Moisture and the Speed of Sound

Many of us first learned about the speed of sound as children when we were taught that counting the seconds between the flash of lightning and the sound of the thunder provided an estimate of the distance to the lightning strike (around 5 seconds per mile).  Later, perhaps, we learned that the speed of sound is calculated by sqrt(k*R*T) where k is the specific heat ratio, R is the gas constant for a particular gas, and T is the temperature.  Obviously, the speed of sound varies with the temperature, but it also varies with the makeup of the air (through k and R).  So, does humidity have a significant effect on the speed of sound?  We’ll explore that question today.

 This figure shows the speed of sound as a function of temperature in dry air.  The variation looks linear, but that is just because it is over such a small temperature range.  You can see that over typical ambient temperatures, the speed of sound doesn’t change a lot.  It is about 337 m/s at 10 °C, and 363 m/s at 55 °C—a difference of about 7.7%.

In considering humid air, it would be unusual to have an ambient humidity ratio much above 0.035.  We can calculate the values for k and for R for various humidity ratios, then use those to calculate the speed of sound as a function of temperature:

 

The lines are different lengths because you can’t have, for example, a humidity ratio of 0.030 at a temperature of 20 °C; the moisture would condense out of the air.  It is clear that the moisture doesn’t have a huge effect on the speed of sound, although it does affect it some.

This figure shows a different view of the same thing--the speed of sound as a function of humidity ratio for four different temperatures.  At 35 °C, the difference between dry air and a humidity ratio of 0.035 is about 1.5%.

Over the small ranges of temperature and humidity that we are considering here, the variations are small and this equation predicts the speed of sound to within 0.06%:

a=331.74 m/s + (0.5732 m/s/°C)*T+(151.33 m/s)*ω

where a is the speed of sound in m/s, T is the temperature in °C, and ω is the humidity ratio.

Most important, these changes are much too small to notice in calculating the distance to a lightning strike.  For example, at 35 °C, the time difference between completely dry air (unlikely during a thunderstorm) and humidity ratio of 0.035 is only about 67 milliseconds per mile.