Waves & Oscillations · Chapter 11

Interference & Diffraction

Waves add together. Where they add constructively, you get bright spots. Where they cancel, darkness. These patterns encode deep information about sources, slits, and the wave nature of light itself.

Prerequisites

Wave properties (Ch. 8) — superposition, wavelength, phase \cdot Trigonometry

Learning Goals

State the superposition principle and express the total displacement of two overlapping waves.
Derive the conditions for constructive and destructive interference from path difference.
$Apply the Young's double-slit fringe spacing formula y_{m} = m\lambdaL/d to calculate fringe posit$ ions.
Predict how fringe spacing changes with wavelength, slit separation, and screen distance.
Explain why single-photon and single-electron double-slit experiments still show interference.

11.1 The Superposition Principle

When two waves overlap in space, the resulting displacement at any point is simply the sum of the individual displacements. This linearity is the superposition principle, and it is what gives rise to interference patterns.

y_\mathrm{total}(x,t)=y_1(x,t)+y_2(x,t)

(11.1)

The superposition principle holds for all linear wave systems — light, sound, water waves, and quantum mechanical probability amplitudes. It breaks down only when the medium responds nonlinearly to large amplitudes (e.g., nonlinear optics, shock waves).

11.2 Two-Source Interference

Two coherent point sources emit identical waves. At any point P in the field, the two waves arrive having traveled different distances r₁ and r₂. The path differenceΔr = |r₂ − r₁| determines whether they interfere constructively or destructively.

Definition 11.1 — Conditions for Two-Source Interference

For two coherent sources in phase (\Delta\phi = 0

Constructive: $\Deltar = m\lambda \qquad (m = 0, \pm1, \pm2, \cdots) — waves arrive in phase, amplitudes add$
Destructive: $\Deltar = (m + \frac{1}{2})\lambda \qquad (m = 0, \pm1, \pm2, \cdots) — waves arrive out of phase, amplitudes cancel$

If the sources have an initial phase offset \Delta\phi, replace \Deltar = m\lambda with \Deltar = m\lambda - \Delta\phi/(2\pi)\lambda

Each source emits:

y=A\sin(kr-\omega t+\phi) \qquad k=\frac{2\pi}{\lambda}

(11.2)

Wavelength λ60 px

Source spacing5 × λ

Phase difference Δφ0°

Interference condition

Constructive (Δφ = 0)

Constructive — waves in phase (Δφ = 0, 2π, …) → bright bands

Destructive — waves out of phase (Δφ = π, 3π, …) → dark bands

Blue = positive amplitude, Red = negative amplitude.

The pattern is the 2D superposition: y = y₁ + y₂

Figure 11.1.

Two-source interference pattern. Blue = constructive interference (waves in phase), red =

destructive (out of phase), black = nodal lines. Adjust source separation, wavelength, and

phase difference to watch the pattern change.

11.3 Young's Double-Slit Experiment

Thomas Young's 1801 experiment showed that light produces an interference pattern, proving its wave nature. A monochromatic source illuminates two narrow slits separated by distance d. On a screen at distance L (with L ≫ d), bright fringes appear at positions:

y_m=\frac{m\lambda L}{d} \qquad m=0,\pm1,\pm2,\ldots

(11.3)

where m is the fringe order. The fringe spacing is constant: Δy = λL/d. Shorter wavelength gives closer fringes; larger slit separation also gives closer fringes.

Example 11.1 — Double-Slit Fringe Spacing

$In Young's experiment: slit separation d = 0.2 mm, screen distance L = 2.0 m, wavelength \lambda$ = 550 nm. Find the fringe spacing and the position of the third bright fringe.

Fringe spacing:

\Deltay = \lambdaL/d = (550\times10^{-9} \times 2.0) / (0.2\times10^{-3}) =

5.5 mm

Third fringe (m=3):

y_{3} = 3 \times 5.5 mm =

16.5 mm from center

Note:

Switching to \lambda = 440 nm (violet): \Deltay = 4.4 mm — fringes pack closer together. Switching to

d = 0.4 mm: \Deltay = 2.75 mm — same effect

11.4 What to Watch in the Simulation

Phase Δ = 0°: Standard double-slit pattern — bright central maximum, symmetric fringes on both sides.
Phase Δ = 180°: Dark central minimum — the two waves arrive out of phase and cancel exactly at the center.
Source spacing: Increasing the spacing makes the fringes closer together (angular separation decreases).
Wavelength: Shorter λ produces more fringes packed in the same angular range.

This experiment is so fundamental that it has been repeated with single photons, electrons, atoms, and even molecules — and it still shows interference. The wave nature is a property of quantum probability amplitudes, not just classical wave superposition.

Definition 11.2 — Common Traps

Constructive does not mean high everywhere: the path difference changes from point to point.
Fringe spacing uses consistent units: $convert nm, mm, and m before using \Deltay = \lambdaL/d$
Increasing slit separation narrows the pattern: $\Deltay is inversely proportional to d$
Single-particle interference is not particle collision: the interference belongs to probability amplitudes.

Exercises — 11.1–11.4 Interference

In Young's double-slit experiment with d = 0.2 mm, L = 2.0 m, and \lambda = 550 nm, find the fri

nge spacing.

Straightforward

Repeat the Young's double-slit calculation with violet light (\lambda = 440 nm), same geometry

. What is the new fringe spacing?

Straightforward

Two coherent sources are 4\lambda apart (same frequency, in phase). Find the angle of the first

dark fringe.

Intermediate

4.Compare the interference pattern from two slits vs. three equally spaced slits (same spacing d). Where are the principal maxima for three slits? How do intensity and fringe width compare?

Intermediate

5.Derive the intensity pattern for a double slit with finite slit width a and separation d. Explain missing orders and when they occur.

Challenging

Key Takeaways

$Superposition: y_{total} = y_{1} + y_{2}. Linear waves add pointwise$ .
$Constructive: \Deltar = m\lambda; destructive: \Deltar = (m+\frac{1}{2})\lambda (for in-phase sources$ .
$Young's fringe spacing: \Deltay = \lambdaL/d. Smaller \lambda or larger d \to closer fringes$ .
$Phase offset \Delta\phi shifts the entire pattern — 180° phase inverts constructive and destructiv$ e.
Single-particle experiments (photon, electron) reproduce the interference pattern — wave nature is intrinsic to quantum amplitude, not a classical field effect.