Question

Explain how the interference of waves is formed.

Answer 1

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

Interference of two sinusoidal waves

Let us consider two harmonic waves having identical frequencies, constant phase difference φ, and same waveform (can be treated as coherent source), but having amplitudes A₁ and A₂, then

y₁ = A₁ sin(kx – ωt) … (1)

y₂ = A₂ sin(kx – ωt) … (2)

Suppose they move simultaneously in a particular direction, then interference occurs (i.e., the overlap of these two waves). Mathematically y = y₁ + y₂ … (3)

Hence by substituting equation (1) and equation (2) in equation (3), we get

y₁ = A₁ sin(kx − ωt)

y₂ = A₂ sin(kx − ωt + φ)

y = `{("A"_1 sin (k"x" - omega"t")),(+ "A"_2 sin(k"x" - omega"t" + φ)):}`

Using trigonometric identity

sin (α + β) = (sin α cos β + cos α sin β), we get

y = `{("A"_1 sin (k"x" - omega"t")),(+ "A"_2 sin(k"x" - omega"t" + φ)),(+ cos (k"x" - omega"t") sin φ):}`

y = `{(sin (k"x" - omega"t")("A"_1 + "A"_2 cos φ)),(+ "A"_2 sin φ cos (k"x" - omega"t")):}`   ...(4)

Let us re-define

A cos θ = (A₁ + A₂ cos φ)    ....(5)

and A sin θ = A₂ sin φ    ....(6)

then equation (4) can be rewritten as 

y = `{("A" sin (k"x" - omega"t") cos theta),(+ "A" cos(k"x" - omega"t")sin theta):}`

y = `{("A" sin (k"x" - omega"t") cos theta),(+ sin theta cos(k"x" - omega"t")):}`

y = A sin (kx - ωt + θ)   ....(7)

By squaring and adding equation (5) and (6), we get,

A² = A₁² + A₂² + 2A₁A₂cosφ … (8)

Since, intensity is square of the amplitude (I – A²), we get,

I = I₁ + I₂ + 2`sqrt("I"_1"I"_2)`cos φ   ....(9)

This means the resultant intensity at any point depends on the phase difference at that point.