Mean free path: Difference between revisions

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In molecular physics, mean free path is the mean distance a molecule, atom, or other gas unit can travel before collision with other particles.

In molecular physics, '''mean free path''' is the mean distance a molecule, atom, or other gas unit can travel before collision with other particles. It determines the largest frequency that can possibly propagate in a gas. Any larger frequencies will not propagate, owing to the fact that molecules will not transfer the pressure wave.

It is defined by

In air at <abbr title="Standard temperature and pressure ">STP</abbr>, mean free path is about 68 nm. Together with the sound speed of 340 m/s, the maximum physically possible frequency is about 5 GHz.

== Definition ==

Mean free path is defined by

:<math>\ell = \frac{\mu}{p} \sqrt{\frac{\pi k_\text{B} T}{2 m}},</math>

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where:

<math>\ell</math> is the mean free path,

: <math>\ell</math> is the mean free path,

<math>m</math> is the mass of the particle

: <math>m</math> is the mass of the particle,

<math>M</math> is the molar mass of the substance

: <math>M</math> is the molar mass of the substance,

<math>\mu</math> is the gas's viscosity

: <math>\mu</math> is the gas's viscosity,

<math>R</math> is the universal gas constant

: <math>R</math> is the universal gas constant,

Kb is Boltzmann constant,

: kB is the Boltzmann constant,

T is temperature

: T is the temperature,

P is ~~pressure~~

: P is the pressure.

~~== Application ==~~

~~Mean free path determines the largest frequency that can possibly propagate in a gas. Any larger frequencies will not propagate, owing to the fact that molecules will not transfer~~ the pressure ~~wave.~~

In air at <abbr title="Standard temperature and pressure ">STP</abbr>, mean free path is about 68 nm. Together with 340 m/s sound speed, this gives a maximum of 5 GHz of frequency. If there is a species which can hear this high of a sound, the maximum EDO which can be applied will be defined by the species' lowest frequency, or the place where the difference between rhythm and pitch occurs.

If the beat rate is ~~less~~ than the lowest frequency,beats are distinct, and the interval sounds like a "wahwah" instead of a true sound pair.

== Analysis with respect to equal-step tunings ==

There exists a point in frequency which marks the transition between rhythm and pitch. The beat rate of an interval is their frequency difference. If the beat rate is lower than the the said lowest frequency, beats are distinct, and the interval sounds like a "wahwah" instead of a true sound pair. It can be argued, by such criteria, that the smallest interval useful as a pitch material has the same beat rate as the lowest frequency.

If ~~humans could hear to~~ the mean free path limit, ~~this would limit our perception range to <math>\frac~~{~~5\cdot10^9 \ln~~{2}}~~{16}</math> or about '''216~~,~~608,494-EDO'''. This corresponds to~~ a ~~step~~ size of about 5.5 x 10-6 cents, ~~which is roughly the size~~ of ~~the [[Unnoticeable comma|unnoticeable]] rascal comma, [~~-~~7470 2791 1312⟩~~. Playing in a ~~temperament~~ which makes a distinction for intervals this small is physically impossible without beating interference on all notes.

If an interval whose base frequency is played at the mean free path limit, and beats at the lowest frequency, given as 16 Hz for human beings{{citation needed}}, the interval would have a size of about 5.5 × 10-6 cents, corresponding to a step of about 200-million-edo. Playing in a tuning system which makes a distinction for intervals this small is physically impossible without beating interference on all notes.

However, higher-pitched sounds will get heavily attenuated the closer their wavelength gets to free path. When the sounds attain levels of just several MHz, some estimate they will fail to travel through the air longer than a few centimeters.

However, higher-pitched sounds will get heavily attenuated the closer their wavelength gets to free path. When the sounds attain levels of just several MHz, some estimate they will fail to travel through the air longer than a few centimeters{{citation needed}}.

In small-atom solids such as dense metals, the sound can travel at much higher frequencies due to both higher sound speed and the periodicity and strength of the lattice. Theoretically, an alien civilization with their "ears" ~~designed to hear~~ through solids rather than gases would be able to make music with these higher frequencies, and perceive intervals as distinct instead of "wahwah".

In small-atom solids such as dense metals, the sound can travel at much higher frequencies due to both higher sound speed and the periodicity and strength of the lattice (→ [[Wikipedia: Speed of sound]]). Theoretically, an alien civilization with their "ears" fit for hearing through solids rather than gases would be able to make music with these higher frequencies, and perceive intervals as distinct instead of "wahwah".

[[Category:~~Extreme~~]]

[[Category:Psychoacoustics]]

[[Category:Limiting cases]]

@@ Line 1: / Line 1: @@
 {{Wikipedia|Mean free path}}
-In molecular physics, mean free path is the mean distance a molecule, atom, or other gas unit can travel before collision with other particles.
+In molecular physics, '''mean free path''' is the mean distance a molecule, atom, or other gas unit can travel before collision with other particles. It determines the largest frequency that can possibly propagate in a gas. Any larger frequencies will not propagate, owing to the fact that molecules will not transfer the pressure wave.
-It is defined by
+In air at <abbr title="Standard temperature and pressure ">STP</abbr>, mean free path is about 68 nm. Together with the sound speed of 340 m/s, the maximum physically possible frequency is about 5 GHz.
+== Definition ==
+Mean free path is defined by
 :<math>\ell = \frac{\mu}{p} \sqrt{\frac{\pi k_\text{B} T}{2 m}},</math>
@@ Line 12: / Line 15: @@
 where:
-<math>\ell</math> is the mean free path,
+: <math>\ell</math> is the mean free path,
-<math>m</math> is the mass of the particle
+: <math>m</math> is the mass of the particle,
-<math>M</math> is the molar mass of the substance
+: <math>M</math> is the molar mass of the substance,
-<math>\mu</math> is the gas's viscosity
+: <math>\mu</math> is the gas's viscosity,
-<math>R</math> is the universal gas constant
+: <math>R</math> is the universal gas constant,
-Kb is Boltzmann constant,
+: k<sub>B</sub> is the Boltzmann constant,
-T is temperature
+: T is the temperature,
-P is pressure
+: P is the pressure.
-== Application ==
-Mean free path determines the largest frequency that can possibly propagate in a gas. Any larger frequencies will not propagate, owing to the fact that molecules will not transfer the pressure wave.
-In air at <abbr title="Standard temperature and pressure ">STP</abbr>, mean free path is about 68 nm. Together with 340 m/s sound speed, this gives a maximum of 5 GHz of frequency. If there is a species which can hear this high of a sound, the maximum EDO which can be applied will be defined by the species' lowest frequency, or the place where the difference between rhythm and pitch occurs.
-If the beat rate is less than the lowest frequency,beats are distinct, and the interval sounds like a "wahwah" instead of a true sound pair.
+== Analysis with respect to equal-step tunings ==
+There exists a point in frequency which marks the transition between rhythm and pitch. The beat rate of an interval is their frequency difference. If the beat rate is lower than the the said lowest frequency, beats are distinct, and the interval sounds like a "wahwah" instead of a true sound pair. It can be argued, by such criteria, that the smallest interval useful as a pitch material has the same beat rate as the lowest frequency.
-If humans could hear to the mean free path limit, this would limit our perception range to <math>\frac{5\cdot10^9 \ln{2}}{16}</math> or about '''216,608,494-EDO'''. This corresponds to a step size of about 5.5 x 10<sup>-6</sup> cents, which is roughly the size of the [[Unnoticeable comma|unnoticeable]] rascal comma, [-7470 2791 1312⟩. Playing in a temperament which makes a distinction for intervals this small is physically impossible without beating interference on all notes.
+If an interval whose base frequency is played at the mean free path limit, and beats at the lowest frequency, given as 16 Hz for human beings{{citation needed}}, the interval would have a size of about 5.5 × 10<sup>-6</sup> cents, corresponding to a step of about 200-million-edo. Playing in a tuning system which makes a distinction for intervals this small is physically impossible without beating interference on all notes.
-However, higher-pitched sounds will get heavily attenuated the closer their wavelength gets to free path. When the sounds attain levels of just several MHz, some estimate they will fail to travel through the air longer than a few centimeters.
+However, higher-pitched sounds will get heavily attenuated the closer their wavelength gets to free path. When the sounds attain levels of just several MHz, some estimate they will fail to travel through the air longer than a few centimeters{{citation needed}}.
-In small-atom solids such as dense metals, the sound can travel at much higher frequencies due to both higher sound speed and the periodicity and strength of the lattice. Theoretically, an alien civilization with their "ears" designed to hear through solids rather than gases would be able to make music with these higher frequencies, and perceive intervals as distinct instead of "wahwah".
+In small-atom solids such as dense metals, the sound can travel at much higher frequencies due to both higher sound speed and the periodicity and strength of the lattice (→ [[Wikipedia: Speed of sound]]). Theoretically, an alien civilization with their "ears" fit for hearing through solids rather than gases would be able to make music with these higher frequencies, and perceive intervals as distinct instead of "wahwah".
-[[Category:Extreme]]
+[[Category:Psychoacoustics]]
+[[Category:Limiting cases]]