Time-harmonic wave ($\psi (\mathbf{r})e^{-i\omega t}$)

A time-harmonic wave has sinusoidal dependence on time at a single real angular frequency. With the conventionthe 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.^[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.