Progressive Waves

A $\text{progressive (or travelling) wave}$ transfers energy through a medium — or through space — without transferring the medium itself. Drop a stone in a pond and the ripples spread outwards carrying energy, yet a floating leaf just bobs up and down: it does not travel with the wave.

Every point in the medium performs the same oscillation as the point next to it, but a little later in time. That tiny time lag from one point to the next is what makes the pattern appear to move forwards.

$\text{Displacement}$ ($x$ or $y$) is the distance of a point on the wave from its equilibrium position, and $\text{amplitude}$ ($A$) is the maximum displacement. Amplitude is linked to the energy the wave carries.

$\text{Wavelength}$ ($\lambda$) is the shortest distance between two points oscillating in phase — for example crest to crest. The $\text{period}$ ($T$) is the time for one complete oscillation, and the $\text{frequency}$ ($f$) is the number of complete oscillations per second, measured in hertz (Hz).

Frequency and period are reciprocals of each other, and the $\text{wave speed}$ ($c$) is how fast a point of constant phase — such as a crest — travels through the medium.

The $\text{phase}$ of a point tells you where it is in its cycle of oscillation. Because one complete cycle corresponds to going all the way round a circle, phase is measured as an angle — usually in $\text{radians}$, with one full cycle equal to $2\pi$ rad.

$\text{Phase difference}$ compares two points (or two waves). Two points exactly one wavelength apart are $2\pi$ rad out of phase — which is the same as being $\text{in phase}$. Points half a wavelength apart are $\pi$ rad out of phase, or in $\text{antiphase}$.

In general, the phase difference between two points a distance $d$ apart is found by scaling that separation to a full wavelength.