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Physical Sciences
Physics
1747
Intermediate

Wave Equation

∂2u∂t2=c2∂2u∂x2\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}

Describes how waves propagate—acceleration in time equals curvature in space times speed squared.

By Jean le Rond d'Alembert, Leonhard Euler

Physical Sciences
Wave Equation
1747 · Jean le Rond d'Alembert
Human Reviewed
84%

Rabbit Hole Mode

Five doors into the universe behind this equation. Choose your path.

Story PortalWho discovered this?Discover how Jean le Rond d'Alembert and others shaped this equation.Visual PortalWhat does it look like?See the equation come alive in the Visual Studio.Machine PortalWhere is it used?Explore machines powered by this equation — sound engineering.Math PortalWhat does it derive from?Trace the mathematical lineage from fourier transform.Future PortalWhere is it going?Gravitational wave detection
Launch collection — full story, visual, sound & share card
Why it matters: Foundation of acoustics, electromagnetism, quantum mechanics, and telecommunications.

Discoverers: Jean le Rond d'Alembert, Leonhard Euler (1747)

What does it mean?

Describes how waves propagate—acceleration in time equals curvature in space times speed squared.

Why should I care?

Foundation of acoustics, electromagnetism, quantum mechanics, and telecommunications.

Equation Compass

North — Prerequisites

West — History

East — Applications

South — Derivations

Variables & Units

SymbolNameUnitMeaning
uuDisplacement—Wave amplitude
ccWave speedm/sPropagation speed
ttTimesTime variable
xxPositionmSpatial coordinate

Worked Example

String fixed at ends vibrates at harmonics f_n = nc/(2L).

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Sources & further reading

Search WikipediaSearch Wolfram MathWorldOpen in Wolfram|AlphaWatch explainers on YouTube

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Equation Universe

Wave Equation

∂2u∂t2=c2∂2u∂x2\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}

Real-world impact

Wireless technology

Electromagnetic waves carry information worldwide.

Photo: Unsplash — network infrastructure

Describes how waves propagate—acceleration in time equals curvature in space times speed squared.

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