# Wave interference

> IB Physics SL · IB Physics SL
> Source: https://www.owlsprep.com/study/ib-physics-sl-u3-wave-interference/

This module covers the core principles of wave interference, including conditions for observable stable patterns, path difference rules, and how constructive and destructive interference arise for coherent wave sources.

**Prerequisites:** [Principle of superposition of waves](https://www.owlsprep.com/study/ib-physics-sl-u3-wave-superposition/); Basic wave properties (wavelength, frequency, amplitude)

## Learning objectives

- Distinguish between constructive and destructive interference of waves
- State the conditions required for observable stable interference
- Apply path difference rules to coherent wave sources
- Solve calculation problems for double-slit interference patterns

## Core Principles & Conditions for Interference

**Interference** — When two or more waves overlap at a point in space, the resultant displacement equals the sum of the individual displacements (from the principle of superposition), creating a new amplitude pattern.

*Example:* Overlapping water waves on a pond produce alternating regions of higher and lower amplitude.

For a stable, observable interference pattern to form, two key conditions must be met. First, the sources must be coherent. Second, for transverse waves, they must have parallel polarisation to produce a high-contrast pattern.

**Coherent Sources** — Sources of waves that maintain a constant phase difference over time, and emit waves of the same frequency.

*Example:* A single laser split across two slits produces coherent sources; two separate light bulbs do not.

> **tip**
>
> A common misconception is that coherent sources need to have the same amplitude — this is not required. Only constant phase difference and matching frequency are required for coherence.

**Worked example:** State whether two separate sodium lamps emitting light of the same wavelength can produce a stable interference pattern. Explain your answer.

1. 1. First recall the conditions for a stable interference pattern:
2. Two separate sodium lamps emit light in random, uncorrelated wave packets. The phase difference between the two sources changes constantly over very short time scales.
3. Because the phase difference is not constant, the sources do not meet the definition of coherent sources.
4. Conclusion: No stable interference pattern can be observed.

## Path Difference Rules for Interference

Path difference (denoted $Δ x$) is the difference between the distance two waves travel from their respective sources to the same observation point. The type of interference depends on the path difference relative to the wavelength $λ$, for coherent in-phase sources.

$$\text{Constructive interference (in-phase sources): } \Delta x = n\lambda \quad n = 0, 1, 2, ...$$

$$\text{Destructive interference (in-phase sources): } \Delta x = \left(n + \frac{1}{2}\right)\lambda \quad n = 0, 1, 2, ...$$

> **info**
>
> If the two sources are exactly 180° (half a wavelength) out of phase, these rules are reversed: constructive interference occurs at $Δ x = (n + 1/2)λ$, and destructive at $Δ x = nλ$. Always check the source phase difference in exam questions!

**Worked example:** Two in-phase coherent point sources emit sound waves of wavelength 40 cm. A listener stands 210 cm from source 1 and 290 cm from source 2. What type of interference occurs at the listener's position?

1. 1. Calculate the path difference between the two waves:
2. $$\Delta x = |290 \text{ cm} - 210 \text{ cm}| = 80 \text{ cm}$$
3. 2. Compare the path difference to the wavelength:
4. $$\frac{\Delta x}{\lambda} = \frac{80}{40} = 2 = n$$
5. 3. For in-phase sources, a path difference equal to an integer multiple of wavelength produces constructive interference.
6. Final answer: Constructive interference

**Check your understanding**

Test your understanding of the path difference rules:

1. Two in-phase coherent sources have a path difference of $2.5\lambda$ at point P. What type of interference occurs at P?

   - Constructive
   - Destructive
   - No interference

   *Why:* Correct! For in-phase sources, a path difference of $(n + 1/2)\lambda$ gives destructive interference, where $n=2$ here.

## Double-Slit Interference Patterns

When coherent light passes through two narrow, closely spaced slits, a regular interference pattern of bright and dark fringes forms on a screen placed behind the slits. Bright fringes (maxima) correspond to constructive interference, and dark fringes (minima) correspond to destructive interference.

The fringe separation $s$ (distance between the centres of two adjacent bright fringes) is given by the double-slit formula:

$$s = \frac{\lambda D}{d}$$

Where: $λ$ = wavelength of light, $D$ = perpendicular distance from the slits to the screen, $d$ = distance between the centres of the two slits.

**Worked example:** A double-slit experiment uses light of wavelength 500 nm, a slit separation of 0.2 mm, and a screen placed 1.0 m from the slits. Calculate the fringe separation.

1. 1. Convert all values to consistent SI units:
2. $$\lambda = 500 \text{ nm} = 500 \times 10^{-9} \text{ m}\\d = 0.2 \text{ mm} = 0.2 \times 10^{-3} \text{ m}\\D = 1.0 \text{ m}$$
3. 2. Substitute into the double-slit formula:
4. $$s = \frac{\lambda D}{d} = \frac{(500 \times 10^{-9})(1.0)}{0.2 \times 10^{-3}} = 2.5 \times 10^{-3} \text{ m} = 2.5 \text{ mm}$$
5. Final answer: Fringe separation = 2.5 mm

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Assuming all same-frequency sources are automatically coherent.
  - Why it fails: Same frequency is required but not sufficient for coherence; sources also need a constant phase difference, which separate independent sources do not have.
  - Correct: Always confirm that the phase difference between sources is constant before classifying them as coherent.
- **Wrong:** Using the in-phase source path difference rules for sources that are 180° out of phase.
  - Why it fails: The initial phase difference of the sources adds to the path difference, reversing the conditions for constructive and destructive interference.
  - Correct: Note the source phase difference at the start of the question, and reverse the rules if sources are half a wavelength out of phase.
- **Wrong:** Mixing units when calculating fringe separation in the double-slit formula.
  - Why it fails: Wavelength is often given in nanometres, slit separation in millimetres, and screen distance in metres, leading to order-of-magnitude errors.
  - Correct: Convert all quantities to the same base unit (usually SI units) before substituting into the formula.
- **Wrong:** Confusing path difference with phase difference.
  - Why it fails: The two quantities are related but not equal, and exam questions often ask for one when you are given the other.
  - Correct: Remember the conversion: $\text{phase difference} = \frac{2\pi}{\lambda} \times \text{path difference}$.

## Cheatsheet

| Condition | Constructive (in-phase sources) | Destructive (in-phase sources) |
| --- | --- | --- |
| Path difference | $n\lambda \quad n=0,1,2...$ | $(n + 1/2)\lambda \quad n=0,1,2...$ |
| Double-slit pattern position | Bright fringe (maximum) | Dark fringe (minimum) |
| Resultant amplitude (equal amplitude sources) | $2A$ | $0$ |
| Reversed rule (180° out-of-phase sources) | $(n + 1/2)\lambda$ | $n\lambda$ |
| Double-slit fringe separation | \multicolumn{2}{c}{$s = \frac{\lambda D}{d}$} |  |

## What's next

Wave interference is a core fundamental wave phenomenon that underpins many other key topics in IB Physics SL. Understanding the rules of interference is essential for analysing diffraction patterns from single slits, thin film interference, and standing waves, all of which are frequently assessed in SL exams. Interference also forms the basis for many real-world experimental techniques in physics, from high-precision distance measurement to optical spectroscopy and signal processing. This sub-topic connects the basic principle of superposition to observable, measurable wave patterns that you will be expected to calculate and explain in exam questions.

- [Standing Waves](https://www.owlsprep.com/study/ib-physics-sl-u3-standing-waves/)
- [Fields](https://www.owlsprep.com/study/ib-physics-sl-u4-overview/)
- [Gravitational fields](https://www.owlsprep.com/study/ib-physics-sl-u4-gravitational-fields/)

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