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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 $$x$$).

Phase difference of two particles on a wave
Phase difference of two particles on a wave

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.

Phase difference of two particles on a wave
Phase difference of two particles on a wave

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.

Phase difference of two particles on a wave
Phase difference of two particles on a wave