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E2.2 · Investigate wave diffraction and interference

Learn to investigate wave diffraction and interference through clear examples and targeted practice.

Ontario Grade 12 Physics

The Wave Nature of Light

How waves spread and combine

In Grade 11, you studied waves using quantities such as wavelength, frequency, and speed. A wave can also spread after passing through an opening, or combine with another wave. These effects are called diffraction and interference. In this lesson, the physical system is a wave source, one or more openings, and the region where the waves are observed. We use a fixed reference frame attached to the apparatus. For diagrams and position measurements, take the direction from the opening toward the observation screen as positive. Position and displacement are vectors when direction matters; wavelength and frequency are scalars. The goal is to describe what an investigation could show and to use simple relationships to interpret it.

What you will learn

1. Prerequisite bridge: describing waves

A wave transfers energy from place to place. In a wave diagram, crests are high points and troughs are low points. Wavelength, represented by λ\lambda, is the distance between matching points on successive cycles. Its SI unit is the metre, m\mathrm{m}. Frequency, ff, is the number of cycles passing a point each second, measured in hertz, Hz\mathrm{Hz}. Wave speed, vv, is measured in metres per second, m/s\mathrm{m/s}.
These quantities are linked by a relationship you can use before studying interference. For a given wave, a higher frequency means a shorter wavelength if its speed stays the same. The equation does not tell you what the wave will do at an opening; it connects the wave's basic measurements.
A wave's amplitude describes the size of its displacement from its undisturbed position. Amplitude is not the same as wavelength. When two waves meet, their displacements combine. This combining rule is called superposition.
v=fλv=f\lambda

2. Diffraction: waves spread at openings

Diffraction is the spreading of a wave as it passes through an opening or around an obstacle. It is noticeable when the opening is about the same size as the wavelength, or smaller. If an opening is much wider than the wavelength, the wave spreads less into the region beyond the opening.
A wavefront is a line or surface joining points at the same stage of a wave cycle. A simple diagram of plane wavefronts approaching a narrow opening would show the wavefronts bending outward after the opening. The outgoing wavefronts can be represented as curved arcs. This diagram is a model, not a measurement.
For a single narrow slit, the spreading can be observed as a broad central bright region on a screen when light is used. Dark regions occur where contributions from different parts of the slit cancel. For the first few dark fringes, the slit-width relationship is asin⁡θ=mλa\sin\theta=m\lambda. Here, aa is slit width in metres, θ\theta is the angle from the forward direction, and mm is a positive whole number identifying a dark fringe. This relationship applies when the geometry and wave behaviour match the single-slit model.
To investigate diffraction, change one factor at a time. For example, compare patterns for different slit widths while keeping the source and screen arrangement fixed. Record slit width, wavelength if known, and the positions or angles of visible features. A wider pattern for a narrower slit is consistent with stronger diffraction. It is a proposed procedure until measurements are actually taken; no result should be reported as measured unless it was observed.
asin⁡θ=mλa\sin\theta=m\lambda

3. Interference: waves combine

Interference is the pattern produced when waves overlap. Constructive interference occurs when the waves arrive in step: crest with crest, or trough with trough. Their displacements reinforce one another. Destructive interference occurs when a crest overlaps a trough of equal size; the displacements cancel at that location.
For a stable two-source pattern, the sources should be coherent. Coherent sources have the same frequency and a constant phase relationship. Phase describes where a source is in its cycle. In a double-slit arrangement, light from one source passes through two narrow slits, which act as coherent sources. Bright fringes are locations of constructive interference; dark fringes are locations of destructive interference.
The path difference is the difference between the distances travelled by the two waves to an observation point. A bright fringe occurs when the path difference is a whole number of wavelengths. A dark fringe occurs when it is a half-integer number of wavelengths. For two slits separated by distance dd, the bright-fringe condition is dsin⁡θ=mλd\sin\theta=m\lambda, where m=0,1,2,…m=0,1,2,\ldots and the central bright fringe has m=0m=0.
A useful diagram labels the slit separation dd, the screen distance LL, the central maximum, and a fringe displaced by distance yy from the centre. When angles are small, sin⁡θ≈y/L\sin\theta\approx y/L. This approximation lets you relate a measured fringe position to the slit spacing and wavelength. Ensure yy and LL use the same units.
dsin⁡θ=mλd\sin\theta=m\lambda

4. Planning and interpreting an investigation

A fair investigation changes one variable while keeping the others as steady as possible. For diffraction, vary opening width and compare the spread. For interference, measure the positions of bright or dark fringes and compare them with the model. State the question, identify the variable being changed, and list the quantities to record before collecting evidence.
A suitable record might include the source type or wavelength, slit width or separation, screen distance, and fringe position. Include units and repeat a position measurement if the setup allows. The recorded values are measured evidence; calculations such as an estimated wavelength are derived from those values. A proposed setup, drawing, or simulation can illustrate a method, but it is not evidence of a completed physical measurement.
In evaluating a pattern, check whether its features fit the model and consider measurement limits. A fringe may be broad, dim, or difficult to locate. Uncertainty in locating its centre affects the calculated angle. Do not claim exact agreement from approximate measurements. A useful conclusion names the observed trend, refers to the recorded evidence, and notes a relevant limitation.
sin⁡θ≈yL\sin\theta\approx\frac{y}{L}

Worked example

1. Predicting diffraction from slit width

A monochromatic wave of wavelength 6.0×10−4 m6.0\times10^{-4}\,\mathrm{m} passes through a slit of width 1.2×10−3 m1.2\times10^{-3}\,\mathrm{m}. In the single-slit model, find the angle to the first dark fringe. The system is the slit and outgoing wave; the reference frame is fixed to the slit, and the forward direction is positive.
  1. Choose the model
    The first dark fringe has m=1m=1. Use the single-slit minimum condition. The slit width and wavelength are both given in metres.
    asin⁡θ=mλa\sin\theta=m\lambda
  2. Substitute and solve
    Isolate the sine of the angle, then use the inverse sine. The angle is measured from the forward direction.
    θ=sin⁡−1((1)(6.0×10−4 m)1.2×10−3 m)=30∘\theta=\sin^{-1}\left(\frac{(1)(6.0\times10^{-4}\,\mathrm{m})}{1.2\times10^{-3}\,\mathrm{m}}\right)=30^\circ
Answer: The first dark fringe is at 30∘30^\circ from the forward direction.
Check: The metre units cancel inside the sine. The ratio is 0.500.50, so the angle is physically possible. A slit width only twice the wavelength produces noticeable spreading.

Worked example

2. Finding the first bright-fringe angle

Two coherent slits are separated by 2.0×10−4 m2.0\times10^{-4}\,\mathrm{m}. They are illuminated by light with wavelength 5.0×10−7 m5.0\times10^{-7}\,\mathrm{m}. Find the angle of the first-order bright fringe. The system is the two slits and screen; the frame is fixed to the apparatus, with the central direction positive.
  1. Identify the bright fringe
    The first-order bright fringe has m=1m=1. Apply the two-slit bright-fringe condition.
    dsin⁡θ=mλd\sin\theta=m\lambda
  2. Calculate the angle
    Substitute the slit spacing and wavelength. Both are in metres, so their ratio is dimensionless.
    θ=sin⁡−1((1)(5.0×10−7 m)2.0×10−4 m)=0.14∘\theta=\sin^{-1}\left(\frac{(1)(5.0\times10^{-7}\,\mathrm{m})}{2.0\times10^{-4}\,\mathrm{m}}\right)=0.14^\circ
Answer: The first-order bright fringe is at approximately 0.14∘0.14^\circ from the central direction.
Check: The small angle is reasonable because the wavelength is much smaller than the slit separation. The calculated sine is positive, so this is on the chosen positive side; a matching fringe can occur on the opposite side.

Worked example

3. Inferring wavelength from fringe position

In a proposed double-slit measurement, the slit separation is 3.0×10−4 m3.0\times10^{-4}\,\mathrm{m}. The screen is 2.0 m2.0\,\mathrm{m} away, and a first-order bright fringe is recorded 4.0×10−3 m4.0\times10^{-3}\,\mathrm{m} from the central maximum. Estimate the wavelength using the small-angle model. Treat these as supplied example measurements, not results from a completed experiment. The positive direction is from the centre toward the recorded fringe.
  1. Relate fringe position to angle
    The fringe displacement is small compared with screen distance, so use sin⁡θ≈y/L\sin\theta\approx y/L. For a first-order bright fringe, m=1m=1.
    dsin⁡θ=mλd\sin\theta=m\lambda
  2. Substitute the geometry
    Replace sin⁡θ\sin\theta with y/Ly/L, then solve for wavelength. The length units reduce to metres.
    λ=dymL=(3.0×10−4 m)(4.0×10−3 m)(1)(2.0 m)=6.0×10−7 m\lambda=\frac{dy}{mL}=\frac{(3.0\times10^{-4}\,\mathrm{m})(4.0\times10^{-3}\,\mathrm{m})}{(1)(2.0\,\mathrm{m})}=6.0\times10^{-7}\,\mathrm{m}
Answer: The estimated wavelength is 6.0×10−7 m6.0\times10^{-7}\,\mathrm{m}.
Check: The units reduce to metres, as required for wavelength. The displacement is much smaller than the screen distance, supporting the small-angle approximation. This is an estimate based on the supplied values.

Common mistakes and how to avoid them

Saying diffraction is strongest when the opening is much wider than the wavelength.
Correction: Diffraction is more noticeable when the opening is comparable to or smaller than the wavelength.
Treating every overlap as destructive interference.
Correction: Waves reinforce when they arrive in step and cancel when they arrive half a cycle apart with equal displacement.
Using the dark-fringe condition for a bright fringe.
Correction: For two slits, bright fringes correspond to whole-wavelength path differences; dark fringes correspond to half-integer wavelength differences.
Presenting a predicted pattern or proposed procedure as measured evidence.
Correction: Label predictions, simulations, and supplied example values clearly. Report measurements only when they have actually been collected.

Lesson summary

Check your understanding

Question 1

Which change generally makes diffraction more noticeable?
  1. Increase the opening width while keeping wavelength fixed.
  2. Decrease the opening width while keeping wavelength fixed.
  3. Change the screen's positive direction.
  4. Increase the screen distance without changing the opening.
Show answer and explanation
Decrease the opening width while keeping wavelength fixed.
A narrower opening is more comparable to the wavelength, so spreading is more noticeable.

Question 2

At a location where two equal waves arrive crest with crest, what type of interference occurs?
  1. Constructive interference
  2. Destructive interference
  3. Diffraction only
  4. No overlap
Show answer and explanation
Constructive interference
Their displacements reinforce, so the overlap is constructive.

Question 3

A first-order bright fringe is measured at a small displacement yy on a screen distance LL away. Which model estimates its wavelength?
  1. λ=dyL\lambda=\frac{dy}{L}
  2. λ=dLy\lambda=\frac{dL}{y}
  3. λ=yLd\lambda=\frac{yL}{d}
  4. λ=dyL\lambda=dyL
Show answer and explanation
λ=dyL\lambda=\frac{dy}{L}
For the first order, m=1m=1, and the small-angle relation gives λ≈dy/L\lambda\approx dy/L. Its units are metres.

Key terms

Diffraction
The spreading of a wave at an opening or around an obstacle.
Interference
The pattern formed when overlapping waves combine.
Superposition
The rule that overlapping wave displacements add.
Coherent sources
Sources with the same frequency and a constant phase relationship.
Path difference
The difference in the distances travelled by two waves to the same point.
Fringe
A bright or dark band in an interference pattern.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Physics (SPH4U), expectation E2.2. It is a study resource, not an official curriculum publication.

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