Time-harmonic wave
= Time-harmonic wave
{title2=$\psi(\mathbf r)e^{-i\omega t}$}
A <time-harmonic wave> has sinusoidal dependence on <time> at a single real <angular frequency>. With the convention
$$
u(\mathbf r,t)=\operatorname{Re}\{\psi(\mathbf r)e^{-i\omega t}\},
$$
the spatial <complex amplitude> $\psi$ contains both local <amplitude> and <wave phase>. Differentiation with respect to <time> becomes multiplication by $-i\omega$. Substitution into the uniform <wave equation> gives the <Helmholtz equation> $\nabla^2\psi+k^2\psi=0$, with $k=\omega/c$.
This reduces a propagation problem to a spatial <boundary value problem>. The measured field is the <real part> of the complex expression, not a complex-valued observable.<The Feynman Lectures on Physics, Volume I by Richard Feynman and others>[Chapter 23, §23–1: real sinusoidal forces represented by complex exponentials]{ref} A radiating solution still needs an outgoing condition; its spatial factors and signs must agree with the selected time convention. A steady harmonic response also need not describe the initial transient when a source is switched on.