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. Inputs are checked without clamping: c=1–5000 m/s, size=0.001–100 m. Changing shape retains m,n and reports an error if they do not suit the new shape. Every edit clears the old result; Calculate starts a new calculation. Reset retains the inputs. Within-page navigation retains the result; reload retains only valid parameters and waits for Calculate.

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.