Membrane nodal patterns

This calculator describes a homogeneous stretched membrane with its entire edge fixed: u_tt = c²Δu and zero displacement at the boundary. It does not calculate the flexural modes of a metal Chladni plate or simulate sand. c is the wave speed on the membrane, not necessarily the speed of sound in air. L is the square side length and R the circle radius, in meters.

Square. With X=x/L and Y=y/L, A=sin(mπX)sin(nπY), B=sin(nπX)sin(mπY) and Φ=(A+bB)/(1+|b|). The integers m,n are 1–8; b is −1 to 1. The two basis modes share f=c√(m²+n²)/(2L). b=0 shows A, b=1 their sum, and b=−1 their difference. When m=n the expression is evaluated as Φ=[(1+b)/(1+|b|)]A, retaining even very small nonzero amplitudes. Only m=n and exactly b=−1 cancel completely: then no active frequency or wavelength is displayed.

Circle. With X=x/R, Y=y/R, ρ=√(X²+Y²) and θ=atan2(Y,X), Φ=J_m(αρ)cos(mθ). m=0–4 counts nodal diameters; n=1–4 selects the nth positive zero α of J_m. The frequency is f=αc/(2πR), and λ=c/f. There are n−1 interior rings with radius/R=α_mj/α_mn and m diameters at θ=(π/2+jπ)/m. Their angles are counterclockwise from +x and given in radians in the details. m=0 has no diameters. The mix b has no role for the circle.

Reading the picture. The axes are dimensionless x/L,y/L or x/R,y/R with equal scales and y upward. Φ is a relative signed mode shape, not a displacement in meters. Colors distinguish positive and negative values; black lines mark nodes. The band |Φ|≤ε, with ε from 0 to 0.1, only highlights proximity to zero. Its threshold uses the stated Φ formula, not a fresh normalization to the plotted maximum, and never changes the mathematical nodes or frequency. The plot is static; a vibrating mode's instant of zero displacement everywhere is not its nodal pattern.

Circle rings and diameters, square single modes and equal-index modes have analytic nodal lines. For mixed square modes with different m,n, contours are linearly interpolated on a 257×257 grid with explicit handling of grid zeros and saddle cells. Geometry finer than one cell remains a plotting approximation. Circle Bessel functions use a direct integer-order power series and 20 tabulated zeros; this is a bounded calculation, not a general special-function library. The neutral cancellation view has no isolated nodal lines.

Use. Calculate starts the first pattern. Thereafter valid edits update it immediately; invalid inputs clear the old plot without moving keyboard focus. Reset restores the default square difference mode. Changing shape keeps m,n and reports values outside that shape’s range. Inputs, including empty or invalid drafts and the inactive square mix, are retained through the shared browser storage; plots and computed fields are not saved. Reload or returning to this tool waits for Calculate. The plot uses equal horizontal and vertical scales, resizes to the available viewport and limits its pixel buffer.

Sources: Penn State: rectangular membranes, Penn State: circular membranes, UT Austin: elastic membranes, and NIST DLMF 10.9.2 for the Bessel integral representation underlying the independently determined zeros.