\section{S2~~Transverse Standing Waves on a Wire~~(Sonometer)}

\indent Do not choose this experiment unless you have a good sense of pitch.


\noindent OBJECT:  To check  \\
$$ f~=~ \frac{1}{2L} \sqrt{ f/ \mu} $$
for the fundamental mode of a transversely vibrating wire, where 
\begin{enumerate} \hspace*{.5in} \epind{
\item f~=~frequency of vibration  
\item L~=~length of wire between the wooden bridges  
\item F~=~tension  
\item {$\mu $}~=~mass per unit length of the wire.}
\end{enumerate}

\noindent APPARATUS: \smallskip\\ \pind{
 Sonometer with piano wire; micrometer; four tuning forks of different 
frequencies; strobe; weight holder \& kg weights; two wooden bridges; 
selection of slotted masses. } \\

\noindent SUGGESTIONS: \smallskip \\ \pind{
To minimize the effect of friction, tap the wire on both sides of bridge 
next to the pulley.  A drop of oil on the bridge and on the pulley bearing 
may also help. To transfer fork vibration to the wire, place bottom of 
fork in contact with the sounding board.  A small V shaped paper placed on 
the wire {\bf near an antinode} may help you in tuning.  At resonance the wire 
vibration may displace the paper.} \\

\noindent PART~I:  \begin{enumerate} \epind{
\item[a)] For an $\sim $~60 cm wire length
between the wooden bridges, adjust the 
tension until the string resonates with the fork of lowest frequency. 
\item[b)] For this tension, find lengths between the bridges which give resonance
with the other forks. Make several settings for each fork to 
estimate the accuracy. 
\item[c)] Compare frequencies calculated from above with those stamped on the forks. 
Is the agreement reasonable?  Explain. }
\end{enumerate}
\noindent PART~II:   \begin{enumerate} \epind{
\item[a)] For a fixed $\sim $~30 cm length find the tensions required to make 
the fundamental frequencies of the wire equal to those of the forks.
\item[b)] Compare calculated values of {\bf f} with those stamped on the forks.
\item[c)] Plot curves for parts I and II. Show that the results agree with (1). }
\end{enumerate}
\noindent OPTIONAL:  Check one or more frequencies with a strobe light. \\

\noindent Note::  Overtones are generally not a problem since excitation is by a tuning 
fork which by design diminishes all overtones. 


\newpage

\section{S3~~Velocity of sound in air}

\noindent OBJECT:  \smallskip \\ \pind{ 
To calculate the velocity of sound from measurement of the 
wavelength in air for sound of a certain frequency. } \\
 
\noindent APPARATUS:  \smallskip \\ \pind{ 
Resonance tube with arrangement for varying water level (use 
only distilled water); rubber tipped hammer; tuning fork; Hg thermometer.} \\

\noindent INTRODUCTION:  \smallskip \\ \pind{ 
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.} \\

\noindent SUGGESTIONS: 
\begin{enumerate} \epind{
\item Find the positions of the water level in the tube for 
the first three of these resonances. Use these readings to calculate the 
velocity of sound, {\bf 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 Since v = \( v~=~\sqrt{ \gamma RT/M }\), correct your value of 
{\bf v} to 
0$^o $C 
(T~=~273.16 K) and compare with that accepted 
for {\bf dry} air at 0$^o $C: 331.29~$\pm $~.07~m/s,
[Wong, {\it J.~Acoust.~Soc.~Am.}, {\bf 79}, 1559, (1986)]. For humid air see 
3.~below. 
\item How should the v.p.~of water, P$_{w}$, in the tube 
affect the measured velocity?
An estimate is not difficult from 
v~=~$\sqrt{\gamma RT/M}$.
Use 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}

\noindent where $\gamma _{air}~=~1.40, \gamma _{w}~=~1.33; P_{a}$
is the partial pressure of air 
and P$_{w}$ is the v.p.~of water; M$_{a}$~$\sim $~29~kg
and M$_{w}$~=~18~kg.  Hence
\begin{center}
$v_{dry}~\cong ~v_{humid}\sqrt (\gamma _{a}/M_{a})/<
\gamma /M>$
\end{center}
\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} 
