\section{S-2~~Velocity of Sound in Air}

\noindent OBJECTIVE: 
\begin{enumerate}
\item[] \latex{\vspace*{-.05in}}To calculate the velocity of sound from measurement of the 
wavelength in air for sound of a certain frequency.
\end{enumerate}
 
\noindent APPARATUS:
\begin{enumerate}
\item[] \latex{\vspace*{-.05in}}Resonance tube with arrangement for varying water level (use 
only distilled water); rubber tipped hammer; tuning fork; Hg thermometer.
\end{enumerate}

\noindent INTRODUCTION:
\begin{enumerate}
\item[] \latex{\vspace*{-.05in}}For a closed tube, resonance occurs at tube 
lengths of an odd multiple of 
one-fourth wavelength, i.e. at  $\lambda $/4, 3$\lambda $/4, 
5$\lambda $/4 etc.
\end{enumerate}

\noindent SUGGESTIONS: 
\begin{enumerate}
\item Find the positions of the water level in the tube for 
the first three of these resonances. Use these readings to calculate the 
speed of sound, $v = |\vec{v}|$.  Initially have enough water that you can 
raise the level above the first resonance position.  The tuning fork frequency 
is on the fork.\\ \\
Since the effective end of the resonance tube is not at the tube's end, 
{\bf do not use the position of the tube's top} in your calculations, 
but rather {\bf take differences between the other readings}.
\item Sound waves in gases have a speed $v = \sqrt{ \gamma RT/M }$.  (Recall the formula for the speed of sound on a string, $v = \sqrt{T/\mu}$ (e.g., Lab SC-1)).  Correct your value of 
$v$ to  0$^\circ$C  (T~=~273.16 K) and compare with that accepted 
for {\bf dry} air at 0$^\circ $C: 331.29~$\pm $~.07~m/s,
[Wong, {\it J.~Acoust.~Soc.~Am.}, {\bf 79}, 1559, (1986)]. For humid air see 
3.~below. 
\item We quantify proportions in gas mixtures by the pressure each gas contributes to the total pressure. This is called the ``partial'' or ``vapor'' pressure.  Think of the speed as resulting from an average $<\gamma/M>$,
\begin{center}
$<\gamma/M>~=~\left[ (\gamma _{a}/M_{a})P_{a}~+~
(\gamma _{w}/M_{w})P_{w}\right]/ (P_{a} + P_{w})\,,$
\end{center}
so that
\begin{center}
$v_{dry}~\cong ~v_{humid}\sqrt{(\gamma _{a}/M_{a})/<
\gamma /M>}\,,$
\end{center}

\noindent where $\gamma _{air}~=~1.40, \gamma _{w}~=~1.33, P_{a}$
is the partial pressure of air, 
P$_{w}$ is the vapor pressure of water, M$_{a}$~$\sim $~29~kg
and M$_{w}$~=~18~kg.

How should the v.p.~of water, P$_{w}$, in the tube 
affect the speed? 
 
\noindent OPTIONAL: Humidity changes will affect tuning of what musical instruments?
\item What effect does atmospheric pressure have on the velocity of sound 
in dry air?  (Assume air at these pressures is an ideal gas.)
\item Viscosity and heat conduction in the tube may reduce {\bf v}
 by $\sim $0.1\%.
See N.~Feather, ``The Physics of Vibrations and Waves'', 
Edinburgh Univ.~Press, (1961), p.~110-120; this reference also has a 
delightful
historical account (including Newton's famous goof). 
\end{enumerate} 
