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IB DP Physics · Theme C Wave behaviour · C.3 · ~7 min read

C.3 Wave phenomena

Wavefronts and rays, reflection, refraction and Snell's law, critical angle and total internal reflection, diffraction, superposition and interference, Young's double slit, and (HL) single-slit diffraction and diffraction gratings.

Wavefronts & raysSnell's lawTotal internal reflectionDiffractionInterferenceDouble slit s = λD/dSingle slit (HL)Gratings (HL)
RSby Riyaz Sir
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No tag = SL and HLHL ONLY HL students onlyDATA BOOKLET given in the examM1A1 how marks are awardedhighlight = comes up again and againTest yourself at the end

In 30 seconds

  • Wavefronts join points in phase; rays show the direction of travel and are perpendicular to wavefronts.
  • At a boundary waves reflect, refract (change speed and direction) and transmit. Snell's law: n1 sin θ1 = n2 sin θ2. Past the critical angle, light is totally internally reflected.
  • Diffraction is spreading through a gap or around an obstacle; it is strongest when the gap is about one wavelength.
  • Superposition of coherent waves gives interference: bright where the path difference is nλ, dark where it is (n + ½)λ. For two slits the fringe spacing is s = λD/d.
  • HL: single slit minima at θ = λ/b, and sharp grating maxima at nλ = d sin θ.

By the end you can

  • Draw wavefront and ray diagrams for reflection, refraction and diffraction.
  • Use Snell's law, refractive index and the critical angle.
  • Explain the conditions for constructive and destructive interference.
  • Use s = λD/d for Young's double slit.
  • Describe and use single-slit diffraction, θ = λ/b, and how it modulates the double-slit pattern. HL
  • Use nλ = d sin θ for diffraction gratings. HL

1Reflection and refraction

  • Law of reflection: angle of incidence = angle of reflection, both measured from the normal.
normalθ₁θ₂air (n₁ = 1.00)glass (n₂ = 1.50)weak reflectionn₁ sin θ₁ = n₂ sin θ₂slower in glass, sobends towards normal
Light slows down in glass, so the ray bends towards the normal; the frequency stays the same.
n1n2 = sin θ2sin θ1 = v2v1 DATA BOOKLET
Snell's law; n = c/v for each medium
Worked example

Light hits glass (n = 1.50) from air at 40° to the normal. Find the angle of refraction and the speed in glass.

  1. sin θ2 = sin 40°/1.50 = 0.429 M1
  2. θ2 = 25° A1
  3. v = c/n = 3.00 × 108/1.50 = 2.0 × 108 m s−1 A1

Critical angle and total internal reflection

glass (dense)airθ < θcθ = θcθ > θcsin θc = n₂/n₁ = 1/1.50, so θc = 41.8°. Beyond it: total internal reflection.
Going from glass to air, the refracted ray bends away from the normal until it skims the surface at the critical angle.
  • Conditions for TIR: light travels from a higher to a lower refractive index, and the angle of incidence exceeds the critical angle.
sin θc = n2n1
from Snell's law with θ2 = 90° (n1 > n2)
Worked example

Find the critical angle for glass (n = 1.50) in air.

  1. sin θc = 1/1.50 M1
  2. θc = 41.8° A1
  • Uses: optical fibres (signals and endoscopes) bounce light along the core by TIR; the cladding has a lower n than the core.
Common mistakeMeasuring angles from the surface instead of from the normal. Every angle in Snell's law is measured from the normal.

2Diffraction

gap ≈ λ: strong spreadinggap ≫ λ: little spreading
A gap similar in size to the wavelength makes almost circular wavefronts.
  • Diffraction does not change the wavelength or speed. You hear round corners because sound wavelengths (about 1 m) are similar to doorways; light (about 500 nm) barely diffracts through doorways.

3Superposition and interference

Principle of superposition — when waves meet, the resultant displacement is the vector sum of the individual displacements.

Coherent sources — sources with the same frequency and a constant phase difference. Needed for a stable interference pattern.

path difference = nλ
constructive interference (sources in phase)
path difference = (n + ½)λ
destructive interference (sources in phase)
Worked example

Two loudspeakers driven in phase emit 680 Hz sound (λ = 0.50 m). At point P the path difference is 1.25 m. What do you hear?

  1. 1.25 m = 2.5λ M1
  2. a half-wavelength path difference: destructive interference, so a quiet spot A1

Young's double slit

screendDss = λD/dbright: path diff = nλdark: (n + ½)λ
Light from two narrow slits overlaps on a distant screen and forms equally spaced bright and dark fringes.
s = λDd DATA BOOKLET
s = fringe spacing, D = slit-to-screen distance, d = slit separation
Worked example

Red laser light (633 nm) passes through slits 0.25 mm apart onto a screen 2.0 m away. Find the fringe spacing.

  1. s = λD/d = 633 × 10−9 × 2.0/0.25 × 10−3 M1
  2. = 5.1 × 10−3 m = 5.1 mm A1
Exam tipTo measure s accurately, measure across many fringes (say 10 spacings) and divide. It cuts the percentage uncertainty.

4Single-slit diffraction HL

θIλ/b−λ/bsecondary maximaCentral maximum is twice as wide as the others. First minimum at θ = λ/b.
Intensity against angle for one slit of width b.
θ = λb DATA BOOKLET
angle (in radians) to the first minimum for a slit of width b
Worked example

Light of 600 nm passes through a slit 0.10 mm wide. Find the angle to the first minimum and the width of the central maximum on a screen 2.0 m away.

  1. θ = λ/b = 600 × 10−9/0.10 × 10−3 = 6.0 × 10−3 rad A1
  2. central maximum half-width = Dθ = 2.0 × 6.0 × 10−3 = 12 mm
  3. full width = 24 mm A1
  • Double slit, real version: each slit has its own width, so the double-slit fringes are modulated by a single-slit envelope. Fringes are brightest in the middle and some fringes go missing where a single-slit minimum falls.

5Diffraction gratings HL

sin θn = -3n = -2n = -1n = 0n = 1n = 2n = 3Many slits: very sharp, bright maxima at d sin θ = nλMore slits ⇒ sharper lines; same positions as a double slit with the same d.
Hundreds of slits per mm give bright, very narrow maxima.
nλ = d sin θ DATA BOOKLET
d = slit spacing (1 mm ÷ lines per mm), n = order
Worked example

A grating has 600 lines per mm. Find the angle of the first-order maximum for sodium light (589 nm) and the highest order visible.

  1. d = 1.0 × 10−3/600 = 1.67 × 10−6 m A1
  2. sin θ = λ/d = 0.353 ⇒ θ = 20.7° M1A1
  3. nmax < d/λ = 2.83, so the highest order is 2 R1
Common mistakeUsing θ = λ/b (single slit) for a grating, or small-angle s = λD/d when angles are large. For gratings, always use nλ = d sin θ.
Linking questionDiffraction of electrons through crystals (E.2) is evidence that matter behaves like a wave; the same nλ = d sin θ idea is used to measure atomic spacings.
Idea for your IA / explorationFind the wavelength of a laser with a diffraction grating, or measure the refractive index of a liquid with a semicircular dish and a protractor, plotting sin θ1 against sin θ2.

Test yourself — cover the answers first

Quick check

  1. What happens to the frequency of light entering glass?
  2. State the two conditions for total internal reflection.
  3. When is diffraction most noticeable?
  4. What does "coherent" mean?
  5. In Young's experiment, what happens to s if d is halved?
  6. HL What happens to a single-slit central maximum if the slit gets narrower?

Exam-style

  1. An optical fibre core has n = 1.48 and cladding n = 1.46. Find the critical angle at the boundary.
  2. HL White light passes through a grating with 300 lines per mm. Find the angular width of the first-order spectrum from 400 nm to 700 nm.
Show answers
  1. It stays the same (speed and wavelength fall).
  2. from higher to lower n, and angle of incidence > critical angle
  3. when the gap is similar in size to the wavelength
  4. same frequency and constant phase difference
  5. s doubles
  6. it gets wider
  7. sin θc = 1.46/1.48 = 0.986
    θc = 80.6°
  8. d = 3.33 × 10−6 m
    400 nm: sin θ = 0.120 ⇒ 6.9°; 700 nm: sin θ = 0.210 ⇒ 12.1°
    width = 5.2°
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Independent notes written by Riyaz Sir. Not produced, endorsed or approved by the International Baccalaureate Organization. Syllabus codes refer to the IB DP subject guides.