# Is there a resonance frequency to air itself?

Obviously a lot of things cause air to vibrate, but does air have an actual resonance frequency?

When you contain air, or any gas for that matter in a bounded volume $V_o$ (container) at static pressure $P_o$ you can define a somewhat linear constant called Elastance . $$E=\gamma\frac{P_o}{V_o}$$

where $\gamma$ is the specific heat ratio of the gas.

The elastance can be thought of as a bulk property of the contained gas that tends to behave as a spring.

Also if you consider gas inside a long slender container we can define another constant, the Inertance

$$I=\frac{\rho l}{A}$$

where $\rho$ i sthe gas density, $l$ is the length and $A$ the cross sectional area of the container respectively. The inertance can be though of as a bulk property of the contained gas that tends to behave as a bulk mass.

So together if we consider these bulk properties acting together within a closed container we can expect to see a natural frequency, resonance of

$$\omega=\sqrt{\frac{E}{I}}$$ given sufficient energy to drive it relative to what energy losses might be also be present.

From these properties, certain geometric assumptions you can derive an expression for the resonant frequency of a Helmholtz Resonator.

So at least by this modeling you need some geometric boundary to establish a resonant condition within the gas - to sustain the properties. Consider the volume $V_o$ larger and larger at constant $P_o$ ; the Elastance and frequency essentially vanish towards zero.

So I don't believe the bulk properties themselves, without a constraining geometry can sustain a resonant system, at least within the domain of acoustic frequencies (sonic and ultrasonic).

I can imagine there are a few ways to answer this question. In the field of environmental acoustics we are often concerned with energy lost due to atmospheric absorption, as a wavefront moves through the atmosphere.

The equations to estimate atmospheric absorption utilize the vibrational energy relaxation frequencies of diatomic oxygen and diatomic nitrogen, which are approximately $$f_{N_2} \simeq 205 Hz$$ $$f_{O_2} \simeq 56 Hz$$ At 55° C, 70% relative humidity, and a standard atmospheric pressure of 101.325 Pa.
(In the lab researchers report a value of $f_{O_2} = 62.9$ Hz the experiments were done at 30° C, so we expect a faster vibration frequency anyway.)

As @Jwalbrecht2000 mentions, the other molecules may also resonate, but as Nitrogen and Oxygen make up an estimated 99% of the atmosphere this is a great starting estimate.

• thanks for this. to clarify: do the resonances you cite mean that a white-noise source propagated through air will eventually develop "dips" in the spectrum at 56 and 205Hz? – niels nielsen Mar 4 '18 at 6:35
• Thanks for pointing that out Niels, I don’t want to be misleading. What I’ve cited are the relaxation frequencies of diatomic molecules in the atmosphere. The actual absorption depends on multiple variables, including acoustic frequency. If folks are interested there is a plot by frequency here: physics.stackexchange.com/questions/364633/… – D. Betchkal Mar 4 '18 at 16:48

According to the wave-particle duality, all particles have a natural frequency; therefore, since air is made up of particles, resonance applies to air as well. We must also take into account, however, that since air is a homogeneous substance composed of nitrogen, oxygen, argon, and other gases, and the percentages of each gas changes locally, the resonant frequency (or frequencies) of air would be affected by its specific composition.

• Why does the wave-particle duality matter here? – probably_someone Mar 4 '18 at 3:41
• The wave-particle duality explains how matter can exhibit the properties of waves; it is part of my explanation. Is there something wrong with using it here? – Jwalbrecht2000 Mar 4 '18 at 3:53
• What does it have to do with the actual mechanism of resonance? – probably_someone Mar 4 '18 at 3:54
• If matter didn't exhibit wave properties, it wouldn't resonate, would it? – Jwalbrecht2000 Mar 4 '18 at 3:55
• A classical mass-spring system resonates, and that has nothing to do with wave-particle duality. – probably_someone Mar 4 '18 at 3:59