Modes du tambour

Explore the shape, fixed nodes and oscillation of a circular membrane.

About this tool

Explore one standing mode

Select an example or enter angular order m, radial index n, form coefficient A and grid resolution. Render prepares a paused mode at t = 0. The time field and slider move through 0…2T₀₁. At playback speed 1×, one fundamental period T₀₁ takes four real seconds. Reset time returns to the initial phase with the same mode. Changing m, n, A or the grid requires a new render; speed only changes playback.

Mode and frequency

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

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 αmn; 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