What is meant by interference of waves?

Explanation

Consider two harmonic waves having identical frequencies , constant phase difference `phi` and same wave form (can be treated as coherent source ) , but having amplitudes `A_(1)` and `A_(2)` ., then
`y_(1) = A_(1) sin ( kx - omega t) " " .... (1)`
`y_(2) = A_(2) sin (kx - omega t + phi)" " ... (2)`
Suppose they move simultaneously in a particular direction , then interference occurs (i.e., overlap of these two waves ) . Mathematically
`y = y_(1) + y_(2) " " ... (3)`

Therefore , substituting equation (1) and equation (3) in equation (3) , we get
`y = A_(1) sin (k x - omega t) + A_(2) sin (k x - omega t + "Times New Roma ")`
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 phi + cos (kx - omega t) sin phi]`
`y = sin (kx - omega t) (A_(1) + A_(2) cos phi ) + A_(2) sin phi cos (kx - omega t) " " ... (4)`
Let us re-define `A cos theta = (A_(1) + A_(2) cos phi) " ".... (5)`
and `A sin theta = A_(2) sin phi " " ...... (6)`
then equation (4) can be rewritten as `y = A sin ( kx - omega t) cos theta + A cos (kx - omega t) sin theta`
`y = (A sin (kx - omega t) cos theta + sin theta ( kx - omega t))`
`y = A sin ( kx - omega t + theta) " " .... (7)`
By squaring and adding equation (5) and equation (6) , we get
`A_(2) = A_(1)^(2) + A_(2)^(2) + 2 A_(1) A_(2) cos phi " " ...... (8)`
Since , intensity is square of the amplitude `(1 = A_(2))` we have
`I = I_(1) + I_2 + 2 sqrt(I_(1) I_(2)) cos phi " " ..... (9)`
This means the resultant intensity at any point depends on the phase difference at that point .
(a) For constructive interference : When cresis of one wave overlap with crests of another wave has a larger 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 phi = +1 implies = 0 , 2pi , 4pi ...... = 2n pi ` where n = 0 , 1 , 2.........
This is the phase difference in which two waves overlap to give constructive interference. Therefore, for this resultant wave,
`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 overlaps with· the crest of another wave, their amplitudes "cancel" each other and we get destructive interference as shown in figure (b). The resultant amplitude is nearly zero. The destructive interference occurs if there is minimum intensity at that point, which means cos `phi = -1 implies phi = pi , 3pi , 5 pi ,..... = (2n - 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)`