This and the following many, many slides show buckling of various kinds of cylindrical shells under various loadings.
This is Figure 3 from the report:
Michael P. Nemeth and Martin M. Mikulas, "Simple formulas and results for buckling resistance and stiffness design of compression-loaded laminated composite cylinders", NASA/TP-2009-215778, August 2009
n is the number of full circumferential waves in the buckle pattern. The critical buckling wavenumber is that which corresponds to the minimum buckling load factor.
The buckling modal displacement in general varies also in the axial direction, as demonstrated in a particular example displayed in the next slide.
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