# What is wave interference? What is its mathematical formula?

## Question:

What is wave interference? What is its mathematical formula?

## Interference of Waves

Waves propagate in accordance with the classical wave equation. The linear nature of this equation suggests that waves can be superposed. That is, any number of waves can coexist in the same region of space at the same time. This has physical consequences and leads to the phenomenon of interference.

Since a wave is an oscillating physical quantity with an amplitude and phase angle it may be mathematically represented by a rotating arrow called a phasor. These are just like vectors with the magnitude replaced by the amplitude and the direction angle replaced by the phase angle. The resultant phasor may be determined using the parallelogram law of addition.

The intensity of a wave is the energy crossing unit perpendicular area per second. It is seen that intensity is proportional to the square of the amplitude. Thus if we have two waves of say the same frequency and of amplitudes {eq}\displaystyle {A_1} {/eq} and {eq}\displaystyle {A_2} {/eq} interfering with the phase difference {eq}\displaystyle {\phi} {/eq} then the resultant wave has an amplitude,

{eq}\displaystyle { A= \sqrt{A_1^2+A_2^2+2A_1A_2 \cos \phi}} {/eq}

Hence if the individual intensities are,

{eq}\displaystyle {I_1=KA_1^2} {/eq} and {eq}\displaystyle {I_2=kA_2^2} {/eq} then the resultant intensity is,

{eq}\displaystyle { I=kA^2=I_1+I_2+2kA_1A_2 \cos \phi} {/eq}.

The last term is called the interference term.

Now intensity detectors usually read the time average of the rapidly oscillating intensities. In the majority of situations, the time average of the interference term will vanish. Then we have the incoherent addition of intensities. If the term survives the averaging process then we have coherent addition. This condition is achieved in Young's double-slit arrangement. Interference will manifest as regions of alternating high and low intensities. Thus clustering and anti-clustering of energy is a telltale sign of interference. 