Consider two harmoic waves having identical frequencies constant phase difference ` varphi ` and same wave form ( can be treated as cohernt source ) , but having amplitudes `A_1` and `A_2` then
`Y_ 1 = A_1 sin (Kx - omega t)`
` Y_2 = A_2 sin ( kx - omegat+varphi )`
suppose they more simulaneously in a particular direction , then interference occurs ( i.e., overlap of these two waves ) . Mathematically
` y= Y_1 +Y_2`
therefore , substiuting equation (1) and equation (3) in eqation (3) , we get
`Y= A_1 sin (Kx - omega t ) +A_2 sin ( k x -omega t + varphi ) `
using trigonometric identify sin ` (alpha +beta) = ( sin alpha cos beta + cos alpha sin beta ) , ` we get
`y=A _1 sin ( kx - omega t ) + A_2 [ sin (kx -omega t cos varphi + cos ( kx - omega t ) sin varphi ]`
` y= sin (kx - omega t ) (A_1 +A_2 cos varphi ) A_2 varphi cos (kx - omega t)`
Let us re - define ` A cos theta = (A_1 +A_2 cos varphi )`
and ` A sin theta = A_2 sin varphi`
then equation (4) can be rewritten as `Y= A sin ( kx - omega t ) theta + A cos ( kx -omega t ) sin theta `
` y=A ( sin ( kx - omega t) cos theta + sin theta cos ( kx - omega t)`
` y=A sin ( kx - omega t + theta ) `
By squaring and adding equation ( 5) and equation ( 6) we get
` A_2 = A_(1)^(2) + A_(1)^(2) + A_(2)^(2) + 2A_1 A_2 cos varphi`
since intensity is square of the amplitude `(I = A_2 ) `, we have
`I=I_1 +I_2 +2 sqrt(I_1 I_2 ) cos varphi`
this means the resultant intensity at any point depends on the difference at that point
(a ) for constructive interference : when depends creasts of one wave overlap with creasts of another wave has lager amplitude than the individual waves as shown in figure (a) .
The constructive interference at a point occurs if there is maximum intensity at that point which means that
` cos varphi = + 1 1 implies varphi = 0 , 2 pi , 4 pi ,... = 2 n pi `
this is the plase difference in which two waves
` I_("maximum ") = ( sqrt(I_1) + sqrt(I_2))^2 = (A_1 +A_2 )^2`
hence the resultant amplitude
` A=A_1 +A_2`
(b) For destructive interference : when the trough of one wave overlap with the creast of another wave , their amplitudes .. cancel .. each other and we get destructive interference as occurs if these is minimum internsity at that point which means ` cos varphi =- 1implies varphi = pi , 3 pi 5 pi ,... = (2 n-1) pi ` , where n.=0,1,2,... i.e, this is the phase difference in which two waves overlap to give destructive interference . therefore ,
`I_("minimum ") = ( sqrt(I_1)- sqrt(I_2))^2 =(A_1 -A_2 ) ^2 `
hence ,the resultant amplitude
` A= |A_1 -A_2|`
