5.6 Interference of Waves
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.
Learning Objectives
- State the principle of superposition of waves.
- Define coherent sources of waves and explain the conditions required for wave interference.
- Describe constructive interference and destructive interference with wave profiles.
- Draw and analyse patterns of wave interference showing antinodal lines and nodal lines for water, light, and sound waves.
- Relate the variables of wave interference: wavelength (\(\lambda\)), separation between two coherent sources (\(a\)), separation between adjacent antinodal or nodal lines (\(x\)), and perpendicular distance (\(D\)) using the formula \(\lambda = \frac{ax}{D}\) or \(x = \frac{\lambda D}{a}\).
- Solve numerical problems involving interference of waves.
- Explain daily life applications of wave interference involving water, light, and sound waves.
Superposition and Young's Double-Slit Interference Geometry
See how crests and troughs combine through superposition, and how the geometric parameters \(a\), \(D\), and \(x\) determine the interference fringe pattern.
Wave Profile Superposition
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.
Young's Double-Slit Geometry
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 \(x\).
Short Explanation
Understand superposition, coherent sources, interference patterns, and the formula that connects them.
Principle of Superposition
When two waves overlap, the resultant displacement is the sum of the individual displacements of the two waves.
Coherent Sources
Two sources of waves are coherent when the frequency of both waves is the same and the phase difference between them is constant.
Constructive Interference
Occurs when crest meets crest or trough meets trough. The displacements add to produce a maximum: \(+A + (+A) = +2A\) or \((-A) + (-A) = -2A\).
Destructive Interference
Occurs when crest meets trough. The displacements cancel: \(+A + (-A) = 0\).
Relationship between variables of interference of waves
\(x = \dfrac{\lambda D}{a}\) or \(\lambda = \dfrac{ax}{D}\)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.
| Relationship | Condition | Effect on fringe separation, \(x\) |
|---|---|---|
| \(x \propto \dfrac{1}{a}\) | \(\lambda\) and \(D\) are constant | As distance between sources increases, \(x\) decreases. |
| \(x \propto \lambda\) | \(a\) and \(D\) are constant | As wavelength increases, \(x\) increases. |
| \(x \propto D\) | \(\lambda\) and \(a\) are constant | At positions further from the coherent sources, \(x\) increases. |
Worked Example
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 \(D = 3.0\text{ m}\):
\(\lambda = \dfrac{ax}{D} = \dfrac{(0.4 \times 10^{-3}\text{ m}) \times (4.5 \times 10^{-3}\text{ m})}{3.0\text{ m}} = 6.0 \times 10^{-7}\text{ m}\)
Try to Answer First
Answer in your mind, then press "Check Answer".
What are the two necessary conditions for two wave sources to be coherent?
Check Answer
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?
Check Answer
How does a bulbous bow on a ship reduce water drag?
Check Answer
Common Mistakes
- Forgetting to convert units to metres before applying \(\lambda = \frac{ax}{D}\) (e.g., leaving \(a\) and \(x\) in millimetres while \(D\) is in metres).
- Confusing constructive interference with "crest meets crest only" — constructive interference also occurs when two troughs meet, yielding a displacement of \(-2A\).
- Measuring \(x\) as the total width across multiple fringes instead of the distance between two adjacent bright (or dark) fringes.
- Believing waves are destroyed or cease to exist after destructive interference — waves actually pass through each other and continue unaffected along their paths.
Concept Misunderstandings
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).
Summary
- Principle of Superposition: Resultant displacement equals the vector sum of individual displacements of overlapping waves.
- Coherent Sources: Waves having identical frequency and constant phase difference.
- Constructive Interference: In-phase superposition (crest + crest = \(+2A\); trough + trough = \(-2A\)) creating antinodes, bright fringes, or loud sounds.
- Destructive Interference: Out-of-phase superposition (crest + trough = \(0\)) creating nodes, dark fringes, or soft sounds.
- Formula: \(\lambda = \frac{ax}{D}\) or \(x = \frac{\lambda D}{a}\), where \(x \propto \lambda\), \(x \propto D\), and \(x \propto \frac{1}{a}\).
- Applications: Bulbous bow (calms ship waves), anti-reflective lens coatings (cancels reflected glare), and noise-cancelling headphones (neutralises engine noise).
Short Activity
Objective: Test your understanding of superposition, coherent sources, interference patterns, and applications.
A. Objective Quiz
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?
B. Fill in the Blanks
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 \(\lambda = \frac{ax}{\text{}}\).
C. Matching / Drag and Drop
Drag each term to the correct description. If using a phone, tap the answer first, then tap the matching box.
