Drum modes

Explore one standing mode

Choose an example to see it immediately. Numeric mode changes update after a short pause, resetting the clock to zero; playback stays paused. Render also starts at zero. Time and probe controls update directly. At speed 1×, one fundamental period takes four seconds.

Reload restores completed views at the exact paused clock, including unfinished inputs and open sections.

Mode and frequency

For a homogeneous circular membrane with a fixed rim, φ = J_m(α_mnr/R) cos(mθ) and u = Aφ cos[2π(f/f₀₁)(t/T₀₁)]. The value α_mn is the nth positive zero of J_m. Frequency and period ratios are f/f₀₁ = α_mn/α₀₁ and T/T₀₁ = α₀₁/α_mn.

m = 0…3 counts nodal diameters; m = 0 is rotationally symmetric. n = 1…4 counts the positive radial zero including the fixed outer rim, so there are n − 1 inner nodal rings. The fixed rim always has u = 0. Inner rings lie at earlier zeros divided by α_mn; the nodal diameters satisfy cos(mθ) = 0.

A = 0…1 is a dimensionless Bessel-form coefficient. It is not a maximum height normalized separately for each mode. At A = 0 the membrane rests, while its structural nodal lines remain marked. A moment when the whole membrane passes through u = 0 creates no additional mode nodes.

Example: for m = 0 and n = 2, α₀₂ ≈ 5.52007811 and f/f₀₁ ≈ 2.29541727. Its single inner nodal ring lies at r ≈ 0.43565064R.

Read the views

The transparent wireframe shows all grid lines in an orthographic projection. Its height is exaggerated to 1.5u; the horizontal coordinates are x/R and y/R. The view-angle field changes the azimuth while elevation stays at 35°. The drawing has no perspective or hidden-surface removal.

The top view uses the same fixed scale −1…1 for every time and mode: blue is negative, orange is positive and the neutral background is zero. Color strength changes with displacement, and the colors reverse after half a cycle. The solid rim, inner rings and diameters mark structural nodes. The cross marks the same probe in both views. Under Probe and view, set r/R = 0…1 and θ = 0…360°, measured counterclockwise from +x. φ reports the spatial shape and u the current displacement there.

This ideal membrane model assumes uniform tension and density, with no bending stiffness, damping, air coupling or sound output. Ratios are dimensionless; absolute frequencies require the physical radius, tension and surface density.

Sources

• NIST DLMF §10.21(i) — positive Bessel zeros and their properties.
• NIST DLMF Eq. 10.9.2 — integral representation for integer-order Bessel functions.
• Dan Russell, Penn State: Vibrational Modes of a Circular Membrane — nodal diameters, rings and frequency ratios.

The displayed clock is rounded for stable fields; the full clock drives calculations and resumes. Click or drag in the top-view disk to move the probe and pause. PNG pauses and captures both views at the same instant.

Only input values are remembered in this browser. Calculated results and experiment histories are not saved.