A Free bar fundamental vibration relationship length to width (1 Viewer)

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When finding the fundamental for thin bars, the approximate equation (http://hyperphysics.phy-astr.gsu.edu/hbase/Music/barres.html) only refers to the length and thickness when calculating. I'm trying to figure out the frequency and node location along the length and width of the fundamental vibration as the width approaches the same value as the length. Basically, I want to extend the accuracy of this equation to include width. Intuitively, it seems as though a square bar will not retain the same node locations or frequency as a rectangle. As the bar's width begins to exceed the length, eventually that direction will begin to define the fundamental resonance. Trying to root out my misconception or understand the relationship of length, width, and thickness. Any resources, intuitive spews, or pure math are welcome. Imagine square vibraphone bars, where would be the best place to put holes? would the same relationship .224L still be the node point for both length and width? I look forward to this discussion.
 
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You are looking for a plate vibration solution, a more complicated matter than a simple beam vibration. Search on the subject "plate vibrations," and I think you will find what you need.
 

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