Pipe Closed at Both Ends

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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

for all

. To get a node at the other end, we require:

$\begin{displaymath} \sin(k_n L) = 0 \end{displaymath}$

(181)

$\begin{displaymath} k_n L = n\pi \end{displaymath}$

(182)

for

. 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