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?
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.
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!
The point of this answer:
Also, less critical but still important:
The key formulas
\begin{align} c = \sqrt{\gamma R T} \\ \gamma = \frac{C_p}{C_v} \\ C_p = 1.005 + 1.820 \times H \\ C_v = C_p - R \end{align}
$C_i$ being the specific heats ($C_p$, $C_v$).
For the speed of sound ($c$) one Taylor expansion of the second factor, centred at $0$ C (for $\theta$.):
\begin{align} c &= \sqrt{\gamma R T} = \sqrt{\gamma R \times 273.15} \sqrt{1 + \theta/273.15} \\ c &\approx \sqrt{\gamma R 273.15} + 0.606 \theta \end{align}
Where $T$ is the temperature in Kelvin and $\theta$ in Celsius.
For dry air the first term is (~331.3 m/s.) See range of validity.
Varying composition changes $R$, $C_v$ and $C_p$ which makes moist air a bit more complex to calculate.
To link it to specific humidity:
\begin{align} H = \large \frac{m_v}{m_d} &= \frac{P_v}{P} \frac{M_w}{M_d}\\ &\approx X_v \times \frac{18.015}{28.9645} \\ &= \frac{\%RH}{100} \times \frac{P_s}{P} \times 0.622 \end{align}
With atmospheric pressure $P$, $P_v$ water vapour pressure, $M_d=28.9645~\rm{g/mol}$ is the molecular weight of dry air, and $M_w = 18.015 ~\rm{g/mol}$ for water.
Note that we need $P_s$.
We need $P_s$ to calculate $RH$ and $H$. The Arden-Buck equation yields $P_s$ at a specific temperature. And then do: $P_v = \frac{\%RH}{100} \times P_s$.
An enhancement factor can be used, see guide here or the Wikipedia link above.
Misc.
We need $M$ to calculate $R$ which requires the partial pressure of water vapour in the mix.
The molecular weight of a mixture ($M$) will be given by the fraction ($f$) of each specie ($i$):
$$\large M_{mix} = \sum f_i M_i $$
For dry air it is $28.96~\rm{ g/mol}$. More details on Wikipedia: Air Composition
The molar fraction is calculated with Dalton's Law as $X_v = \frac{P_v}{P}$ of water in air, but we need $P_v$. This makes $P_s$ essential (next section.)
Finally, \begin{align} M_{mix} (X_v) &= (1 - X_v) \times M_{dry} + X_v\times 18.015 \\ &= - X_v \times 10.945 + 28.96 \end{align}
And for each molar fraction we get $\large R = \frac{R_u}{M}$.
An answer can be found starting from a different assumption, as done in this research article:
$$\gamma = \frac{C_p + h}{C_v+h}$$
where the heat capacities are of dry air and $h$ is the fraction of it that is water vapour. The addition comes from the heat capacity of water being $\frac{8}{2}R$. This ends up adding $h$.
This expression is interesting because you could use it for multiple different gases other than water vapour.
So for the speed of sound we get:
$$\sqrt{\frac{C_p + h}{C_v+h} \times \frac{R}{M_{moist}} T}$$
Where $R$ is the universal gas constant and $M_{moist} = M_{dry} \times (1 - h) + M_{water} \times h = M_{dry} \times (1-0.378h)$
A useful approximation chops a Taylor expansion of $\sqrt{1+x}$ in the first term, after switching from Kelvin to Celsius (Wikipedia: Speed of sound: Details):
\begin{align} \sqrt{\gamma R T} &= \sqrt{\gamma R 273.15 \times (1 + \frac{\theta}{273.15})} \\ &= \sqrt{\gamma R 273.15 \times (1 + 0.00183 \theta)} \end{align}
An extra assumption can be made for $h \approx 0$ so that $\gamma \approx 1.4$ . Using the previous formula for $M_{moist}$:
\begin{align} \sqrt{\gamma R T} &\approx \frac{331.296}{\sqrt{1-0.378\times h}} \times (1 + 0.00183 \theta) \end{align}
An analogous Taylor expansion gives the new approximation:
\begin{align} \sqrt{\gamma R T} &\approx \frac{331.296}{\sqrt{1-0.378\times h}} \times (1 + 0.00183 \theta)\\ & \approx (331.296 + 0.60627~\theta) \times (1+0.189 \times h) \end{align}
If one does not make the assumption, the formula for gamma is can still be rearranged as:
$$\sqrt{\frac{R}{M_{dry}}\times\frac{7}{5}} \times (1+0.00183\theta) \sqrt{\frac{1+h/7}{1+h/5}\times \frac{1}{1-0.378h}}$$
This corrects the $0.189$ to $0.1607$ which appears in the article.
As it was stated in my other answer $h$ still needs an estimation of the vapour pressure of saturation.
There are many ways to do it, but one that seems accurate is the procedure for the Arden-Buck equation, which was detailed in the previous answer.
