The two-dimensional wave equation on a stretched membrane, ∂²u/∂t² = c²∇²u with fixed boundary conditions on a rim, describes drums, loudspeaker cones, and thin plates in a bending-stiffness-free idealization. Normal modes are standing waves labeled by two integers; their eigenfrequencies depend on geometry (circle vs rectangle), tension, and areal mass density. Energy injected at one mode frequency excites that Chladni-like pattern of nodal lines where motion vanishes. The simulator integrates the PDE or superposes analytic modes so students hear/see how spectrum and symmetry arise, comparing low modes of a circle (Bessel J_m zeros) to simpler sine modes on a rectangle. Damping, stiffness, and nonlinear stretching are omitted for clarity. Learners connect boundary conditions to quantization of k, beating between nearby modes, and why a drum's timbre is not harmonic like a string.
Who it's for: Wave physics and partial differential equation students who completed the 1D string demo and need a 2D eigenmode picture before full plate theory.
Key terms
Wave equation
Membrane
Normal modes
Nodal lines
Chladni figures
Bessel functions
Boundary conditions
Superposition
How it works
Explicit finite-difference Laplacian on a rectangle with Dirichlet boundaries (clamped rim). A Gaussian bump spreads as circular wavefronts that reflect from the edges. Small damping stabilizes long runs; reduce Δt scaling via c for stability in this demo.
Frequently asked questions
Why are drum overtones not integer multiples of the fundamental?
Unlike a 1D string, the 2D eigenvalue problem on a circle yields transcendental frequency ratios from Bessel function zeros; only special geometries approximate harmonic spacing.
Is a real drumhead exactly described by the pure 2D wave equation?
Thin membranes are close, but bending stiffness, air loading, and nonuniform tension shift frequencies and damping.
What boundary condition is assumed at the rim?
Usually u = 0 (fixed edge), analogous to a string's pinned ends. Other edges (free, partially damped) change the mode shapes.
How does this relate to the cymatics membrane page?
Both visualize 2D standing waves; the rectangular case may use sines while circular drums invoke radial Bessel structures. Pedagogical emphasis may differ.