Phase Difference ($\phi$) between two particles or two waves tells us how much a particle (or wave) is in front or behind another particle (or wave).

- Value ranges from 0 to $2 \pi$ radians

**Referring to the diagram above,**

- P1 and P2 are in phase. They are in exactly the same state of disturbance at any point in time.(have same displacement and velocity)
- Phase difference : 0 radians (or multiples of $2 \pi$)
- Distance between 2 particles (path difference) is an integer multiple of the wavelength.

- P1 and P3 are $\pi$ radian out of phase. They are $\frac{1}{2}$ a cycle apart from each other at any point in time.
- They have velocities in the opposite direction
- Phase difference: $\pi$ radians (or $\pi$, $3 \pi$, $5 \pi$, …)
- Path difference: odd multiple of half a wavelength (i.e. $\frac{1}{2} \lambda$, $\frac{3}{2} \lambda$ , …)

**Referring to the graph above,**

$\phi = 2 \pi \frac{x}{\lambda}$ OR $\phi = 2 \pi \frac{t}{T}$

Phase difference, $\Delta \phi$ between 2 particles is just the difference in phase between them.

$\Delta \phi$ between A and B: $\Delta \phi = 2 \pi \frac{\Delta t}{T}$ or $\Delta \phi = 2 \pi \frac{\Delta x}{\lambda}$

**Wave equation:**

- If wave start from equilibrium, use sin
- If wave start from extreme displacement, use cos
- If wave starts below equilibrium, put negative sign in front

Examples:

$y = y_{o} \, sin \left( x \frac{2 \pi}{\lambda} \right)$

$y = – y_{o} \, cos \left( t \frac{2 \pi}{T} \right)$