in a gas is:

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

where $\gamma = c_p/c_v$ is the ratio of specific heats, $R$ is the specific gas constant and $T$ is temperature. Both $\gamma$ and $R$ depend on the composition of the gas, which includes humidity in air.

The specific heats are $c_p = 1.005+1.82H$ (see this answer) where $H$ is the absolute humidty and $c_v = c_p - R$. Finally, $R = R_{univ}/M_{gas}$ where $M_{gas}$ is the molecular weight of the gas (which depends on humidity).

To get it all in terms of relative humidity is just an exercise in unit conversion.

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