Spring lattice.
A thirty-by-thirty grid of point masses, each tethered to its four neighbours by a Hookean spring. Pluck any node and a transverse wave radiates outward across the membrane. Drag a node and release; switch the boundary between clamped, free, and a sinusoidally driven left edge. In the continuum limit the lattice obeys the wave equation; tune the drive to a resonance and the chatter resolves into a standing wave.
What you're looking at.
Each interior node obeys m·z̈ = k·(z_N + z_S + z_E + z_W − 4z) − c·ż: a Hookean restoring force from the sum of its four neighbour displacements, minus a small velocity drag. Take the lattice spacing h to zero and the bracket becomes the Laplacian, leaving the wave equation z̈ = v²∇²z with wave speed v = h·√(k/m). Everything interesting — interference, reflection, dispersion, standing modes — falls out of that one line.
Boundaries decide what the wave does when it hits the edge. A clamped edge pins z = 0, so the reflected pulse comes back inverted — the wall enforces a mirror image of opposite sign. A free edge enforces zero gradient instead; the reflection keeps its sign. It's the same distinction that separates a guitar string (clamped at both ends, harmonics at n·v/2L) from an open organ pipe — the same finite lattice, two completely different mode spectra.
Switch to driven, push the frequency slider up, and the left column oscillates at A·sin(2πft). Most frequencies produce a busy interference mess. At the right ones the lattice locks into a standing wave — fixed nodal lines, antinodes ringing in place. Those are the drumhead's eigenmodes, the two-dimensional cousins of the Chladni patterns that sand grains trace on a bowed metal plate.