Chladni patterns

Question

So I was watching this video on Chladni figures and thought that it would be nice to replicate a few of these, especially the more complicated, high frequency ones.

So I know that the general formula for creating such patterns is \begin{equation} u(x,y,t)=\sin\left(\frac{\pi mx}{a}\right) \sin\left(\frac{\pi ny}{b}\right)\left[B_{mn} \sin\left(\omega_{mn}\pi kt \right) + A_{mn} \cos\left(\omega_{mn}\pi kt \right)\right] \end{equation} where \begin{equation}\omega_{mn}=\sqrt{\frac{m^2}{a^2}+ \frac{n^2}{b^2}} \end{equation} and have a pair of integers such that $\omega_{mn}$ produced are rationally related. Now I have been experimenting and playing around with this on Maple but I was wondering if there was a way to obtain specific figures such as the ones on the video (for example, the pattern produced at 5284 hz) and how that is determined.

Answer 1

Using the formula from page $5$ of this reference, I did this in Mathematica.

L=2;
f[x_,y_,m_,n_]=Cos[n π x/L]Cos[m π y/L]-Cos[m π x/L]Cos[n π y/L];
TableForm[
Table[ ContourPlot[f[x,y,m,n]==0,{x,-1,1},{y,-1,1},PlotPoints->40],
{m,7,9},{n,13,17,2}]]

to get these figures. Maybe you can adapt this to Maple. I don’t know how the frequency relates to the values of $m$ and $n$, but I didn’t read the whole paper I linked.

And here are some more, for $m=2,3,4$ and $n=13,15,17,19$.

The full contour plots are nice, too. This is for $m=17$,$n=9$, with $L=4$.