Chladni Plate Standing Waves - 2D Nodal Lines, Resonance Spectrum and Sand Migration

Presets

Speed

Live readouts

Graphs

Numerical checks

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How to use

Choose a preset: the plate is driven exactly at one of its resonances and the sand drifts away from the strongly vibrating regions until it outlines the nodal lines. Try: (1) Lowest mode (1,2) - the simplest figure; (2) High mode #20 - a dense lattice; (3) Between resonances - detuned drive gives a blend of several modes and a blurred pattern; (4) Steel, auto sweep - the pattern morphs as frequency climbs through each resonance. Change material, size or thickness and read the new mode frequencies. Space pauses, R scatters fresh sand.

The science

Plate modes. A thin elastic plate obeys the Kirchhoff bending equation D nabla^4 w + rho h d2w/dt2 = F(x,y,t), D = E h^3 / (12 (1 - nu^2)) Its normal modes are standing patterns; for a square plate the classic free-edge approximation is phi_nm(x,y) = cos(n pi x) cos(m pi y) - cos(m pi x) cos(n pi y) which is zero on the diagonal and changes sign across every nodal line.

Frequencies. Dimensional scaling gives f_nm = (pi/2) sqrt(D/(rho h)) (n^2 + m^2) / a^2 so thicker or stiffer plates ring higher and larger plates lower; doubling the thickness doubles every frequency.

Driving and sand. Near resonance each mode responds as a damped oscillator, amplitude 1/sqrt((1 - r^2)^2 + (r/Q)^2) with r = f/f_nm. Sand grains are thrown about in proportion to the local displacement and so accumulate where the displacement is zero.

Checks. (1) the diagonal x = y is a node of every mode; (2) f(2,3)/f(1,2) = 13/5; (3) different modes are orthogonal over the plate; (4) a random walk with step proportional to |phi| lowers the mean |phi| of the grains by a factor of three.

Simplifications. The shape formula is an approximation to true free-edge modes (exact ones need a numerical Ritz or finite-element solve and give slightly different frequencies); the frequency law is the simply-supported scaling, so real absolute frequencies differ by tens of percent; sand is non-interacting and ignores air streaming that clusters it in practice; amplitudes of the 40 lowest modes are weighted by a 1/(n^2+m^2) drive profile.