Timeharmonic

Timeharmonic refers to a physical quantity that oscillates at a single temporal frequency. In practice, time-harmonic signals are often represented as the real part of a complex exponential, f(x,t) = Re{F(x) e^{i ω t}}, where ω is the angular frequency and F(x) is a complex amplitude that encodes spatial variation. The frequency f is related by f = ω/(2π).

In mathematical modeling, time-harmonic behavior is used to convert time-dependent problems into frequency-domain problems. Substituting a time-harmonic ansatz into a linear PDE replaces time derivatives by i ω factors. For example, a wave equation ∂^2 u/∂t^2 = c^2 ∇^2 u becomes ∇^2 U + (ω^2/c^2) U = 0, where U(x) is the complex spatial amplitude. This is the Helmholtz equation. Similar reductions apply to electromagnetism and quantum mechanics via Maxwell's equations or the Schrödinger equation, yielding frequency-domain formulations that are often easier to solve numerically or analytically.

Applications span acoustics, optics, radio and microwave engineering, and quantum mechanics. Time-harmonic analysis underpins phasor methods in circuit theory and is a cornerstone of finite element and boundary element techniques used to solve boundary-value problems at a given frequency.

Limitations include its restriction to linear, time-invariant systems and a single dominant frequency; real signals are rarely perfectly time-harmonic, and transient or broadband phenomena require full time-domain analysis or superposition of many harmonics.