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\begin{document}

\( \gamma =\frac{c_{p}}{c_{v}}=1.4 \), and \( c_{s}=\sqrt{\gamma \frac{kT}{\mu }}=343m/s \). There is an implicit assumption that \( P\propto \rho ^{\gamma } \) (adiabatic, no heat
transfer).

Time to equilibriate \( \frac{(\lambda /2)^{2}}{\kappa } \)= \( 5*(\frac{\lambda }{2}) \). \( \nu \sim \frac{c_{s}}{\lambda } \) which means \( t\sim \frac{c_{s}^{2}}{\nu ^{2}} \) so \( t\nu \sim \frac{c_{s}^{2}}{\nu } \). This means a high
enough frequency to be nonadiabatic is \( \nu _{0}=5*10^{9}Hz=\frac{1}{t_{coll}} \).

I can hear from about 100 Hz-16000Hz.

human range is about \( 30Hz-10or20KHz \).

For a piano, 

\begin{itemize}

\item A1=27.5 Hz, \( \lambda =12m \)

\item middleC 262Hz \( \lambda =1.3m \)

\item A \( 440Hz \) \( \lambda =.78m \)

\item \( C7 \)= \( 4186Hz \) = \( \lambda 8cm \)

\end{itemize}

A sound wave causes a change in volume of a bunch of particles by
\( \frac{\triangle V}{V}\sim \frac{x}{\frac{\lambda }{2}} \), also \( \frac{\triangle P}{P}=\gamma \frac{\triangle V}{V}\sim \frac{x}{\frac{\lambda }{2}} \)

the fluctutation velocity is \( u=2\pi \nu x \), so \( \frac{\triangle V}{V}\sim \frac{x}{\lambda }\sim \frac{u}{c_{s}}\sim \frac{\triangle P}{P_{0}} \).

\( \triangle p\sim p\frac{x}{\lambda }\sim p\frac{u}{c_{s}}\sim \rho c_{s}u \).

the quantity, \( \rho c_{s}=Z \) is the acoustic impedance.



\end{document}
