What is the pattern for resonant frequencies of a tube closed at BOTH ends?
The cases for the string, the open tube and the tube closed at one ends are well known.
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The rule for tubes is that you need a node on each closed end of the tube. This means that the tube closed on both ends is EXACTLY equivalent to a string. The equation that applies will thus be $f_n=n(v/2L)$, where $n=1, 2, 3, ...$.