# Waveforms and buckling modes

Can anything be said about the expected shape of a specific buckling mode? Or more precisely, is is true for example that for the first buckling mode the second derivative of lateral deflection w.r.t the length $d^2w / dx^2$ maintains its sign over the whole span $L$?

If yes, how can one show that mathematically? For a beam, one can easily work it out from $w(x) = A\sin(n\pi x/L)$ with $n=1$, but how about a plate for example?

I found the solution i was looking for in this MIT paper. The out-of-plane deflection equation that satisfies the governing differential equation is of the form:

$w(x, y)=sin(\frac{mπx}{a})sin(\frac{nπy}{b})$

with a & b being the plate length and width respectively.

The equation for the critical load (11.4 in the aforementioned paper) has a minimum when n=1 (only a half-wave in the direction perpendicular to the load application direction) but the for the value of m, it depends on the a/b ratio as figure 11.2 beautifully demonstrates. To put it simply, the bigger the ratio, the more waves it can fit on its length as intuitively assumed.

Suprisingly and almost fascinatingly the ratios for which the number of halfwaves increases by 1 is:

• $\sqrt{2}$ from 1 to 2
• $\sqrt{6}$ from 2 to 3
• $\sqrt{12}$ from 3 to 4
• $\sqrt{20}$ from 4 to 5 and so on..