What is meant by interference of waves?

Explanation

Interference is a phenomenon in which two waves superimpose to form a resultant wave of greater,lowe or the same amplitude.

Let us consider two harmonic waves having identical frequencies constant phase difference `varphi` and same dwave form (can be trated as coherent source), but having amplitudes `A_(1) and A_(2)` then
`y_(1)=A_(1)s in(kx-omegat) " "....(1)`
`y_(1)=A_(1)s in(kx-omegat+varphi) " "....(2)`
Suppose they move simultaneously in a particular direction, then interference occur (i.e., overlap of these two waves). Mathematically
`y=y_(1)+y_(2) " "...(3)`
Hennce by substituting equation (1) and equation (2) in equation (3) we get
`y={{:(A_(1)sin(kx-omegat)),(+A_(2)sin(kx- omegat+varphi)):}`
Using trigonometric identity
`sin (alpha+beta)=(sin alpha cos beta+ cos alpha sin beta)`, we get
`y={{:(A_(1)sin(kx-omegat)),(+A_(2)sin(kx- omegat+varphi)),(+cos(kx-omega t) sin varphi"]"):}`
`{{:(sin (kx- omega t)(A_(1)+A_(2)cos varphi)),(+A_(2)sin varphi cos(kx-omegat)" "....(4)):}`
Let us re-define
`Acos theta=(A_(1)+A_(2)cos varphi) " "...(5)`
and `A sin theta=A_(2) sin varphi " "....(6)`
Then equation (4) can be rewritten as
`{{:(A sin (kx-omegat)cos theta),(+A cos (kx- omegat) sin theta):}`
`{{:(A "("sin (kx-omegat)cos theta),(+sin theta cos (kx- omegat) sin theta):}`
`A=A sin (kx- omegat+theta) " "...(7)`
By squaring and adding equation (5) and (6), we get
`A^(2)=A_(1)^(2)+A_(2)^(2)+2A_(1)A_(2)cos varphi...(8)`
Since intensity is square of the amplitude `(I=A^(2))`, we get
`I=I_(1)+I_(2)+ 2sqrt(I_(1)I_(2)) cos varphi " " ...(9)`
This means the resultant intensity at any point depends on the phase difference at that point.