Wave

A wave is a propagating or standing disturbance that can carry energy and momentum. Its oscillations are described by quantities such as wavelength, frequency, and wave speed.

A complex number can encode both the wave amplitude and phase of a harmonic disturbance: $a(x,t)=\mathrm{Re}\{\hat{a}{e}^{i(kx-\omega t)}\}$. Its modulus is the real oscillation amplitude and its complex argument fixes the wave phase.

In a reciprocal medium, appropriately normalized source and receiver responses can be exchanged: the scalar Green function satisfies $G(\mathbf{r},{\mathbf{r}}^{\prime })=G({\mathbf{r}}^{\prime },\mathbf{r})$. Reciprocity alone does not imply absence of wave absorption.

Frequency is the number of repetitions of a periodic phenomenon per unit time. For a wave of period $T$, its frequency is $\nu =1/T$, measured in hertz. The angular frequency is $\omega =2\pi \nu$.

The Doppler effect changes the observed frequency of a wave when source and observer move relative to one another. For light and a small radial recession speed $v_r\ll c$, $\Delta \lambda /\lambda \simeq v_r/c$, where $\lambda$ is wavelength. This is the basis of Doppler spectroscopy.