Dear Ismail,

sound waves are usually described by classical physics, but they also display quantum behaviour; particularly at high frequencies or in specialised settings. The quantum particles associated with sound waves are phonons! Which represent quantised lattice vibrations and act as the true quantum carriers of sound energy. Quantum acoustics is the field that examines how these phonons interact with quantum systems.

Recent advances suggest that traditional classsical approaches are insufficient to fully explain sound wave phenomena; especially regarding volume control and scattering in high-frequency audio environments. These effects can be better understood using advanced quantum frameworks. For instance a modern form of the Schroedinger eqn. developed after 2020.

Sound waves exhibit wave-particle duality similar to light, meaning they can behave both as waves and as particles. While the particle nature of sound is less apparent in everyday experience, recognising this quantum aspect is essential to accurately describe and study sound at a fundamental level.

Hope, it hleps ;-}

The core of the statistical theory, called Cairo techniques, is based on matrix/tensor B transition chains in a closed volume or control volume, which allow for an exact simulation of nature in all time-dependent physical and mathematical processes.

In conclusion, the hypothesis that the statistical theory, called Cairo techniques, is the unified theory for all domains, or the theory of everything, has been successfully proven since 2020, generating more than 50 original papers [1,2] in various fields of classical physics, quantum physics, and mathematics.

The theory asserts that there is nothing new in quantum mechanics. The solution to the Schröيهnger equation is simply the square root of the solution to the heat diffusion equation. And the transition matrix of the quantum system Q is the square root of the transition matrix of the heat diffusion system B.

In other words, Q = √B. The simplest 8x8 matrix B for a three-dimensional cube is therefore:

The simplest3D matrix B for 8 free vertices or nodes is:

B8x8=

0 1/6 1/6   0 1/6   0   0   0

1/6   0   0 1/6   0 1/6   0   0

1/6   0   0 1/6   0   0 1/6   0

0 1/6 1/6   0   0   0   0 1/6

1/6   0   0   0   0 1/6 1/6   0

0 1/6   0   0 1/6   0   0 1/6

0   0 1/6   0 1/6   0   0 1/6

0   0   0 1/6   0 1/6 1/6   0

Quantum mechanics implies that the quantum transition matrix Q is the square root of the classical transition matrix B,

Q = √B

Therefore, the transition matrix Q is given by the following matrix,

((3+3*i)*104166670^0.5+(1+i)*312500010^0.5)/200000    ((1-i)*104166670^0.5+(1-i)*312500010^0.5)/200000    ((1-i)*104166670^0.5+(1-i)*312500010^0.5)/200000   ((-1-i)*104166670^0.5+(1+i)*312500010^0.5)/200000    ((1-i)*104166670^0.5+(1-i)*312500010^0.5)/200000   ((-1-i)*104166670^0.5+(1+i)*312500010^0.5)/200000   ((-1-i)*104166670^0.5+(1+i)*312500010^0.5)/200000 ((-3+3*i)*104166670^0.5+(1-i)*312500010^0.5)/200000

((1-i)*104166670^0.5+(1-i)*312500010^0.5)/200000 ((3+3*i)*104166670^0.5+(1+i)*312500010^0.5)/200000   ((-1-i)*104166670^0.5+(1+i)*312500010^0.5)/200000    ((1-i)*104166670^0.5+(1-i)*312500010^0.5)/200000   ((-1-i)*104166670^0.5+(1+i)*312500010^0.5)/200000    ((1-i)*104166670^0.5+(1-i)*312500010^0.5)/200000 ((-3+3*i)*104166670^0.5+(1-i)*312500010^0.5)/200000   ((-1-i)*104166670^0.5+(1+i)*312500010^0.5)/200000

((1-i)*104166670^0.5+(1-i)*312500010^0.5)/200000   ((-1-i)*104166670^0.5+(1+i)*312500010^0.5)/200000 ((3+3*i)*104166670^0.5+(1+i)*312500010^0.5)/200000    ((1-i)*104166670^0.5+(1-i)*312500010^0.5)/200000   ((-1-i)*104166670^0.5+(1+i)*312500010^0.5)/200000 ((-3+3*i)*104166670^0.5+(1-i)*312500010^0.5)/200000    ((1-i)*104166670^0.5+(1-i)*312500010^0.5)/200000   ((-1-i)*104166670^0.5+(1+i)*312500010^0.5)/200000

((-1-i)*104166670^0.5+(1+i)*312500010^0.5)/200000    ((1-i)*104166670^0.5+(1-i)*312500010^0.5)/200000    ((1-i)*104166670^0.5+(1-i)*312500010^0.5)/200000 ((3+3*i)*104166670^0.5+(1+i)*312500010^0.5)/200000 ((-3+3*i)*104166670^0.5+(1-i)*312500010^0.5)/200000   ((-1-i)*104166670^0.5+(1+i)*312500010^0.5)/200000   ((-1-i)*104166670^0.5+(1+i)*312500010^0.5)/200000    ((1-i)*104166670^0.5+(1-i)*312500010^0.5)/200000

((1-i)*104166670^0.5+(1-i)*312500010^0.5)/200000   ((-1-i)*104166670^0.5+(1+i)*312500010^0.5)/200000   ((-1-i)*104166670^0.5+(1+i)*312500010^0.5)/200000 ((-3+3*i)*104166670^0.5+(1-i)*312500010^0.5)/200000 ((3+3*i)*104166670^0.5+(1+i)*312500010^0.5)/200000    ((1-i)*104166670^0.5+(1-i)*312500010^0.5)/200000    ((1-i)*104166670^0.5+(1-i)*312500010^0.5)/200000   ((-1-i)*104166670^0.5+(1+i)*312500010^0.5)/200000

((-1-i)*104166670^0.5+(1+i)*312500010^0.5)/200000    ((1-i)*104166670^0.5+(1-i)*312500010^0.5)/200000 ((-3+3*i)*104166670^0.5+(1-i)*312500010^0.5)/200000   ((-1-i)*104166670^0.5+(1+i)*312500010^0.5)/200000    ((1-i)*104166670^0.5+(1-i)*312500010^0.5)/200000 ((3+3*i)*104166670^0.5+(1+i)*312500010^0.5)/200000   ((-1-i)*104166670^0.5+(1+i)*312500010^0.5)/200000    ((1-i)*104166670^0.5+(1-i)*312500010^0.5)/200000

((-1-i)*104166670^0.5+(1+i)*312500010^0.5)/200000 ((-3+3*i)*104166670^0.5+(1-i)*312500010^0.5)/200000    ((1-i)*104166670^0.5+(1-i)*312500010^0.5)/200000   ((-1-i)*104166670^0.5+(1+i)*312500010^0.5)/200000    ((1-i)*104166670^0.5+(1-i)*312500010^0.5)/200000   ((-1-i)*104166670^0.5+(1+i)*312500010^0.5)/200000 ((3+3*i)*104166670^0.5+(1+i)*312500010^0.5)/200000    ((1-i)*104166670^0.5+(1-i)*312500010^0.5)/200000

((-3+3*i)*104166670^0.5+(1-i)*312500010^0.5)/200000   ((-1-i)*104166670^0.5+(1+i)*312500010^0.5)/200000   ((-1-i)*104166670^0.5+(1+i)*312500010^0.5)/200000    ((1-i)*104166670^0.5+(1-i)*312500010^0.5)/200000   ((-1-i)*104166670^0.5+(1+i)*312500010^0.5)/200000    ((1-i)*104166670^0.5+(1-i)*312500010^0.5)/200000    ((1-i)*104166670^0.5+(1-i)*312500010^0.5)/200000 ((3+3*i)*104166670^0.5+(1+i)*312500010^0.5)/200000

At first glance, we notice that √2 is scattered among the elements of the vector Q, and that its eigenvalue/energy eigenvector is based on √2.

The factor √2 represents the Pythagorean theorem or the mutual perpendicularity of x, y, z, and t.

At first glance, we notice that √2 is scattered among the elements of the vector Q, and that its eigenvalue/energy eigenvector is based on √2.

The factor √2 represents the Pythagorean theorem or the mutual perpendicularity of x, y, z, and t.

The eigenvalue equation of the matrix/tensor Q is:

Q . IC = Constant or eigenvalue * IC . . . (1)

where IC is an arbitrary initial condition or eigenstate, simply assumed to be arbitrary like [1,1,1,1,1,1,1,1]

Solution to Equation 1 or energy vector in arbitrary units for the case IC arbitrary chosen as,

IC=[1 1 1 1 1 1 1 1 ]

Is,

0.70710679

0.70710679

0.70710679

0.70710679

0.70710679

0.70710679

0.70710679

0.70710679

This means that the transition matrix  Q has an energy/eigenvector eigenvalue of ,

0.70710679.

Which is exactly equal to √2/2.

(0.707106769)

This remarkable 8-digit precision is quite striking.

The Q matrix states that, when two or more energy/eigenvector states are superimposed, one must multiply by 0.70710679.

Is this convincing enough?

The sound room can be described by a classical transition matrix and/or the square of a quantum transition matrix.

Sound waves, although often treated classically, exhibit quantum effects, particularly in volume control in high-frequency audio rooms.

Sound waves fall partly under classical wave physics and partly under quantum mechanical theory.

1-Book, Fundamentals of Artificial Intelligence.

2-Book, Foundations of Quantum Mechanics.

https://youtu.be/eNmjWjpxUOM

https://youtu.be/QxCeHLTfN1k

https://www.youtube.com/shorts/2sW_IKP382Q?feature=share

The three video examples [Google search] above of sound waves emitted by a brass pot show that:

1- The upward-directed sound wave is reflected downward by the upper edge of the pot.

2- The downward-directed, or reflected, sound wave is reflected by much less sound energy, which is a quantum effect.

3- The interference from the upward-directed wave (452 ​​Hz) interferes with the downward-reflected wave (450 Hz), resulting in beats of 452-450 = 2 Hz, easily and clearly recognizable to the human ear.

4- The same phenomenon occurs in church bells, but with higher intensity and without beats. The sound beats are virtually eliminated thanks to the shape of the bell.

Have you looked at acoustic metrics?

This might be what you're looking for:

Article Acoustic black holes: Horizons, ergospheres and Hawking radiation

Acoustic metrics seem to be "dual" with quantum mechanics, in that they support the classical counterpart of Hawking radiation.

So yes, there would seem to be a significant overlap between the statistic modelling of acoustics and the behaviours shown by quantum mechanics.