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 *a*_{1} and *a*_{2} are the amplitudes of the waves and φ is the phase
difference between them, then their instantaneous displacements are

y_{1}
= a_{1} sin ωt ?.(1)

y_{2}
= a_{2} sin (ωt + φ) ?.(2)

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

y
= y_{1} + y_{2}

= a_{1} sin ωt + a_{2} sin (ωt
+ φ)

=
a_{1} sin ωt + a_{2} (sin ωt. cos φ + cos ωt.sin φ)

=
(a_{1} + a_{2} cos φ) sin ωt + a_{2} sin φ cos ωt
...(3)

Put
a_{1} + a_{2} cos φ = A cos θ ...(4)

a_{2}
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)

A^{2}cos
^{2} θ + A^{2} sin^{ 2} θ

=
(a_{1} +a_{2} cos φ) ^{2} + (a^{2} sin φ) ^{2}

∴A^{2}
= a_{1} ^{2} + a_{2} ^{2} + 2a_{1}a_{2}
cos φ

∴ A = root[ a_{1}^{2} + a_{2}^{2}
+ 2a_{1} a_{2} cosφ ] ... (7)

Also
tan θ = ( a_{2} sin φ ) + ( a_{1}
+a_{2} cos φ ) ...(8)

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

(i.e)
I α A^{2}

I
α (a_{1} ^{2} + a_{2} ^{2} + 2a_{1}a_{2}
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) I_{max} 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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