# Wave equation
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## Microsims — p5.js
### Wave equation (p5.js) · `∂²u/∂t²=c²∇²u`
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<iframe src="https://editor.p5js.org/sciencenibber/full/rGvUcgG8H" width="100%" height="480" frameborder="0" loading="lazy" sandbox="allow-scripts allow-same-origin" title="Wave equation — p5.js microsim"></iframe>
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*The differential equation every wave obeys, tying its curvature in space to its acceleration in time.*
**Open in the editor:** [▶ fork this sketch](https://editor.p5js.org/sciencenibber/sketches/rGvUcgG8H) · movement *VII · Antennas, waves & RF propagation* · library `p5js`
### Related microsims
Live sims on neighbouring articles — 2 of them inside this article's own Wikipedia link tree:
- [[Convolution]] *(in tree)*
- [[Impulse_response]] *(in tree)*
- [[Analog_signal]]
- [[Autocorrelation]]
- [[Bifurcation_theory]]
- [[Calculus]]
*Sim hosted off-article; the article owns the reference, not the runtime (WIKI_RULES §10.4). Placed by `g08_place_microsims.py`.*
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## Links (Wikipedia order)
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`Acoustic_attenuation` · `Acoustic_wave_equation` · `Acoustics` · [[Angular_frequency]] · `Bateman_transform` · `Bessel_function` · [[Cartesian_coordinate_system]] · `Chirp` · `Classical_physics` · `Coefficient` · [[Convolution]] · `Current_density` · `D'Alembert's_formula` · `D'Alembert_operator` · `David_Hilbert` · [[Density]] · `Differential_operator` · `Dirac_delta_function` · `Dirac_measure` · `Directional_derivative` · `Dispersion_(optics)` · `Dispersion_relation` · `Duhamel's_principle` · [[Earth]] · `Eigenmode_expansion` · `Electric_field` · `Electromagnetic_radiation` · `Electromagnetic_wave_equation` · `Electromagnetism` · `Elliptic_partial_differential_equation` · `Eric_W._Weisstein` · [[Fluid_dynamics]] · `Fourier_transform` · `Green's_function` · `Green's_theorem` · `Half-integer` · `Heaviside_step_function` · `Helmholtz_equation` · `Homogeneity_(physics)` · `Hooke's_law` · `Huygens–Fresnel_principle` · `Hyperbolic_partial_differential_equation` · `Improper_integral` · [[Impulse_response]] · `Infinitesimal` · `Inhomogeneous_electromagnetic_wave_equation` · `Initial_value_problem` · `Inverse-square_law` · `Lamé_parameters` · `Laplace_operator` · `Lars_Gårding` · `Lawrence_C._Evans` · `Leonhard_Euler` · `Light` · `Light_cone` · `Linear_combination` · `Linear_differential_equation` · `Linear_elasticity` · `Longitudinal_wave` · `MathWorld` · `Maxwell's_equations` · `Method_of_characteristics` · `Michael_Atiyah` · `Multiple_integral` · `N-sphere` · [[Ordinary_differential_equation]] · `Partial_derivative` · [[Partial_differential_equation]] · `Phase_factor` · `Phase_velocity` · `Physical_quantity` · `Plane_wave` · `Point_source` · `Poisson_kernel` · `Pressure` · `Pulse_(physics)` · `Raoul_Bott` · `Real_number` · `Reflection_phase_change` · `Residue_theorem` · `Richard_Courant` · `Scalar_field` · [[Schrödinger_equation]] · `Second_derivative` · `Separation_of_variables` · `Sign_function` · `Sinc_function` · `Spacetime` · `Spherical_harmonics` · `Standing_wave` · `Stephen_Wolfram` · `Stiffness` · `Stress_(mechanics)` · `String_vibration` · `Sturm–Liouville_theory` · `Superposition_principle` · `Tension_(physics)` · `Trigonometric_functions` · `Ultrasound` · `Vector_calculus` · `Vector_field` · `Volume_element` · `Wake_(physics)` · `Water` · [[Wave]] · `Wave_function` · `Wave_packet` · [[Wayback_Machine]] · `Wheeler–Feynman_absorber_theory` · `Wolfram_Demonstrations_Project` · `Young's_modulus`
> Signal Processing concept · part of the Signal Processing Portal · movement VII · !76 送電 sōden.svg
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*Rendered from the live microsim (▶ motion).*
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## See it next
[](Electromagnetic_wave)
*→ Electromagnetic wave*
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---
Back to Signal Processing Portal · the room · Semiotic gateway
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## From the Real GENERATIVE library
*Animated: Wave equation — placed from the Real G.E.N.E.R.A.T.I.V.E. course library (Electronics room). Source: Wikimedia Commons (via Wikipedia article media). [Details & license](https://commons.wikimedia.org/wiki/File:Wave_equation_1D_fixed_endpoints.gif).*
> The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves). ([Wikipedia](https://en.wikipedia.org/wiki/Wave_equation))
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## Media (PD/CC)
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!Gif Library/Wave equation/Wave equation 1D fixed endpoints.gif
*Wave_equation_1D_fixed_endpoints.gif · Public domain*
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## What it is
The wave equation is the second-order partial differential equation $\partial^2 u/\partial t^2 = c^2 \nabla^2 u$ whose solutions describe how a disturbance $u$ propagates through space at speed $c$.
## How it works / why it matters
It states that the acceleration of the field at each point is proportional to the local curvature, which forces disturbances to travel outward rather than stay put. Its one-dimensional d'Alembert solution is any pair of shapes moving left and right at speed $c$, and by superposition any waveform can be built from sinusoids — the link to Fourier analysis. Because it governs sound, light, radio, and vibrations alike, it is the unifying mathematics behind every propagating signal.
## Signs & universals
Instantiates: wave · oscillation · field — it is the law by which an oscillating field propagates as a wave.
## Related
The master equation obeyed by the Electromagnetic wave, Sound wave, Radio wave, and plane wave; its solutions define Phase velocity and Group velocity and the modes inside Waveguides.
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*Built to the [[WT!P5_js_Microsim_Master_Class|p5.js Master Class]].*
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<!-- ACOUSIM:BEGIN g22 — Acoustics portal microsim (framework build, specs/acoustics/variants/Wave_equation.json); do not hand-edit inside -->
**Microsim — three.js (Wikitube framework):** *Wave equation*
<div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/acoustics/Wave_equation.html" data-title="Wave equation"></div>
*Built from `MICROSIM_GUIDE/specs/acoustics/variants/Wave_equation.json`; part of the [[PORTAL_Acoustics|Acoustics portal]] spine (section sims and See-also variants).*
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## Wikipedia : Wikitube
**Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Wave_equation) : [Wikitube](https://en.wikitube.io/wiki/Wave_equation)
## Previous hub tags
Tree parent: [[Helium]].
Legacy hubs: none.
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*Sources: 1 legacy note. Minted wave 1, 2026-07-30 (v1.6 order).*