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# Interference of waves

When two waves of same frequency travelling in the same direction in a medium superpose with each other, their resultant intensity is maximum at some points and minimum at some other points. This phenomenon of superposition is called interference.

Interference of waves

When two waves of same frequency travelling in the same direction in a medium superpose with each other, their resultant intensity is maximum at some points and minimum at some other points. This phenomenon of superposition is called interference.

Let us consider two simple harmonic waves of same frequency travelling in the same direction. If a1 and a2 are the amplitudes of the waves and φ is the phase difference between them, then their instantaneous displacements are

y1 = a1 sin ωt    ?.(1)

y2 = a2 sin (ωt + φ)   ?.(2)

According to the principle of superposition, the resultant displacement is represented by

y = y1 + y2

= a1 sin ωt + a2 sin (ωt + φ)

= a1 sin ωt + a2 (sin ωt. cos φ + cos ωt.sin φ)

= (a1 + a2 cos φ) sin ωt + a2 sin φ cos ωt ...(3)

Put a1 + a2 cos φ = A cos θ ...(4)

a2 sin φ = A sin θ ...(5)

where A and θ are constants, then

y = A sin ωt. cos θ + A cos ωt. sin θ

or

y = A sin (ωt + θ) ...(6)

This equation gives the resultant displacement with amplitude A. From eqn. (4) and (5)

A2cos 2 θ + A2 sin 2 θ

= (a1 +a2 cos φ) 2 + (a2 sin φ) 2

A2 = a1 2 + a2 2 + 2a1a2 cos φ

A = root[  a12 + a22 + 2a1 a2 cosφ ] ... (7)

Also tan θ = ( a2 sin φ )   + ( a1 +a2 cos φ ) ...(8)

We know that intensity is directly proportional to the square of the amplitude

(i.e) I α A2

I α (a1 2 + a2 2 + 2a1a2 cos φ) ... (9)

Special cases

The resultant amplitude A is maximum, when cos φ = 1 or φ = 2mπ where m is an integer (i.e) Imax α (a1+ a2) 2

The resultant amplitude A is minimum when

cos φ = ?1 or φ = (2m + 1)π

I min α (a1 ? a2) 2

The points at which interfering waves meet in the same phase φ = 2mπ i.e 0, 2π, 4π, ... are points of maximum intensity, where constructive interference takes place. The points at which two interfering waves meet out of phase φ = (2m + 1)π i.e π, 3π, ... are called points of minimum intensity, where destructive interference takes place.

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11th 12th std standard Class Physics sciense Higher secondary school College Notes : Interference of waves |