Wave Equation vs Standing Waves: PDE Simulation vs Harmonic Mode Visualizer
Both simulations end up showing a standing pattern, but they get there very differently: the Wave Equation simulator solves the general PDE from any disturbance you make, while Standing Waves in Pipes draws the closed-form harmonic solution directly so you can dial in an exact mode n.
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⚡ Quick answer
Both simulations end up showing a wave that doesn't travel, but they get there differently. The Wave Equation simulator numerically integrates ∂²u/∂t² = c²∇²u with a finite-difference leapfrog scheme on a grid: you click to create disturbances on a 1D string or 2D membrane, and standing waves, reflection and diffraction all emerge naturally from the same update rule. Standing Waves in Pipes skips the general PDE and draws the closed-form solution directly — y(x,t) = 2A·sin(kx)·cos(ωt) for a string clamped at both ends — letting you pick an exact harmonic n with a slider and see its nodes, antinodes and component travelling waves labelled explicitly.
📊 Wave Equation vs Standing Waves in Pipes — PDE Simulation vs Harmonic Mode Visualizer
| Wave Equation | Standing Waves in Pipes | |
|---|---|---|
| Underlying method | Finite-difference leapfrog integration of ∂²u/∂t² = c²∇²u on a discrete grid, step by step | Closed-form superposition of two counter-propagating waves, A·sin(kx−ωt) + A·sin(kx+ωt), rendered by a GLSL shader |
| What you interact with | Click anywhere to pluck the string or strike the membrane and watch the disturbance evolve | Sliders for harmonic n (1–8), amplitude, tension/wave speed and a "pluck" mix control |
| Dimensionality | Both a 1D plucked string (600 points) and a 2D drum membrane (100×100 grid) | 1D string only, clamped at both ends |
| Boundary conditions | Fixed (reflecting) or absorbing (open) edges; slits can be added to see diffraction | Always fixed at both ends, so only wavelengths λₙ = 2L/n are allowed |
| Phenomena on display | Propagation, reflection, diffraction and interference — standing waves are one emergent special case among many | Nodes, antinodes, the harmonic series and the two travelling-wave components are labelled and isolated explicitly |
| Best for learning | How a single differential equation generates every wave phenomenon — propagation, reflection, diffraction and standing waves alike | Why a fixed string can only ring at discrete harmonic frequencies, and what nodes and antinodes physically are |
The Wave Equation simulator is a general-purpose numerical solver: it discretises ∂²u/∂t² = c²∇²u with a leapfrog finite-difference scheme, so every grid point's next displacement comes from its current and previous values plus the curvature contributed by its neighbours. Because it solves the equation itself rather than a pre-worked-out solution, it can show you anything the wave equation predicts: a single pulse racing outward and reflecting off a wall, two ripples crossing and interfering, a wave diffracting around a slit — and, if you let reflections build up between two fixed ends, a standing wave appears as one emergent outcome among many. You can switch between a 1D plucked string and a 2D drum membrane, and choose fixed or absorbing boundaries.
Standing Waves in Pipes takes a shortcut straight to that one emergent outcome. Instead of integrating the PDE from an arbitrary disturbance, its GLSL shader draws the exact algebraic solution for a string clamped at both ends: two travelling waves, A·sin(kx−ωt) moving right and A·sin(kx+ωt) moving left, sum by a trigonometric identity into y(x,t) = 2A·sin(kx)·cos(ωt) — a fixed spatial envelope that only pulses in time. Because the string is pinned at both ends, the wavenumber must satisfy kL = nπ for a whole number n, so only discrete harmonics λₙ = 2L/n and frequencies fₙ = n·v/2L can exist. Dragging the harmonic slider jumps directly between these modes and shows their nodes, antinodes, and — if you flip on the component toggle — the two travelling waves that add up to make them, which is exactly how a real plucked guitar or organ pipe string produces its overtone series.
Want to explore more? Browse the full library of 1000+ interactive, browser-based simulations, or see other side-by-side comparisons.