Drag the frequency slider and the plate jumps to whichever mode resonates nearest. The sand is not drawn: each grain is pushed away from the parts of the plate that move and comes to rest where it does not.
Ernst Chladni sprinkled sand on a metal plate and drew a violin bow along its edge. At most speeds nothing happened. At certain ones the plate rang, and the sand fled the moving parts to gather along the lines that stood still. Those lines are the nodes of a standing wave, and the picture they make is a Chladni figure.
The square plate here is a steel sheet 1 mm thick, free at its
edges. Its figures are the zero set of
cos(nπx) cos(mπy) −
cos(mπx) cos(nπy): two modes of the same frequency, superposed so that they cancel. The
superposition slider sweeps that cancellation, which is why one
frequency can show several figures.
The drumhead is an ideal circular membrane fixed at its rim. Its modes
are Bessel functions,
Jn(αr) cos(nθ), giving nodal
diameters and nodal circles. The Bessel functions are computed from
their series and their zeros are found by bisection, so nothing here
comes from a table.
The frequencies are the real ones for these idealised objects: a stiff
plate rings at frequencies proportional to
m² + n², while a membrane, having
tension but no stiffness, rings in proportion to the Bessel zero itself.
A real plate of brass or glass, with its own thickness, mounting, and
imperfections, will not match these numbers exactly. The shapes are the
honest part; treat the frequencies as those of the model.