Obtain the formula of intensity if the phase difference at a point from two sources is `phi`.

Explanation

Suppose point G as shown in figure and let the path difference between two displacement be `phi` at that point.
If the displacement produced by `S_(1)` at G is
`y_(1)=acosomegat` then the displacement produced by `S_(2)` at G would be `y_(2)=acos(omegat+phi)`.
The resultant displacement according to superposition principle
`y=y_(1)+y_(2)=acosomegat+acos(omegat+phi)`
`=a[cosomegat+cos(omegat+phi)]`
`=2acos""(phi)/(2).cos(omegat+(phi)/(2))`
`[:.cosA+cosB=2cos((A+B)/(2))cos((A-B)/(2))]`
Amplitude of resultant displacements `=2acos((phi)/(2))` and and intensity of point G, `Iprop4a^(2)cos^(2)""(phi)/(2)`
`I=4I_(0)cos^(2)((phi)/(2))" "[:."t=where "I_(0)propa^(2)]`
If this equation is obeyed, then the constructive interference will formed at point G and interference leading to maximum intensity. For maximum intensity from this equation,
`phi=0,pm2pi,pm4pi`...... for the value `phi`, then constructive interference will formed and interference leading to maximum intensity.
Reverse from this equation, if at point G `phi=pmpi,pm3pi,pm5pi.....,` for the value of o, then destructive interference will formed and leading to zero intensity.