Simulate 1D and 2D wave equations with pluck, strike, and boundary condition controls. Visualize standing waves and resonant modes.
The wave equation, u_tt = c^2 (u_xx + u_yy) in two dimensions, governs strings, membranes, and most other things that vibrate. This tool sets that equation on a one-dimensional string or a two-dimensional rectangular membrane, applies the excitation you choose - a pluck, which sets the initial shape, or a strike, which imparts an initial velocity - animates the result under the boundary condition you select, and shows the standing waves that build up. Everything computes in the browser. The usual surprise: plucking a string fixed at both ends exactly at its center silences every even harmonic.
A concrete input and expected output from the current implementation.
Input
1D string: length 1 m, wave speed 2 m/s, both ends fixed, pluck at x = 0.5 m (the center)
Expected output
A symmetric standing wave rings on the string. The fundamental sits at f = v/(2L) = 2/(2 x 1) = 1 Hz with a single antinode at the center; odd harmonics appear at 3 Hz (nodes at x = 1/3 and x = 2/3 of the length) and 5 Hz. Even harmonics, 2 Hz and 4 Hz, never appear.
With both ends fixed, the nth mode has frequency n x v/(2L) = n Hz for these settings, and every even mode has a node at the midpoint - exactly where the pluck never moves the string. A centered pluck can therefore drive only the odd modes.
The string or membrane is represented by displacement values at a grid of sample points. Each animation frame advances those values by a small time step, then re-applies the chosen boundary condition at the edges; a pluck initializes the displacements, a strike initializes the velocities. The result is a numerical approximation: fine grids track the analytic modes closely, and the shortest wavelengths are the first to go visibly wrong.
With both ends fixed, the only 1D motions that persist are half-sine standing modes: wavelength 2L/n, frequency n x v/(2L), with n - 1 interior nodes. In 2D, a fixed-edge membrane of sides a and b supports modes sin(m pi x / a) x sin(n pi y / b) at frequency (v/2) x sqrt((m/a)^2 + (n/b)^2). On a square, the (1,2) and (2,1) modes share one frequency, so square membranes beat and swirl as rectangles do not.
No - symmetry forces it. Every even mode of a fixed-fixed string has a node at the midpoint, and an excitation that never moves the string there cannot drive those modes. Only the odd harmonics ring. Move the pluck off-center and the even harmonics come back.
No - free means unclamped, not pinned. A free end keeps zero slope, so it moves, and every mode has an antinode at it; a fixed end, by contrast, is a node. Toggle the boundary and both the visible shape and the fundamental frequency change, because the allowed wavelengths change.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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