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Pipe Closed at Both Ends

There are displacement nodes at both ends. This is just like a string fixed at both ends:

\begin{displaymath}
s(x,t) = s_0 \sin(k_n x )\cos(\omega_n t)
\end{displaymath} (180)

which has a node at $x =
0$ for all $k$. To get a node at the other end, we require:
\begin{displaymath}
\sin(k_n L) = 0
\end{displaymath} (181)

or
\begin{displaymath}
k_n L = n\pi
\end{displaymath} (182)

for $n = 1,2,3...$. This converts to:
\begin{displaymath}
\lambda_n = \frac{2L}{n}
\end{displaymath} (183)

and
\begin{displaymath}
f_n = \frac{v_a}{\lambda_n} = \frac{v_a n}{2L}
\end{displaymath} (184)



Robert G. Brown 2004-04-12