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14.10.1 Superposition of Two Waves of Same Frequency but Constant Phase Difference
Resultant intensity I which is proportional to the square of the resultant amplitude, is given as
(Here we have taken the constant of proportionality as 1, for simplicity).
The intensity I is maximum when cos d = + 1, that is, when phase difference is given by
d = 2n p (even multiple of p).
The intensity I is minimum when cos d = –1, i.e., when d is given by
(odd multiple of p).
We have from eq.
The resultant intensity is thus less than the sum of two separate intensities. If I1 = I2 then Imin= 0, which means that there is no light.
In general, for the two waves of same intensity and having a constant phase difference of d, the resultant intensity is given by
Interference in physics refers to a phenomenon that occurs when two or more waves combine to produce a new wave pattern. This can happen with various types of waves, including electromagnetic waves (such as light), water waves, sound waves, and even matter waves associated with particles like electrons.
The key principle behind interference is the superposition principle, which states that when two or more waves overlap in space and time, the resulting displacement of the medium at any point is the sum of the displacements that would have been produced by each wave individually. Depending on whether the individual waves are in phase or out of phase, interference can result in different outcomes:
Constructive Interference: When two waves are in phase, meaning their crests and troughs align, they combine to create a new wave with a larger amplitude. This results in a wave that appears brighter (in the case of light waves), louder (for sound waves), or with increased intensity.
Destructive Interference: When two waves are out of phase, meaning their crests align with the troughs of the other wave, they combine to create a new wave with a smaller amplitude. In this case, the waves can cancel each other out, leading to regions where the waves partially or completely extinguish each other. This can result in dark or quiet regions.
Partial Interference: In many cases, interference is not purely constructive or destructive. Instead, it can be partial, with some regions experiencing constructive interference and others experiencing destructive interference. This leads to a complex pattern of alternating bright and dark regions known as an interference pattern.
Interference is a fundamental concept in many areas of physics and has practical applications in various technologies. For example:
- In optics, interference is used in devices like interferometers to measure tiny changes in length or detect small differences in wavelength.
- In acoustics, interference can be used to create noise-canceling headphones or improve sound quality in concert halls.
- In quantum mechanics, matter waves exhibit interference, leading to phenomena such as the double-slit experiment, which illustrates the wave-particle duality of particles like electrons.
Interference is a crucial concept for understanding and manipulating wave behavior in the physical world, and it has wide-ranging applications across different scientific disciplines and technologies.
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