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Though vibration is a common concept in sound and “high temperature”, the essential difference between the two is mentioned to be the way the vibration takes place. The former has vibrations that follow certain characteristics (has got an amplitude, frequency, wavelength, speed, etc.) while the latter is random.

The definition is crystal clear in differentiating sound and “high temperature”. But what is the physical reason of our ears not perceiving random vibration as “sound” when the ears are not aware of the difference in the definitions?

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    $\begingroup$ The threshold of hearing is close to thermal noise at 300 K. Possible duplicate of How loud is the thermal motion of air molecules? $\endgroup$ – Pieter Jun 28 '19 at 11:35
  • $\begingroup$ Typical collision frequency of air molecules is 10 GHz, For condensed phases even more. Our ears are extremely insensitive in Gigahertz acoustic band, aside of minimal spectral density of the thermal noise. $\endgroup$ – Poutnik Jun 28 '19 at 15:59
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It is very important to distinguish between vibration and oscillation.

Vibration is basically defined as mechanical oscillation.

Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point.

https://en.wikipedia.org/wiki/Vibration

Now you are asking what the difference between temperature and sound is.

You are assuming that both temperature and sound are caused by vibrational motions of the molecules (let's assume air in your case.)

A molecular vibration occurs when atoms in a molecule are in periodic motion while the molecule as a whole has constant translational and rotational motion. The frequency of the periodic motion is known as a vibration frequency, and the typical frequencies of molecular vibrations range from less than 10^13 to approximately 10^14 Hz, corresponding to wavenumbers of approximately 300 to 3000 cm^−1.

https://en.wikipedia.org/wiki/Molecular_vibration

Now the misconception is, that in air, you assume, that vibrational motions of the molecules is dominant. In reality in gases, translational and rotational energies of the molecules is dominant, and this is what we identify with internal energy (temperature) in the gases. Vibrational motion of the gas molecules is minimal relatively.

Finally, you were right: in most solids, vibration and translation are prominent in internal energy, while in common gases, translation and rotation are prominent while vibration is negligible. Yet, in both cases, relatively complex models are needed to explain what is measured experimentaly.

vibrational motion in gases

In solids, it is the opposite. Vibrational motions of the molecules are dominant in determining temparature.

But you are talking about sound, and you are assuming air.

Now you are asking why we hear sounds, but not the translational and rotational motion of the air molecules (not vibrational).

In physics, sound is a vibration that typically propagates as an audible wave of pressure, through a transmission medium such as a gas, liquid or solid.

https://en.wikipedia.org/wiki/Sound

You are correct to say, that sound is defined as a vibration that propagates as a wave in air.

Humans can only hear sound waves as distinct pitches when the frequency lies between about 20 Hz and 20 kHz.

In air, sound only propagates as a longitudinal wave, and it is important to understand that the particles of air themselves do not propagate with the sound.

Now when we hear with our ears, we are able to hear waves of air, that is our ears can hear from 20 Hz to 20 kHz. Internal energy (temperature) vibrations are usually between 10^13 to 10^14 Hz.

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    $\begingroup$ You are using the term heat at various points in your answer where you should be using the either the term internal energy if referring vibrational, rotational or translational kinetic energy or temperature if referring to the measurement of translational kinetic energy. Heat is energy transfer due solely to temperature difference $\endgroup$ – Bob D Jun 28 '19 at 11:22
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    $\begingroup$ @BobD thank you I edited. $\endgroup$ – Árpád Szendrei Jun 28 '19 at 11:26
  • $\begingroup$ Not the right answer. Thermal noise is at any frequency. Also in the audio range. $\endgroup$ – Pieter Jun 28 '19 at 11:43
  • $\begingroup$ @Pieter what do you base your statements on? can you give references? Any that shows thermal based vibration of air molecules is anywhere inside the 20Hz-20KHz? Wiki specifically states molecular vibrational frequency is only between 10^13-10^14 Hz. $\endgroup$ – Árpád Szendrei Jun 28 '19 at 11:52
  • $\begingroup$ @ÁrpádSzendrei If you have a volume (box, tube, room) with air, there will be resonant modes at many frequencies, I do not think you need a reference for that? Those modes will have an average amplitude that depends on temperature. $\endgroup$ – Pieter Jun 28 '19 at 11:57
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Thermodynamic principles say that molecular motions should produce a white noise background, but a back-of-envelope calculation shows that the resulting white noise will be far below the threshold of audition for any given tone.

The physiological threshold of audition for a well-defined tone a little over 0 dB, or 1 pW/m² in air. If the capture area of the ear shell is roughly 1 cm² , the minimum audible signal would be -160 dBW.

The thermal-noise background would produce a signal kTB, where B denotes the effective bandwidth of cochlear detector cells, reciprocal to their time resolution, say 125 ms, the duration of notes in a fast run on a piano. (This crude estimate of B is also consistent with quarter-tone pitch discrimination in the range of 256-512 Hz, where the threshold is lowest.) Using kT = -204 dBJ and B = +9 dBHz, we get kTB = -195 dBW.

The actual frequency range of audition is almost 10 kHz, but there is not a shred of evidence that neural machinery integrates subliminal signals over such a broad bandwidth. (If it could do so efficiently, the thermal noise might be almost audible.) It is important to understand the difference between coherent and non-coherent integration. Coherent integration sums amplitudes; non-coherent integration sums powers. Signals with different frequencies (hence random phase relationships) could in principle be summed non-coherently, but the inefficiency of non-coherent integration is well known to radar engineers. Pulsed radars typically use signal bandwidths around 1 MHz (reciprocal to pulse width and/or time resolution), but integration bandwidths as low as 100 Hz (reciprocal to beam dwell time). Coherent integration of N pulses gives an integration gain of N, non-coherent integration an integration gain ~ $\sqrt{N}$.

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