Interference of waves, intensity, practice problems, FAQs

When you throw two stones in a pound at some distance the wave pattern will form shown in figure. The thick lines represent crests and thin lines represent troughs. When these wave patterns overlap then you will find at some places the amplitude of the wave is higher than individual (point A and B) and some place the amplitude is zero or minimum(point C). This is called interference of waves.interference of waves is just the superposition of waves at a point.

Table of content

• Coherent waves
• Interference of wave
• Special Cases of Interference
• Intensity of Resultant wave
• Practice problem
• FAQs

Coherent waves

The waves which have the same frequency, wavelength and constant phase difference between them are known as the coherent waves, and the source that produces the coherent waves is known as the coherent source. Example: y1 = A1 cos (ωt - kx) and y2 = A2 cos (ωt - kx + ) are the two coherent waves.Where is phase difference.

Interference of wave

The phenomenon of superposition of two or more waves having the following properties is known as interference of the waves.

The properties are as follows: 

1. The waves must move along the same direction.
2. The waves must have the same wavelength.
3. The waves must have the same frequency.
4. There must be a constant phase difference between the waves.

Now Consider of two coherent waves are as follows:

y1 = A1 sin (ωt - kx)

y2 = A2 sin (ωt - kx + ) 

From the principle of superposition, we get,

 In this case, ynet = y = y1+ y2

⇒  y = A1 sin⁡(ωt - kx) + A2 sin⁡(ωt - kx + )

⇒  y = A1 sin⁡(ωt - kx) +[A2 (sin(ωt - kx) cos + ⁡cos(ωt - kx)sin ] 

⇒  y = (A1+ A2cos ) sin⁡(ωt - kx) + A2 ⁡cos(ωt - kx)sin …….(i)

Let (A1+ A2cos ) = A cos α…..(ii)

A2 sin = A sin α …….(iii)

By substituting these values in equation (i), we get the following:

⇒ y = A cos α sin⁡(ωt - kx) + A sin α ⁡cos⁡(ωt - kx)

⇒ y = A[cos α sin⁡⁡(ωt - kx) + sin α ⁡cos(ωt - kx)]

⇒ y = A sin⁡(ωt - kx + α) (Since sin(A + B)= sin A cos B + cos A sin B)

By squaring and adding equations (ii) and (iii), we get,

⇒ A2 sin2α + A2 cos2 α = (A1+ A2cos )2 + (A2 sin )2 

By dividing equations (iii) by equations (ii), we get the following:

Therefore, it can be concluded that the interference of the two sinusoidal coherent traveling waves moving in the same direction gives rise to another sinusoidal coherent traveling wave moving in that same direction.

Special Cases of Interference

Case 1: cos = 1 (Constructive interference)

For cos = 1,

⇒ cos = cos(2nπ)

⇒ = 2nπ

Where n = Integer = 0, 1, 2, 3…

Therefore, if the phase difference is an even multiple of π, then constructive interference takes place.

For this case, the amplitude of the resultant wave becomes,

⇒ A = Amax = A1 + A2

Interference that produces the greatest possible amplitude is known as constructive interference.

Path difference for constructive interference:

Phase difference Path difference

Path difference

If path difference is in integral multiple of wavelength then interference is constructive.

Case 2: cos = - 1 (Destructive interference)

For cos = -1,

⇒ cos = cos[(2n - 1)π] (odd multiple of π)

⇒ = (2n - 1)π

Where n = 1, 2, 3…

Therefore, if the phase difference is an odd multiple of π, then destructive interference takes place.

Thus, the amplitude of the resultant wave becomes,

⇒ A = Amin = A1 - A2

Interference that produces the minimum possible amplitude is known as destructive interference.

Path difference for destructive interference:

Phase difference Path difference

Path difference

If path difference is odd multiple of half of wavelength then interference is destructive.

Intensity of Resultant wave

We know that the intensity of any sinusoidal wave is proportional to the square of its amplitude, i.e.,

Now we know that

Therefore the net intensity

The intensity of resultant wave is given by the above equation

Now If

For constructive interference: 

Constructive interference takes place when the phase difference between the interfering waves becomes an even multiple of π, i.e., = 2nπ, where n = Integer = 0, 1, 2, 3...

In this case, the intensity of the resultant wave becomes, 

[By putting = 2nπ or cos = 1]

Therefore, the maximum intensity becomes, 

Now, if I1 = I2 = I, then, 

For destructive interference: 

Destructive interference takes place when the phase difference between the interfering waves becomes an odd multiple of π, i.e., = (2n - 1)π, where n = 1, 2, 3…

In this case, the intensity of the resultant wave becomes,

[By putting = (2n - 1)π or cos = - 1]

Therefore, the minimum intensity becomes, 

Now, if I1 = I2 = I, then 

Practice problem

Q 1.Two waves have equations x1 = a sin⁡(ωt - kx + ɸ1) and x2 = a sin⁡(ωt - kx + ɸ2). After interference of the waves the resultant wave has the amplitude equal to the superimposing wave , then find the phase difference between them?

a. 
b. 
c. 
d. 

A. Given,

x1 = a sin⁡(ωt - kx + ɸ1)

x2 = a sin⁡(ωt - kx + ɸ2)

Also,

The phase difference is given as follow:

Thus, option (B) is the correct answer. 

Q 2.Two sinusoidal waves of the same frequency travel in the same direction along a string. If Amplitude of waves A1 = 3.0 cm, A2 = 4.0 cm, And initial phases ϕ1 = 0, and ϕ2 ,Find the amplitude of the resultant wave?

A.The amplitude of the resultant wave is given by,

The phase difference between the given waves is as follows:

By substituting all the given values in this equation, we get,

Therefore, the amplitude of the resultant wave is 5 cm.

Q 3.Consider two wave sources and as shown in the figure. Both sources emit simple harmonic waves of same frequency but of different amplitudes,but in phase. Let O be any point equidistant from and as shown in the figure. Comment on the intensity at points O, Y and X.

A. The distance between and are the same and so, the waves starting from and reach O after covering equal distances.The path difference between two waves at O is zero.

− = 0

Because of the same phase, at the point O, the phase difference between two waves is zero. Thus, the intensity at the point O is maximum.

Now, consider a point Y, such that the path difference between two waves is λ. The phase difference at Y is

At the point Y, the two waves are in phase, hence, the intensity will be maximum.

Now, consider a point X, such that the path difference between two waves is λ/2.

The phase difference at X is

At the point X, the waves meet and are out of phase, Hence, due to destructive interference, the intensity will be minimum.

Q 4. If the ratio of the intensities of two interfering wave is . Find the value of .

A.Given

We know that

Putting the values in expression

FAQs

Q 1. Why brilliant colors in a thin film of oil on water observed when exposed to sunlight?
A. This is observed due to interference of reflected light from the upper and lower surface of the film.

Q 2. Does the sound wave undergo interference?
A. Yes, sound waves also cause interference.

Q 3. Which of the following has the best monochromatic light?

a. Bulb
b. LED bulb
c. Leser
d. Mercury Tube

A. Lasers produce the best monochromatic light.

Q 4. What are the uses of interference of waves?
A. Surface quality testing,measuring light intensity in Space application, etc

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