eigenmodes

An eigenmode is a pattern or configuration of a system that evolves independently of other patterns under linear dynamics. When a system is linear, any initial state can be expressed as a combination of eigenmodes, and each mode changes in time only by a simple scaling or phase factor determined by its eigenvalue. Mathematically, an eigenmode is a function or vector φ that satisfies an eigenproblem of the form L φ = λ φ, where L is a linear operator and λ is a scalar called an eigenvalue. The time evolution or response of that mode is then governed by factors such as e^{λ t} or e^{i ω t}, depending on the context.

In continuous systems, eigenmodes are eigenfunctions of differential operators with appropriate boundary conditions. For example, the Laplacian operator Δ on a domain with fixed boundaries yields spatial eigenfunctions φ_n that satisfy Δ φ_n = -k_n^2 φ_n, with boundary conditions. In time-dependent problems, the solution can often be written as a sum of spatial eigenfunctions each multiplied by a temporal factor, such as sinusoids e^{i ω_n t}. Classical examples include vibrating strings and drums, where the eigenmodes are standing wave patterns, and more generally normal modes in mechanical, acoustic, and optical systems.

Applications of eigenmodes include modal analysis, where complex dynamics are decomposed into a small number of dominant modes to simplify analysis and design. They provide natural coordinates for solving partial differential equations and help connect observable patterns to intrinsic properties like natural frequencies and mode shapes in physics, engineering, and beyond.