Phase Difference

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)
  1. Phase difference : 0 radians (or multiples of $2 \pi$)
  2. 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.
  1. They have velocities in the opposite direction
  2. Phase difference: $\pi$  radians (or $\pi$, $3 \pi$, $5 \pi$, …)
  3. 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:

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

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