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8.1 Stationary waves

When waves overlap, the resultant displacement at each point is the algebraic sum of the individual displacements: y=y₁+y₂.

Keep signs and phases, then add the contributions at the same position and time. The waves continue afterwards rather than permanently merging.

Two equal pulses with the same sign reinforce to double displacement; opposite pulses can cancel temporarily.

Superposition does not mean one wave destroys the other or that energy disappears when displacement is zero.

System	Set-up and adjustment	What reveals the stationary pattern	Interpretation
microwaves	transmitter faces a metal reflector; move a detector between them	alternating signal minima and maxima	incident and reflected microwaves superpose; minima are nodes and maxima antinodes
stretched string	vibration generator drives a tensioned string with reflection at a fixed end; vary frequency, vibrating length or tension until resonance	stable loops with fixed zero-motion points	fixed points are displacement nodes; centres of loops are antinodes
air column	loudspeaker or tuning fork drives air in a tube; vary frequency or air-column length until resonance	large response; fine powder may gather where air motion is least	closed end is a displacement node; open end is a displacement antinode

In each experiment, a wave and its reflection of the same frequency travel in opposite directions and superpose. A clear stationary pattern appears at resonance when the boundary conditions fit an allowed mode.

Moving a microwave detector or locating successive string/air displacement nodes gives a repeated spatial pattern: adjacent nodes or adjacent antinodes are λ/2 apart. Measure across several intervals where possible to reduce percentage uncertainty.

Node and antinode refer here to displacement amplitude. Pressure nodes and antinodes in an air column are reversed relative to displacement. Treat an open end as a displacement antinode and a closed end as a displacement node; end corrections are neglected and their theory is not required.

Take two coherent waves of the same type, frequency, wavelength, speed and amplitude travelling along the same line in opposite directions. At every position, add their signed displacements. Repeating this graphical addition at later times produces a stationary pattern rather than a travelling profile.

$y₁ = A sin(kx − ωt), y₂ = A sin(kx + ωt), y = y₁ + y₂ = 2A sin(kx) cos(ωt)$

Time	Result of adding the two profiles
0	maximum profile on one side of equilibrium
T/4	every non-node point passes through equilibrium
T/2	same maximum profile inverted
3T/4	every non-node point passes through equilibrium again
T	original profile returns

Nodes are fixed positions where the two waves always cancel and resultant amplitude is zero: x = nλ/2. Antinodes lie midway between nodes, where resultant amplitude is maximum, 2A: x = (2n + 1)λ/4. An antinode is not always at maximum displacement; its particle passes through equilibrium every half-cycle.

All particles between the same pair of adjacent nodes oscillate in phase. Particles in neighbouring loops oscillate 180° out of phase. Every oscillating particle has the same frequency, but amplitude changes continuously from zero at a node to maximum at an antinode.

The envelope does not move along the medium, and there is no net energy transfer along a stationary wave. Fixed nodes do not mean every particle is permanently at rest—only particles exactly at nodes have zero amplitude.

In a stationary wave, adjacent nodes or adjacent antinodes are separated by λ/2; a node to the nearest antinode is λ/4.

Measure over several intervals when possible, divide by the number of half-wavelength gaps, then multiply by two.

Four adjacent node gaps spanning 12 cm give λ=2×(12/4)=6.0 cm.

Counting positions rather than gaps gives an off-by-one error; the end-to-end distance must be divided by the number of intervals.

8.2 Diffraction

Diffraction is the spreading of a wave after it passes through an aperture or past an obstacle/edge, including spreading into the geometrical shadow.

The effect is most noticeable when aperture width or obstacle size a is comparable to wavelength λ. If a≫λ, most wavefronts continue nearly straight and only the edges spread strongly.

Plane wavefronts through a very narrow gap emerge approximately semicircular; a wider gap leaves a flatter central region with curved edges. Around an obstacle, waves bend into the region geometric rays would not reach.

If the wave remains in the same medium, diffraction does not change its frequency, speed or wavelength. It redistributes direction and therefore amplitude/intensity across positions.

Long-wavelength sound/radio can reach around doors, buildings or hills more readily than visible light because the relevant obstacles are closer to those wavelengths.

Reflection is bouncing at a boundary; refraction is direction/speed change on entering another medium; interference is superposition. Diffraction can help waves overlap, but it is the spreading at the aperture or edge.

Choose one wave source and geometry; vary only aperture width a or wavelength λ, then compare angular spread, wavefront curvature or central-maximum width at a fixed observation distance.

$smaller a/λ → greater spreading larger a at fixed λ → less spreading larger λ at fixed a → greater spreading$

Ripple tank: use straight incident wavefronts and adjustable gap; strobe/freeze the image if needed. A gap near one wavelength gives strongly curved emerging wavefronts while spacing between successive fronts remains λ.

Laser and single slit: keep slit-screen distance fixed. A narrower slit or longer wavelength gives a wider central maximum; observe safely on a screen and never look into the beam.

Sound/microwave: move a detector behind an aperture/obstacle and map received amplitude. Longer wavelength spreads farther into the geometrical shadow for the same opening.

A broader pattern does not mean the wave travelled faster or its wavelength changed after the gap. It shows that a/λ is smaller; compare like observations with source, medium and distances controlled.

8.3 Interference

Interference is the variation/redistribution of resultant amplitude and intensity when waves overlap and superpose. Coherent waves have the same frequency and a constant phase difference.

$phase difference Δφ=2π(Δr/λ) constructive: Δr=nλ ⇒ Δφ=2nπ destructive: Δr=(n+1/2)λ ⇒ Δφ=(2n+1)π$

At each point add instantaneous displacements. In-phase arrivals reinforce; antiphase arrivals oppose. For equal amplitudes the maximum amplitude doubles and a perfect minimum has zero amplitude.

$I∝A² Two equal in-phase waves: A_max=2A₀, so I_max=4I₀ Unequal antiphase waves leave amplitude |A₁−A₂|, not zero.$

For in-phase sources, equal path lengths give Δr=0 and a central maximum. Moving across the observation region changes the two distances and passes through alternating integer and half-integer wavelength differences.

Same frequency alone is insufficient for coherence: the phase difference must remain constant. Interference redistributes energy spatially; destructive regions do not mean energy has been destroyed.

Create two coherent sources from one oscillator/source, keep their separation and frequency fixed, and sample the overlap region with a screen, microphone, microwave probe or ripple-tank observation.

Light: illuminate two narrow nearby slits with one monochromatic laser. Diffraction at each slit makes the beams overlap; bright/dark bands mark constructive/destructive interference. Observe safely on a screen.

Sound/microwave: drive two emitters from the same signal generator and move a detector along a line. Repeated high/low amplitude or intensity positions correspond to integer/half-integer wavelength path differences.

Ripple tank: two in-phase dippers produce antinodal lines where crest meets crest/trough meets trough and nodal lines where crest meets trough. Use a strobe to make the pattern easier to see.

Mark several successive maxima/minima, measure across multiple intervals and divide by the number of gaps. Identify the central maximum from equal paths before assigning order.

The bands or nodal lines are positions of repeated superposition, not barriers or permanent wave tracks. A reflected single-source pattern may interfere too, but it is not the same two-source geometry.

Stable positions require coherent sources: same frequency and constant phase difference. Splitting one source with two slits provides coherence; two independent lamps have rapidly changing relative phase and wash out fringes.

Both waves must reach each observation point. Narrow slits provide diffraction and overlap; suitable slit separation and a sufficiently distant screen produce resolvable path-difference changes.

Comparable arriving amplitudes give high visibility: bright maxima are strong and dark minima can approach zero. Reducing one slit intensity makes maxima dimmer and minima brighter, so contrast decreases.

$fringe spacing x=λD/a Changing amplitude does not change x. Larger λ or D increases x; larger slit separation a decreases x.$

A narrow wavelength range prevents different fringe spacings from overlapping. Mechanical/thermal stability preserves path difference during observation; align screen/detector so the intended geometry is measured.

Coherence controls stability, amplitude balance controls contrast, and λ/a/D controls spacing. Do not use one of these to explain an effect governed by another.

For slit separation a and screen distance D, the fringe spacing x satisfies λ=ax/D in the small-angle approximation.

Use consistent units and identify x as adjacent bright-fringe spacing, not distance from the centre unless the count is included.

With a=0.25 mm, D=2.0 m and x=5.0 mm, λ=6.25×10⁻⁷ m.

Increasing slit separation reduces fringe spacing; increasing screen distance or wavelength increases it.

8.4 The diffraction grating

Each grating slit diffracts light; waves from many equally spaced coherent slits overlap. A principal maximum forms when the path difference between adjacent slits is an integer number of wavelengths.

$d sinθ=nλ n=0, ±1, ±2, … θ is measured from the grating normal/central (n=0) direction.$

$For N lines per metre: d=1/N For N lines per mm: d=10⁻³/N metres 600 lines mm⁻¹ ⇒ d=1.67×10⁻⁶ m$

Orders appear symmetrically at +θ and −θ. If the question gives the angle between opposite +n and −n maxima, each diffraction angle is half that total angle.

$Because |sinθ|≤1: nλ≤d n_max=floor(d/λ) Longer λ gives larger θ for the same n but usually fewer possible orders.$

n labels the order, not the number of slits. The central maximum is n=0 and cannot determine λ from d sinθ=nλ because both sides are then zero.

Direct monochromatic light normally onto a grating of known line density, identify the central maximum, then locate a labelled non-zero order on both sides using a screen or spectrometer.

Convert line density N to spacing d=1/N in metres per line. Record the order n; higher orders can improve angular sensitivity but must be bright, separated and physically allowed.

$Measure directions α₊ and α₋ for +n and −n: θ=(|α₊−α₋|)/2 λ=d sinθ/n$

Align the incident beam with the normal, use a narrow beam/slit and read angles without parallax. Repeat, average symmetric pairs and, where possible, calculate λ from several orders to check consistency.

Angular uncertainty matters most when θ is small; use the highest clear allowed order and symmetric readings to enlarge the measured separation and reduce zero/alignment bias. Quote λ with justified significant figures.

Do not use the angle from the grating plane, confuse lines-per-length with spacing, or use the full +n-to−n angle as θ. The essential observation is at least one non-zero order.

8. Superposition

Exam analysis

Most tested topics

8.1 Stationary waves

Superposition means overlapping waves add their instantaneous displacements

Three experiments reveal stationary waves through fixed patterns

Add opposite-travelling profiles to build fixed nodes and antinodes

Find wavelength from node or antinode spacing in a stationary wave

8.2 Diffraction

Diffraction spreads a wave through an aperture or around an edge

Compare diffraction patterns by changing one side of the a/λ ratio

8.3 Interference

Path difference fixes phase difference, resultant amplitude and intensity

Map two-source maxima and minima with a screen or moving detector

Separate the conditions for stable, visible and well-spaced fringes

Double-slit fringe spacing follows λ=ax/D for small angles

8.4 The diffraction grating

Use grating spacing, order and angle to test every possible maximum

Determine wavelength from calibrated non-zero grating orders