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\subsubsection*{PHSX 221L\hfill Standing Waves\hfill Name:\underline{\hspace*{2in}}}

% \underline{Title:} Standing Waves

\underline{Object:} (1) To study the relationship between resonant 
frequency and tension for standing waves in a string.  (2) To measure 
the speed of sound in air by observing standing waves in a pipe.

\underline{Theory:} If a uniform string of linear density $\mu$ is 
put under a tension $\tau$ and stretched between two supports it can be 
caused to oscillate in resonance with a driving frequency $f$, if 
that driving frequency matches one of the natural modes of vibration 
of the string.  A natural mode of vibration corresponds to a standing 
wave pattern in which a node exists at each fixed support.  Since the 
distance between two adjacent nodes is equal to one half wavelength, 
the length of the string must be equal to an integral number of half 
wavelengths at resonance.

The speed at which a wave moves through a string can be calculated 
from the tension in the string and the mass per unit length.  This 
relationship is Equation 16.4 in Serway.  Since the speed of a wave 
can also be expressed as the product of wavelength and frequency, it 
is possible to derive a relationship between the string tension, the 
linear density, the string length, and the resonant frequency.  Derive 
this relationship.  This relationship will indicate the theoretical 
slope of the straight line produced when resonant frequency is plotted 
versus resonance mode index.

Standing waves can also be set up in tubes filled with air.  If both 
ends of the tube are open, the length of the tube is equal to an 
integral number of half wavelengths (at resonance).  Again it is 
possible to derive an expression for the resonant frequency in terms 
of the wave speed, the length of the tube, and the mode index.  A plot 
of resonant frequency versus mode index should be a straight line 
with slope equal to the wave speed divided by twice the tube length. 
Verify this derivation.

The derivation for an open-closed pipe is similar but there will be a 
node at the closed end.  See section 18.5 in Serway.

It can be shown theoretically that the speed of sound in air depends 
only on the air temperature and is given by 
\begin{equation}\label{temp}
v=20.1\sqrt{T}
\end{equation}
where $T$ is the air temperature in kelvins.


\underline{Apparatus:} You will use an oscilloscope today to detect
the resonances in the tube of air.  See
\verb=<http://www.physics.udel.edu/wwwusers/watson/phys345/lab/scope.html>=
for an introduction.  As always, sketch all relevant equipment.

\underline{Procedure:} 
\vspace{-.25in}
\begin{enumerate}

\item Set up a string stretched with a known tension and drive it with 
an oscillator of variable frequency.  Adjust the frequency to obtain a 
sequence of resonances.  Plot resonant frequency versus mode index 
number, compute the slope (least squares fit), and compare with theory.

\item\label{tube} Set up a loudspeaker driven by a variable frequency
oscillator and use it to produce resonances in an open-open tube. 
(Please keep the amplitude knob on the frequency generators below 1/3
full value so as not to overdrive and damage the speakers.)  Plot
resonant frequency versus mode index.  From the slope of this plot
compute the speed of sound in air, and compare it with the theory in
equation \ref{temp}.

\item Repeat procedure \ref{tube} for an open-closed tube.

\item Measure the wavelength directly by moving the microphone along 
the open-open tube when the oscillator is set to one of the resonance 
frequencies (say $n= $ 4).  Use $v=\lambda f$ and compare to theory 
for $v$.

\item Draw a few resonance patterns for a square metal plate.

\end{enumerate}


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