Author: Luuk ★2017
Date: 2023-08-23 14:30
A quick look into some physics books learns me that the speed of sound (and thus the frequency) is influenced by temperature, and to a lesser degree by pressure. And, maybe surprising, humidity has an influence on frequency dependent damping, not on frequency itself.
The speed of sound in air v (m/s) as function of temperature T (C) is given by
v = 331.5 + 0.6 T
Here we see that v changes in a linear way with temperature, which is an important effect, but for this discussion I will focus on pressure and humidity, and assume that temperature is constant.
Now v in a gas is depending on its pressure P (Pa) and its density d (kg/m3).
v = (square root (1.4 P)) / d
But the density of a gas is also proportional to pressure:
d = P / RT
where R is the specific gas constant and T is, of course, temperature.
Combining the last two equations gives us
v = (square root (1.4 P)) / (P / RT) = ((square root (1.4 P)) RT) / P
This means that the speed of sound increases with the square root of the pressure, but it deminishes in a linear way with the same. In the end, the linear dependency will win from the square root and thus v will decrease with growing pressure: sound goes slower at sea level because air is more compressed there.
Now frequency is defined by f = v / l, where l = wavelenght (m).
Let's assume a clarinet does not change it's length when pressure is changing. Thus, the wavelength is always the same for the same fingering. However, since v changes with pressure, f is also changing, according to the last formula. With increasing v we will get increasing f. And we have seen that v decreases with P. Thus, the frequency should become lower at higher pressures (= lower altitudes).
I have filled in all constants in the equations, and a comparison of the frequency of a given wavelength at different heights above sea level yields:
f @ sea level = 100%
f @ 1000m or 3280 ft = 106%
f @ 2000m or 6561 ft = 113%
f @ 3000m or 9842 ft = 120%
This means a clarinet which is tuned at sea level, will be about 5% too high at 3000ft, which is somewhat less than a half tone. This result seems too large to me, but it indicates there should be a remarkable effect.
Now humidity only has a very small effect on frequency (think about 0.3% when going from 0 to 100% RH), but it does influence damping. The interesting part is that this damping depends on frequency, working in such a way that higher frequencies are damped more than lower frequencies. I have found no simple formulas to explain and/or compute this effect, but I found lots of graphics on the internet illustrating this (for instance https://www.engineeringtoolbox.com/air-speed-sound-attenuation-humidity-frequency-d_2161.html or https://ebrary.net/134853/environment/absorption). In the end, the effect is that dry air makes the sound more bright, especially over large distances.
Of course all this is at best half the story: there is no mention of a reed or a human body. But these formulas may offer part of the explanation.
Regards,
Luuk
Philips Symphonic Band
The Netherlands
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