modedecomposition

Mode decomposition is a concept used in mathematics, physics, and signal processing to represent a complex system or signal as a superposition of simpler, orthogonal components called modes. Each mode typically captures a distinct aspect of the system’s behavior, such as frequency components in a time series, spatial patterns in a field, or eigenmodes of a differential operator.

The term originated from studies of differential equations and wave phenomena in the late 19th and early 20th centuries. Early works by Fourier, Helmholtz, and others established that solutions to linear partial differential equations could be expressed as sums of sinusoidal or exponential terms. Later, the modal analysis of mechanical systems and electrical circuits further formalized mode decomposition as a tool for simplifying dynamic problems.

Mathematically, mode decomposition often involves diagonalising an operator or applying orthogonal transformations. Fourier series provide the classic example, where a periodic function is expanded in terms of sine and cosine functions, each representing a harmonic mode. In linear algebra, eigenvector decomposition of a square matrix separates a vector space into invariant subspaces associated with eigenvalues. In structural engineering, normal mode analysis decomposes the vibration of a structure into independent oscillatory modes.

Practical applications span many fields. In acoustics, Fourier analysis separates sound into its constituent pitches. In fluid dynamics, proper orthogonal decomposition identifies coherent structures within turbulent flows. In electromagnetic theory, vector spherical harmonics expand fields around spherical bodies. Signal separation techniques, such as independent component analysis, build on mode decomposition principles to isolate underlying sources from recorded data.