\input{labpreamble}
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\subsubsection*{PHSX 101\hfill Experiment 12\hfill Name:\underline{\hspace*{2in}}}

\underline{Title:} Standing Waves in Pipes

\underline{Object:} (1) To observe standing waves in pipes, and  (2) To measure 
the speed of sound.

\underline{Theory:} When a wave is produced in a medium that has a 
well-defined boundary that wave will be reflected at the boundary and 
create a situation in which waves will be traveling in both 
directions through the same medium.  If the dimensions of the medium 
fulfill certain conditions, the wave will reflect back on itself so as to create a 
stationary or \emph{standing} wave with nodes and anti-nodes.

In the case of a pipe that is open on both ends, if it is to 
resonate with an external sound source, the length $L$ of the pipe must 
be equal to a whole number $n$ of half-wavelengths of the incoming 
sound wave:
\begin{equation}
L=n\lambda /2
\end{equation}
where $\lambda$ is the wavelength.  Since $\lambda = v/f$ then $L=nv/2f$.

Thus we see that a pipe has a series of ``resonant frequencies'' 
which can be found by solving the above equation for $f$:
\begin{equation}\label{nv}
f={v\over 2L}n\ \ \ \ \ \ n = 1,2,3,\ldots
\end{equation}

The slope of a graph of $f$ vs. $n$ is therefore $v/2L$.  If we wish to 
determine the speed of sound in air $v$, it should be the {\it slope} $\times\ 2L$.

In the case of a pipe that is open on one end and closed on the other 
the length must be equal to an odd number $(2n-1)$ of quarter 
wavelengths of the incoming sound wave for resonance to 
occur:
\begin{equation}
L=(2n-1)\lambda /4.
\end{equation}
Again, since $\lambda = v/f$ then $L=(2n-1)v/4f$, which gives resonant 
frequencies of
\begin{equation}\label{4L}
f = {(2n-1)v\over 4L} \ \ \ \ \ \ n = 1,2,3,\ldots
\end{equation}
and the speed of sound by 
\begin{equation}
v=4fL/(2n-1).
\end{equation}

\underline{Apparatus:}  Draw a diagram of the standing wave patterns 
for the first two harmonics ($n=1,2$) of both the pipe open on both ends and 
the pipe open on one end and closed on the other.

Pipe open on both ends:\hfill Pipe open on one end:\ \ \ \ \ \ \ \ 
\vspace{.2in}


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

\item Connect the speaker to a variable frequency oscillator and the
microphone to an oscilloscope.  Keep the amplitude low enough to avoid
blowing the speaker (about 1/3 of maximum).  Measure room temperature
in kelvins and compute the expected value of $v$ from $v=20\sqrt{T}$.

\item Starting at a low frequency, slowly increase it until a 
resonance is observed.  Continue in this manner, recording the 
frequency for each resonance ($ n = 1,2,3,\ldots$).

\item Plot the resonances on a graph of frequency~vs.~$n$.  Compare 
the measured slope of the best-fit straight line multiplied by $2L$ to 
the expected value for $v$.  (See equation \ref{nv}.)

\item Now repeat the above steps for the open-closed pipe.  The 
best-fit straight line will not go through the origin in this case.  (See 
equation \ref{4L}; in fact, put it in the form $y=mx+b$.)

\item\label{lambda} Remove the plunger and set the oscillator to one 
of the resonance frequencies (say $n=4$) for the open-open pipe and 
move the microphone along the tube from node to node to measure the 
wavelength directly.  (You must skip one node to get a full 
wavelength.)  Use $v=\lambda f$ to compute the speed of sound measured 
in this manner and compare with the expected value.

\end{enumerate}

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

\item Room temperature in kelvins:  \underline{\hspace{1.2in}};{\hspace{.2in}} Expected 
value of $v$: \underline{\hspace{1.2in}}

\item Record the resonance frequencies and lengths for the two pipes:

\hspace{.1in}
\begin{tabular}{|c|c|}
\multicolumn{2}{c}{open-open}            \\  \hline
 \ \    $n$   \ \   & \ \ Frequency \ \  \\  
                    &       (Hz)         \\  \hline
         1          &                    \\  \hline
         2          &                    \\  \hline
         3          &                    \\  \hline
         4          &                    \\  \hline
         5          &                    \\  \hline
         6          &                    \\  \hline
\end{tabular}
\hspace{.1in}L = \underline{\hspace{.8in}}       
\hspace{.4in}
\begin{tabular}{|c|c|}                
\multicolumn{2}{c}{open-closed}          \\  \hline
 \ \    $n$   \ \   & \ \ Frequency \ \  \\  
                    &       (Hz)         \\  \hline
         1          &                    \\  \hline
         2          &                    \\  \hline
         3          &                    \\  \hline
         4          &                    \\  \hline
         5          &                    \\  \hline
         6          &                    \\  \hline
\end{tabular}       
\hspace{.1in}L = \underline{\hspace{.8in}}       

\item Compare the {\it slope} $\times\ 2L$ with the expected value of $v$.
\vspace{.1in}

For open-open pipe:\hfill {\ \it slope} $\times\ 2L$ = \underline{\hspace{1.2in}}\hfill 
\% diff = \underline{\hspace{1.2in}}

\item Compare the {\it slope} $\times\ 2L$ with the expected value of $v$.
\vspace{.1in}

For open-closed pipe:\hfill {\it slope} $\times\ 2L$ = \underline{\hspace{1.2in}}\hfill 
\% diff = \underline{\hspace{1.2in}}

\vspace{.1in}
\item Compute the speed of sound from procedure \ref{lambda} and 
compare to the expected value.
\vspace{.1in}

$\lambda=$ \underline{\hspace{1.in}} \hfill $f=$  
\underline{\hspace{1.in}}\hfill
$v=$  \underline{\hspace{1.in}}\hfill \% diff = \underline{\hspace{1.2in}}

\end{enumerate}

\vspace{-.2in}
\underline{Conclusions:} 
\vspace{.1in}
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