# Q & A: upper limit to sound frequencies

Q:
Is there an upper frequency limit to ultrasound (or mechanical vibrations)? (in all media, gas, liquid, solid, plasma etc.) if so why? and what determines an upper limit? what are the limiting factors/mechanisms?
- Anonymous
A:

That's a nice question.

Let's think about solids first. The higher the frequency, the shorter the wavelength.  That holds even though the speed of sound may depend somewhat on wavelength. Now think of frequencies so high that the the wavelength is about equal to the spacing between the atoms in the crystal. If you try to describe a sound wave that consists of waves in the positions of the atoms, it doesn't make sense to talk about wavelengths shorter than the spacing. There's no atom in-between to vibrate the opposite direction. So there's a maximum frequency, the Debye frequency, for any particular type of sound waves in any direction in a given crystal. That frequency varies a bit depending on direction and a lot depending on crystal type.

The argument for a liquid is similar to that for a crystal, although there's not quite the same regularity in the spacings.

What about a gas? Molecules or atom in a gas keep moving freely for a little while, until they bump into another particle. Thus the restoring forces that are needed to make a wave propagate can't take effect on time scales shorter than that typical collison time. So you can't have sound waves with frequencies greater than about the inverse of that collison time.

Mike W.

(published on 07/29/2014)

## Follow-Up #1: Why sound propagates

Q:
Thank you, Mike W. for your explanation. so, the maximum frequency (the Debye frequency) exists for "propagating" sound. but if it is "not propagating", what is the upper limit for the frequency? For example, a single atom (or an elementary particle) can vibrate (=back and forth, without propagating) at higher frequencies beyond the Debye frequency? What is the upper frequency limit for this kind of vibration or oscillation? And, how is this model applicable (or not applicable) to other waves in nature? For examples, if 1) an upper frequency limit to photon (electromagnetic wave) or 2) a (on average) non-propagating photon (at extremely high frequencies) is observed, does it indicate electromagnetic field is made of something granular or atomized or particle-like constituents with finite size? Does the Debye frequency (or the maximum frequency) exits for each wave in (fundamental?) fields (=photon, electron, gluon, gravitational, Higgs fields etc.)?
- Anonymous
A:

The picture you suggest of a single atom vibrating isn't physically realistic. (That doesn't mean it's useless. It's the picture Einstein used for the first crude picture of the heat capacity of solids, before Debye fixed it up.) As the atom moves say upward, the next atom up will push down on it and the next one below will push less. It's precisely those forces that cause it to oscillate. Newton's Third Law tells us that if they are pushing on it, it's pushing back on them. So they'll accelerate, push on their neighbors, etc. The actual stable modes of oscillation in solids are not individual vibrating atoms but collective propagating sound waves.That's something the laws of nature tell us, not something we choose for descriptive purposes.

As for the other types of waves you mention, they're all "fundamental" in that they exist in fields which are, so far as we now know, continuous. The familiar wave-like properties would only be lost on frequency scales so high that our current laws of physics fail. For example, ordinary gravity waves should exist at frequencies up to around the Planck frequency, around 1044 Hz. Above that frequency, the non-quantum description of gravity should break down, and there's not yet a good quantum theory of gravity. Some people (not me) even think that on that scale spacetime may be described by a sort of discrete set of little pieces following some sort of cellular automata rules. If they're right, then there would be something quite a bit like the Debye frequency even for gravity waves.

Mike W.

(published on 08/03/2014)