Harmonic waves

Water waves, light waves, and sound waves are all functions of position and time. As they move they can carry energy and information. The simplest wave is a harmonic wave. Harmonic waves have the form,

Here $A$ is the amplitude, $\lambda$ is the wavelength, $T$ is the period, and $\varphi$ is the phase. To determine the period of the wave, focus on one position (like $x=0$). For a harmonic wave, the displacement of the wave at any position exhibits harmonic motion. The period of the wave is the period of the harmonic motion. A wave moves one wavelength forward every period. The wave velocity is therefore,

The formula for a harmonic wave can be written more compactly if we define the wavenumber $k=\frac{2\pi}{\lambda}$ and the angular frequency $\omega=\frac{2\pi}{T}=2\pi f$, where $f$ is the frequency.

The wave speed can also be expressed as $c=\frac{\lambda}{T}=\lambda f=\frac{\omega}{k}$. If $k\omega >0$ the wave moves in the $+x$ direction and if $k\omega <0$ the wave moves in the $-x$ direction.