Consider two harmonic waves having identical frequencies, constant phase difference `omega` and same wave form (can be treated as coherent source), but having amplitude `A_1` and `A_2` then
`y_1 = A_1 sin (kx - omega t) ` …..(1)
`y_2 = A_2 sin (kx - omega t + omega) ` ....(2)
Suppose they move simultaneously iri 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 (kx - omega t) + A_2 sin (kx - omega t - psi)`
Using trigonometric identity `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 psi + cos (kx - omega t) sin psi]`
`y = sin (kx - omega t) (A_1 + A_2 cos psi) +A_2 sin psi cos (kx - omega t ) ` ......(4)
Let us re-difine `A cos theta = (A_1 + A_2 cos psi) ` .....(5)
and `A sin theta = A_2 sin psi` .....(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 cos (kx - omega t) `
` y = A sin (kx - omega t + theta ) ` ....(7)
By squaring and adding equation (5) and equation (6), weget
`A_2 = A_1^2 + A_2^2 + 2A_1 A_2 cos psi` .....(8)
Since, intensity is square of the amplitude `(I = A_2)` , we have
` I = I_1 + I_2 + 2 sqrt(I_1i_2) cos psi` .....(9)
This means the resultant intensity at any point depends on the phase difference at that point.
(i) For constructive interference: When crests of one wave overlap with crests of another wave, their amplitudes will add up and we get constructive interference. The resultant 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 psi = +1 rArr psi = 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_(max) = ( sqrtI_1 + sqrti_2)^2 = (A_1 + A_2)^2`
hence , the resultant amplitude
` A = A_1 + A_2`
(ii) 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 psi = - rArr rArr 1 rArr psi = pi , 3pi , 5pi ,...... = (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_(max) = (sqrtI_1 - sqrt I_2)^2 = (A_1 -A_2)^2`
Hence, the resultant amplitude
`A = |A_1 - A_2 |`
