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What do you Mean by Constructive and Destructive Interferences?
Principle of Superposition of Waves
Simultaneous propagation of a number of waves through the same space in a medium is called superposition of waves. During superposition, while one wave superposes on another, individual properties remain unaltered.
Let us consider the situation where at any point in a medium a number of waves are incident at the same time. Displacement of the point would have been different if the waves passed through it individually. But as all the waves are incident at the same time, the point undergoes a resultant displacement (since displacement is a vector quantity). Clearly, resultant displacement is the vector sum of the displacements produced by each wave. This is the principle of superposition of waves. The principle can he expressed as follows.
At any instant, resultant displacement of a point in a medium due to the influence of a number of waves is equal to the vector sum of displacements produced by each individual wave at that point at that instant.
Interference of Light
Principle of superposition of waves is applicable to all types of waves, that is for sound waves, light waves, and other electromagnetic waves, as well. An example of superposition of waves is interference which was first demonstrated experimentally by Young in 1801.
Young’s Double Slit experiment: Young’s experimental arrangement is as follows. Two narrow slits A and B are made in close proximity to each other on an obstacle O placed in front of a source of a monochromatic light M. Being placed symmetrically about source M, slits A and B act as a pair of coherent sources when illuminated by the source [Fig.(a) and (b)]. If the laboratory is sufficiently dark,
alternate bright and dark lines can be seen on the screen S placed behind O [Fig.]. These alternate dark and bright lines are called interference fringes.
Definition: When two light waves of the same frequency and amplitude (or nearly equal amplitude) superpose in a certain region of a medium, the intensity of the resultant light wave increases at certain points and decreases at some other points in that region. This phenomenon is known as interference of light.
Increase in intensity of light is due to constructive interference and decrease in intensity is due to destructive interference. Increase or decrease of intensity in the resultant light wave depends on the phase difference of the superposing light waves at that point.
Constructive and destructive Interferences: Assume that amplitude of each of the two light waves of same frequency is A. While propagating in the same direction through a medium they superpose at a point in the medium. Resultant amplitude at the point is equal to the vector sum of the amplitude of the original waves (by the principle of superposition of waves).
If superposition takes place in the same phase, then the resultant amplitude = A + A = 2A. As intensity is directly proportional to the square of amplitude, the resultant light will be four times as intense as the individual wave. This is called constructive Interference. On the other hand if superposition taks place in opposite phases, the resultant amplitude = A – A = 0 and intensity of light is also zero. This is called destructive interference.
If the phase difference or phase relation between two waves remains the same then the interference pattern at every point of the medium remains the same.
It may be noted that destructive interference does not imply destruction of energy. No loss of energy takes place, only the energy of the dark points is transferred to that of the bright points, so that the total energy of the incident waves remains constant. In other words, there is only redistribution of light energy over the region of superposition.
Analytical treatment of interference: Let C and D be two sources of monochromatic light. Amplitude of each wave = A, wavelength = λ and speed = c [Fig.]. Two light waves moving in the same direction superpose at the point E. The resultant displacement of point E due to superposition is the algebraic sum of two individual displacements produced by the two waves. Distances of the point E from the two sources are x and (x + δ), respectively. So the path difference of the waves at that point is δ. If y1 and y2 are the displacements of point E due to the waves produced from sources C and D in time t,
y1 = A sin\(\frac{2 \pi}{\lambda}\)(ct – x)
y2 = A sin\(\frac{2 \pi}{\lambda}\) [ct – (x + δ)]
Resultant displacement of point E
y = y1 + y2
Sine function in equation (1) suggests that the resultant wave is also a wave of velocity c and wavelength λ.
Again equation (2) shows, the amplitude A’ of the resultant wave is not equal to the amplitude of the two superposing waves but modified by their path difference δ.
i) Condition for destructive interference: If δ = \(\frac{\lambda}{2}\), \(\frac{3 \lambda}{2}\), \(\frac{5 \lambda}{2}\),…… = (2n + 1)\(\frac{\lambda}{2}\) (where n = 0 or any integer), cos\(\frac{\pi \delta}{\lambda}\) = 0 and hence A’ = 0. Amplitude being zero, intensity is also zero. Thus at points where path difference between the waves are odd multiples of , intensity of light is zero. These points are dark points. The two waves produce destructive interference at such points.
Thus phase difference between the two waves at points where destructive interference takes place is
Δϕ = \(\frac{2 \pi}{\lambda}\)(2n + 1) = (2n + 1)π
[since phase difference, Δϕ = \(\frac{2 \pi}{\lambda}\) × path difference]
ii) Condition for constructive Interference: If δ = o, \(\frac{2 \lambda}{2}\), \(\frac{4 \lambda}{2}\) = 2n\(\frac{\lambda}{2}\) (where n = 0 or any integer), cos\(\frac{\pi \delta}{\lambda}\) = ±1 that is A’ = ±2A . Amplitude being the highest, intensity is also maximum. Thus at points where path difference between the waves is an even multiple of \(\frac{\lambda}{2}\), the intensity of light is maximum. These points are bright points. Two waves produce constructive interference at these points.
Thus phase difference between the two waves at points where constructive interference takes place is
Δϕ = \(\frac{2 \pi}{\lambda} \cdot \frac{2 n \lambda}{2}\) = 2nπ
It is to be noted that, the point E in Fig. is not a single point that satisfies equation (1). That is, either of the conditions of destructive and constructive interferences is obeyed not only at a single point in space, but for a set of points. The locus of the point E is, in general, hyperbolic.
But when E is at a large distance from the sources C and D, compared to their mutual distance CD, the locus of E is effectively a straight line. As a result, dark and bright straight lines are obtained [Fig.] instead of dark and bright points, due to destructive and constructive interference, respectively. These are called dark and bright interference fringes.
Intensity: Clearly, interference of two waves results in a variation of intensity of light at different points. Amplitude of the resultant wave, A’ = 2A cos\(\frac{\pi \delta}{\lambda}\) and therefore can vary from 0 to ± 2A.
As intensity is directly proportional to the square of amplitude, value of intensity increases from 0 to 4A2. That is maximum intensity can be four times the intensity of a single wave. If I is the intensity at a point on the screen then,
I ∝ A’2
∴ I ∝ 4A2cos2\(\frac{\pi \delta}{\lambda}\)
or, I = k4A2cos2\(\frac{\pi \delta}{\lambda}\) or, I = I0cos2ϕ
[I0 = 4A2k, (is the maximum intensity) and ϕ = \(\frac{\pi \delta}{\lambda}\)]
Hence the intensity in the region of superposition follows cosine-squared rule. Variation of intensity with phase difference is shown below in Fig.
1. Let waves coming from two coherent sources of equal frequency be of amplitudes A1 and A2, intensities I1 and I2 with phase difference ϕ.
(i) On superposition of two coherent waves, expression for resultant amplitude is
A’2 = \(A_1^2\) + \(A_2^2\) + 2A1A2cosϕ
(ii) As intensity is directly proportional to the square of amplitude, the resultant intensity is,
I = I1 + I2 + 2\(\sqrt{I_1 I_2}\)cosϕ [∵ I ∝ A2]
∴ Imax = I1 + I2 + 2\(\sqrt{I_1 I_2}\) = (\(\sqrt{I_1}\) + \(\sqrt{I_2}\))2
and Imin = I1 + I2 – 2\(\sqrt{I_1 I_2}\)(\(\sqrt{I_1}\) – \(\sqrt{I_2}\))2
(iii) Moreover, as the amplitude of bright fringe = A1 + A2 and amplitude of dark fringe = |A1 – A2|,
so, \(\frac{I_{\max }}{I_{\min }}\) = \(\frac{\left(A_1+A_2\right)^2}{\left(A_1-A_2\right)^2}\)
2. If the sources are not coherent, resultant intensity at any point will just be the sum of Intensities of the individual waves.
Conditions of sustained interference:
- Two light sources must be monochromatic and should emit waves of the same wavelength.
- Amplitude of the waves should be equal or nearly equal.
- There must always be a constant phase difference between the two waves. With the change in phase of any wave there should be a simultaneous change in the other to the same extent. Such pair of sources are called coherent sources.