# Phase

The phase $$(\phi)$$ of a particle in a sinusoidal wave is the fraction of a complete cycle of a wave (i.e. a fraction of the angle $$2\pi$$) that has been completed (from the beginning of a cycle) when the particle is at a certain displacement.

If one knows the phase of a wave at a given time, one can calculate the displacement of a particle in the wave at that time. This is because the displacement of a particle is a function of the phase.

Like in oscillations, the phase has **no physical significance** (i.e. it cannot be physically measured). It is not defined for non-sinusoidal waves.

The phase of a sinusoidal wave is given by $$$\phi=\frac{2\pi d}{\lambda}.$$$ $$d$$ is the distance of the particle (along the direction of wave motion) from the starting position of a wave cycle (not to be confused with the oscillation displacement $$x$$).

The phase difference $$(\Delta\phi)$$ is the difference in phase between two particles in a wave. It is given by: $$$\Delta\phi=\frac{2\pi\vert d_{2}-d_{1}\vert}{\lambda}$$$ $$d_{2}$$ and $$d_{1}$$ are the distances of the particles from the starting positions of the wave cycles. The starting position can be defined arbitrarily.

$$\lambda=$$wavelength.

Particles with the same phase are at the same displacement from the equilibrium position at the same time. These particles are said to be **in phase**. Two particles at the crests of a wave at a particular time are in phase.

Particles that are in phase are sometimes denoted as wavefronts.

Ocean waves are sometimes represented as a series of parallel lines (i.e. the wavefronts) moving in a direction perpendicular to each line. The lines typically represent the crests of the wave.

Particles with different phases at a particular time are said to be **out of phase**.

A particle on a crest and a particle on a trough are out of phase by $$180^{\circ}$$ or $$\pi\text{ rad}$$. These particles are said to be in **anti-phase**.