I came across this video on YouTube. Here is an image from this video:
In this experiment, sand is scattered on a vibrating surface and one can see some beautiful patterns being formed. I think this happens when the surface is in a normal mode
$$u_n(x,y,t) = v_{n}(x,y)\phi_n(t)$$
where $u(x,y,t)$ is the deplacement of the surface at point $(x,y)$ and time $t$. $v_n$ must obey the Hemholtz equation $\Delta v_n = \lambda_n v$ and the pattern we see is the set $\{(x,y): v_n(x,y) = 0\}$.
The problem is that I don't know how to choose an appropriate boundary condition to solve $v_n$.
Can any one suggest a solution? If possible, please plot the curves to see if they look like ones in the video. Thanks in advance.