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A flute is a wind instrument, which could be modelled as a resonance cylinder open at both ends. Any cylinder resonates at multiple frequencies. A skilful player produces a standing wave in the flute with anti-nodes at the ends of the tube and nodes forming inside the tube. The fundamental frequency contains exactly half of a standing wave. Higher resonances correspond to the wavelength that are integer divisions of the fundamental wavelength. The harmonics generated by a flute and all the wind instruments on a molecular level are causing a periodic, collective excitation of an extremely large number of molecules of gas, of the order close to the Avogadro Number. In a condensed matter physics this type of oscillations are called phonons. They are a quantum mechanical description of an elementary motion in which a lattice of atoms or molecules oscillates uniformly at a single frequency.T he geometrical constraints of the tube create a one-dimensional lattice or harmonic chain consisting of air molecules. To the first approximation, they could be considered as a uniformly distributed mixture of nitrogen and oxygen originally remaining in the ground state, with pressure variation acting as a restoring force. This would be analogues to the crystalline structure in a solid. If so than Quantum Field Theory could be used to analyse the formation of this collective excitation by the creation and annihilation operators? Air molecules comprise mostly of nitrogen which are bosons, https://www.quora.com/Is-nitrogen-14-a-boson So the collective excitation could be a subject to Bose-Einstein statistics.

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Yes, any kind of normal sound excitations are phonons, obeying Bose-Einstein statistics. The fact that the microscopic excitations are made up of molecules in a gaseous state, rather than fixed into a solid lattice does not affect change things in any fundamental way. Of course, for macroscopic sound waves such as produced in a flute, the phonon occupation numbers are enormous, so the classical theory is quite satisfactory.

However, the fact that the gas particles are not bound together by lattice forces does change the number of phonon modes that can exist. Phonons in a solid can be either transverse (like electromagnetic waves), with the atoms oscillating in either of the two directions perpendicular to the wave propagation direction; or they may be oscillating longitudinally, as compression waves. However, for sound waves in a gas, only the compression waves are possible. In fact, one way of giving a theoretical definition of the difference between a solid and a fluid is that a solid is capable of supporting transverse phonons, while a fluid is not.

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  • $\begingroup$ Can we consider the formation of acoustic waves in the flute as a process similar to the population inversion in lasers? $\endgroup$ – Stan Tarka Dec 16 '18 at 19:45
  • $\begingroup$ The behavior of oscillating tubes full of air can, however, be treated satisfactorily by use of conventional acoustics theory, without resorting to phonon-based analysis. $\endgroup$ – niels nielsen Dec 16 '18 at 21:09
  • $\begingroup$ Conventional acoustics does not explain how in milliseconds the harmonic series emerges from the random oscillations. On the molecular level, the process is similar to synchronization of metronomes youtube.com/watch?v=5v5eBf2KwF8 process, involving Avogadro number of particles. Classical acoustics ignores particle spin. Assume that air molecules are bosons. So if n bosons are already in a quantum state it will enhance the probability of one more joining them by a factor of (1 + n ) in accordance with Bose-Einstein statistics. $\endgroup$ – Stan Tarka Dec 19 '18 at 20:04

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