What is the exact nature of molecular vibration in solids and how does it relate to an object's natural fundamental resonant frequency?

Question

This is my understanding and trail of thought on the topic in the title, and I would like to hear from others with a better understanding of the subject if I am on the right track or not.

I will begin with two presumptions, that I believe are correct:

• An iron tube is in a solid state and is made of up atoms in a perceptibly static state, but at the molecular level each individual atom is vibrating within the confines of its position, with surface level atoms having more movement than deeper level atoms (I imagine the restrictiveness of the motion increases the deeper the atom is placed within the mass of the tube, possibly to a threshold beyond which there is no further change). The molecular vibration of the individual iron atoms is only affected by the amount of kinetic energy they possess as determined by the temperature of the iron tube.
• An iron tube has a natural resonant fundamental frequency that is determined by its composition (density and elemental makeup), shape, size, and temperature. You can determine this frequency simply by striking the tube with another solid object and listening to the sound the tube emits. There is also effect of physical stress and tension, as applied in something like a guitar string, which varies the apparent frequency, but not my focus here.

My question comes down to the relationship between these two properties as they apply to, for example, an iron tube: is the molecular vibration of the atoms within the iron tube in anyway related to the natural fundamental resonant frequency of the iron tube?

I was initially leaning towards "yes" because I thought the natural fundamental resonant frequency might be present at inaudible levels before the tube is struck in order for the energy from the strike to amplify the frequency and make it audible, because you can observe how no matter how hard you strike an iron tube, it will always emit the same audible fundamental frequecy (as well as overtones, but those are not my concern right now).

But I believe that reasoning is flawed. For it seems more possible that the effect of mechanical force is producing the natural fundamental resonant frequency and that frequency is not actually present until outside mechanical forces, such as a strike, are applied, and that the fundamental frequency is defined by the object only being able to vibrate a certain way, regardless of the molecular vibration of the atoms.

It is worth noting that the molecular vibration of the iron atoms within the tube are vibrating at frequencies much higher than the audible sounds we hear, but is the fundamental resonant frequency possibly an overtone/harmonic of the molecular vibrational frequency? I suppose this would assume that the molecular vibration of atoms in a solid are in sync, or in sync to some degree...

Or more resonably, perhaps the structural and resulting acoustic properties are related to the molecular vibrations? Perhaps the molecular vibrations are following a pattern of standing waves, in which case the molecular vibrations could be responsible for the natural nodes that the object has, which effects the fundamental frequency, at the macromolecular level where we observe sound at.

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Answer 1

The core issue is the indirect link between microscopic atomic vibrations (phonons, $\omega = \omega(\mathbf{k})$) and macroscopic resonant frequencies in a solid, specifically an iron tube. It's not a direct harmonic relationship. Resonant frequencies are potential modes, excited by external force, not pre-existing vibrations.

Atomic vibrations, at frequencies far exceeding the audible range, determine macroscopic properties: elastic moduli (E) and density (ρ). These properties govern the material's response to external forces and the resulting standing wave patterns.

Resonant frequencies manifest as standing waves, with nodes and antinodes. The relationship is:

$$ f = \frac{v}{\lambda} $$

where f is the resonant frequency, v is the speed of sound, and λ is the wavelength. The speed of sound is:

$$ v = \sqrt{\frac{E}{\rho}} $$

Both E and ρ are functions of the interatomic potential and thermal vibrations.

The connection is indirect: atomic vibrations, through their influence on E and ρ, dictate the object's response and the standing wave patterns it supports. Resonant frequencies are a macroscopic consequence of these patterns. Atomic vibrations are not the source of resonant frequencies, but a fundamental aspect determining macroscopic response. The material's response is governed by the stress-strain relationship, a function of interatomic potential and thermal vibrations.

In essence, atomic vibrations (phonons) influence material properties (E, ρ), which in turn determine the macroscopic resonant frequencies and standing wave patterns. The macroscopic behavior is an emergent property of collective microscopic interactions.

Answer 2

The thermal motion in a solid object is indeed sound waves, but the 'fundamental' sound waves are those that have wavelengths comparable to the object dimensions; those low-frequency modes have comparatively little real heat energy, and there are few such modes... both the energy per phonon and the density of allowable modes rise with frequency, up to a limit; one such limit is the atomic spacing; atomic spacing is variable enough to support a wave, but inside an atom... those electron shells are too hard to flex with modest temperatures, so there's a minimum wavelength limit (i.e. maximum momentum and maximum energy) to the sound quanta we call phonons.

The closest connection of the heat energy to the fundamental modes is that harmonics of the fundamentals give rise to a prediction, the 'Debye approximation' spectrum of thermal phonon energies.

The ringing-like-a-bell oscillation you create by tapping on the pipe is a relatively small contribution to the thermal motion energy of the pipe; unless you hammer it hard, the iron won't get much warmer. The bell eventually quiets, though, partly by emitting sound into the air, but mainly by 'umklapp' processes that create a thermal-equilibrium mix of frequencies by reflections of partial wave energy into non-fundamental frequencies. The grains of impurity in cast iron make the thermalization happen more rapidly than it would in steel.

Even with a stethoscope, you'd have a hard time hearing the thermal vibrations, partly because your ear has its own temperature animation; a conch shell, though, makes an effective echo chamber when placed over your ear; you can hear some of the heat's sonic spectrum that way.