Interference of waves is a phenomenon that occurs when two or more waves overlap while travelling through the same medium. Based on the principle of superposition, the resultant displacement at any point is the vector sum of the individual displacements of the combining waves. This interaction produces fascinating patterns of alternating constructive and destructive interference, observable in water ripples, laser light through double slits, and audible sound from dual speakers, with practical applications ranging from noise-cancelling headphones to anti-reflection coated lenses.
See how crests and troughs combine through superposition, and how the geometric parameters \(a\), \(D\), and \(x\) determine the interference fringe pattern.
When two crests meet, their displacements add to produce a larger crest. When two troughs meet, they add to produce a deeper trough. When a crest meets a trough of equal amplitude, the displacements cancel to zero.
Two coherent slit sources separated by distance \(a\) project interference fringes onto a screen at perpendicular distance \(D\). Adjacent bright fringes are separated by distance \(x\).
Understand superposition, coherent sources, interference patterns, and the formula that connects them.
When two waves overlap, the resultant displacement is the sum of the individual displacements of the two waves.
Two sources of waves are coherent when the frequency of both waves is the same and the phase difference between them is constant.
Occurs when crest meets crest or trough meets trough. The displacements add to produce a maximum: \(+A + (+A) = +2A\) or \((-A) + (-A) = -2A\).
Occurs when crest meets trough. The displacements cancel: \(+A + (-A) = 0\).
Relationship between variables of interference of waves
Where \(\lambda\) = wavelength, \(a\) = separation between two coherent sources, \(x\) = separation between adjacent antinodal or nodal lines, and \(D\) = perpendicular distance from the sources to the position where \(x\) is measured.
Given \(a = 0.4\text{ mm} = 0.4 \times 10^{-3}\text{ m}\), \(x = 4.5\text{ mm} = 4.5 \times 10^{-3}\text{ m}\), and \(D = 3.0\text{ m}\):
\(\lambda = \dfrac{ax}{D} = \dfrac{(0.4 \times 10^{-3}\text{ m}) \times (4.5 \times 10^{-3}\text{ m})}{3.0\text{ m}} = 6.0 \times 10^{-7}\text{ m}\)
Answer in your mind, then press "Check Answer".
What are the two necessary conditions for two wave sources to be coherent?
In Young's double-slit experiment, what happens to the distance between adjacent bright fringes (\(x\)) if the separation between the two slits (\(a\)) is decreased?
How does a bulbous bow on a ship reduce water drag?
Destructive interference means energy is lost or destroyed permanently.
Energy is conserved; it is simply redistributed into the constructive interference regions where the intensity is higher.
Any two identical light bulbs or independent speakers can produce an interference pattern.
Independent light sources emit light with random, rapidly changing phase differences and are not coherent; interference requires coherent sources with a constant phase difference.
Nodes correspond to the maximum sound or bright light.
Nodes are points of destructive interference (minimum displacement: soft sound, dark fringe, calm water), whereas antinodes are points of constructive interference (maximum displacement: loud sound, bright fringe).
Objective: Test your understanding of superposition, coherent sources, interference patterns, and applications.
1 What type of wave interaction occurs when a wave crest of amplitude \(+A\) meets an identical wave trough of amplitude \(-A\)?
2 In an interference experiment using monochromatic light, what will happen to the fringe separation \(x\) on the screen if the distance between the two slits \(a\) is increased while \(\lambda\) and \(D\) remain constant?
3 Two sources of waves are defined as coherent when both have the same frequency and a constant difference.
4 In Young's double-slit experiment, the wavelength of light can be calculated using the formula \(\lambda = \frac{ax}{\text{ }}\).
Drag each term to the correct description. If using a phone, tap the answer first, then tap the matching box.
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