luckis11 said:
Therefore, someone must find the correct equation. If I am wrong show me why.
My mistake could be that we can consider the speed of the molecule as zero when there is no sound, but why-how? Vin, don't say again that the average speed is zero "in all directions", because according to this logic, the speed in the above equations should also be zero.
I was just walking back from the library restrooms and noticed a book called, "A Textbook of Sound" and immediately thought back to this thread. I have just picked out some passages that I looked quickly at. Take it for what it's worth.
"A Textbook of Sound" 3rd Rv. A.B Wood cpyrt 1955. G.Bell and Sons Ltd.
Pg 248 - 249 "velocity of small amplitude waves" excerpts
\sqrt{\overline{u^2}}=482 metres/sec
= root mean square velocity ;
\bar{u}=447metres/sec
= mean velocity of molecules
The Kinetic Theory of Gases shows very simply that the pressure in a gas is given by
P=\frac{1}{3}Nm\overline{u^2}=\frac{1}{3}\rho\overline{u^2}
where \\N is the number of molecules per c.c., \\m the mass of a molecule, and \rho the density of the gas.
This relation expresses the molecular velocity in terms of the pressure and density of the gas, thus
\overline{u^2}=\frac{3P}{\rho}
Since the energy of the motion of the molecules in a given (i.e. x-axis) we may write
speed of sound = \sqrt{\overline{u_{x}^2}}=\sqrt{\frac{P}{\rho}}
Which is Newton's "Isothermal Velocity" of wave-propogation in the gas. If we employ the mean molecular velocity \bar{u} instead of \sqrt{\overline{u^2}}
we find
\sqrt{\overline{u^2}}=\bar{u}\sqrt{\frac{3\pi}{8}}
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Pg 275-276 "Velocity of Sounds of High Frequency -- in Gases" excerpt
H.O. Kneser, in a number of theoretical and experimental papers dealing with anomalous absorption and dispersion of sound, has derived the following expression for the velocity in terms of frequency, \\w=2\pi\\N, and molecular constants:
\\v=\sqrt{\frac{P}{\rho}}\left(1+R\frac{c_{v}+w^2\beta^2c_{va}}{c_{v}^2+w^2\beta^2c_{va}^2}\right)
where \\c_{v} is the molecular heat at constant volume, \\c_{va} is the specific heat of the translation degrees of freedom, \\R is the universal gas constant, and \beta is the mean life of the energy-quantum, that is the time involved in the quantum transformation -- translational-intramolecular-translational energy.
at low frequencies the velocity becomes \sqrt{\frac{P}{\rho}}\left(1+\frac{R}{c_v}\right)
whereas at high frequencies it becomes \sqrt{\frac{P}{\rho}}\left(1+\frac{R}{c_va}\right)
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Anyways it appears to me that you should be able to substitute around and solve for molecular velocity
correct?
