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Wave equation (PDE)
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The partial differential equation governing waves.
∂²y/∂x² = (1/v²)∂²y/∂t²∂²y/∂x² = (1/v²)∂²y/∂t² governs non-dispersive waves. Any function f(x − vt) or g(x + vt) solves it.
Plucked string: d'Alembert's solution.
It's linear, so superposition holds.
In more detail
The wave equation is a partial differential equation, ∂²y/∂t² = v² ∂²y/∂x², that describes how waves propagate in one dimension. Its solutions include travelling and standing waves.
Real-world uses
- Engineers use the wave equation to model vibrating strings and pipes.
- Geophysicists use it to model seismic wave propagation.
- Engineers use the wave equation to predict how a string, a pipe or an electromagnetic wave changes shape over time.
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Frequently asked questions
What are its solutions?
Any function of x minus vt or x plus vt, which represent waves travelling in each direction.
Does it apply in three dimensions?
Yes, with the Laplacian operator replacing the second derivative in x.
What does the equation say?
The curvature of the wave in space is proportional to its acceleration in time, which fixes the wave speed v.
Related topics in Waves & Sound
Acoustics of rooms · Musical scales and intervals · Fourier analysis · Solitons · Noise cancellation