Coupled system
Two fields exchange spatial derivatives.
uₜ+cvₓ=0, vₜ+cuₓ=0
Wave Mode Analysis
Eigenvalues, dispersion, and propagation
Wave-mode analysis diagonalizes coupled linear PDEs into independent patterns with their own propagation speeds.
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The essential idea: the eigenvalues of the coefficient matrix determine the allowed wave speeds.
One Concept · One Example
Wave Mode Analysis
Presented by Charlotte Moser · Coming soon
What to notice
- The coupled variables hide two independent modes.
- w+=u+v travels right.
- w−=u−v travels left.
The mathematical structure
Decoupled modes
A change of variables reveals two transport equations.
(w+)ₜ+c(w+)ₓ=0, (w−)ₜ−c(w−)ₓ=0
Split one disturbance into two wave modes
Follow right- and left-traveling packets, then reconstruct the physical variables u and v.
w+ rightw− leftu=(w++w−)/2v=(w+−w−)/2
Diagonalization reveals independent wave modes hidden inside coupled equations.