Interference

Overview

Interference is a wave phenomenon in which two or more waves overlap and combine through Superposition of Waves.

The overlapping waves produce regions of reinforcement and cancellation. It is important evidence for the wave nature of sound, light, and matter waves.

Definition

Interference is the redistribution of wave amplitude and intensity when two or more waves superpose.

Strictly, displacements are vectors, so the resultant displacement is:

$${\vec{s}}_{\text{resultant}}={\vec{s}}_1+{\vec{s}}_2+\cdots$$

For one-dimensional wave diagrams, this is usually written as a signed-component equation:

$$y=y_1+y_2$$

where $y_1$ and $y_2$ are signed displacement components due to the two waves after choosing a positive displacement direction.

Why It Matters

Interference explains bright and dark fringes, loud and soft regions in sound, ripple tank nodal lines, microwave patterns, and many optical experiments. It is the bridge between basic wave motion and Young double slit, diffraction gratings, and wave-particle duality.

Key Representations

Coherent Sources

Stable interference patterns require coherent sources. Coherent sources have:

• the same frequency;
• constant Phase Difference.

Same frequency alone is not sufficient.

Constructive Interference

Constructive interference occurs when waves arrive in phase. Examples include crest meeting crest or trough meeting trough.

For coherent sources that start in phase, the path difference condition is:

$$\Delta x=n\lambda$$

The corresponding phase difference condition is:

$$\Delta \varphi =2n\pi$$

where:

$$n=0,1,2,3,\dots$$

Destructive Interference

Destructive interference occurs when waves arrive in antiphase, such as crest meeting trough.

For coherent sources that start in phase, the path difference condition is:

$$\Delta x=\left(n+\frac{1}{2}\right)\lambda$$

The corresponding phase difference condition is:

$$\Delta \varphi =(2n+1)\pi$$

Path Difference

Path difference is the difference in distances travelled by two waves to reach the same point:

$$\Delta x=|l_2-l_1|$$

It determines whether a point experiences constructive or destructive interference.

The phase difference due to path difference is:

$$\Delta \varphi =\frac{2\pi }{\lambda }\Delta x$$

Intensity

Intensity depends on amplitude squared:

$$I\propto A_0^2$$

If two equal waves of amplitude $A_0$ combine constructively, the resultant amplitude is $2A_0$, so:

$$I\propto (2A_0{)}^2=4A_0^2$$

If complete cancellation occurs:

$$A_{\text{resultant}}=0$$

so:

$$I=0$$

Examples

Two dippers in a ripple tank produce antinodal lines and nodal lines.

Two loudspeakers driven by the same source produce loud and soft regions. See Sound Waves.

Light interference produces bright and dark fringes. See Light Waves and Young Double Slit.

Initial Source Phase

If sources start in phase:

• constructive: $\Delta x=n\lambda$;
• destructive: $\Delta x=\left(n+\frac{1}{2}\right)\lambda$.

If sources start in antiphase, these conditions swap:

• constructive: $\Delta x=\left(n+\frac{1}{2}\right)\lambda$;
• destructive: $\Delta x=n\lambda$.

Always check the wording of the question.

Conditions for Observable Two-Source Interference

To obtain clear fringes:

• waves must overlap;
• sources must be coherent;
• amplitudes should be similar;
• for transverse waves, the same Polarisation plane is usually needed;
• source separation should be suitable relative to wavelength.

Formula Summary

Resultant displacement:

$$y=y_1+y_2$$

Constructive interference for in-phase sources:

$$\Delta x=n\lambda$$

Destructive interference for in-phase sources:

$$\Delta x=\left(n+\frac{1}{2}\right)\lambda$$

Phase difference:

$$\Delta \varphi =\frac{2\pi }{\lambda }\Delta x$$

Intensity:

$$I\propto A_0^2$$

Links

• Core hub: Superposition of Waves
• Prerequisite: Phase Difference
• Related: Young Double Slit
• Related: Diffraction and Gratings
• Related: Polarisation
• Related: Superposition Common Exam Traps