8. Superposition

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.

Reflected and incident microwaves can form a stationary pattern. A probe finds alternating nodes of minimum signal and antinodes of maximum signal.

The distance between adjacent nodes or adjacent antinodes is λ/2, so scan spacing must be doubled to obtain wavelength.

If adjacent signal minima are 1.5 cm apart, the microwave wavelength is 3.0 cm.

A node is not half a wavelength apart from the next antinode; node-to-antinode spacing is λ/4.

A stationary wave results when two coherent waves of the same frequency, speed and amplitude travel in opposite directions and superpose.

The pattern has fixed nodes and antinodes, so there is no net energy transfer along the medium even though particles oscillate between them.

A string driven at one end and reflected at the other can settle into a pattern with several loops separated by nodes.

A stationary wave is not a wave travelling slowly; its nodes remain fixed while local oscillations continue.

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.

Diffraction is the bending or spreading of waves around an obstacle or through an opening. It is most noticeable when aperture size is comparable to wavelength.

Compare aperture width with λ: narrower gaps produce broader spreading, while much larger gaps give weaker edge effects.

Water waves spread strongly through a gap about one wavelength wide but remain nearly straight through a gap many wavelengths wide.

Diffraction is not refraction: no change of medium is required, and the wave speed need not change.

In a diffraction experiment, observe the angular spread or fringe pattern while varying slit width or wavelength; larger λ or smaller slit gives more spreading.

Keep source, detector and geometry controlled, and compare the pattern qualitatively unless a quantitative model is required.

A laser through a narrower slit produces a wider central maximum than through a wider slit, while colour-dependent wavelength changes the spread.

A wider central maximum does not mean the waves travelled faster; it reflects the geometry-to-wavelength ratio.

Interference is the redistribution of intensity caused when waves superpose. Coherent sources have a constant phase difference and the same frequency.

Constructive interference occurs for in-phase arrivals; destructive interference occurs for opposite phase. Track path difference or phase difference.

Two equal coherent waves can produce bright and dark regions because their displacements add or cancel at different positions.

Interference does not require waves to be identical in every detail, but stable fringes do require a fixed phase relationship.

Place two coherent sources a known separation apart and observe alternating reinforcement and cancellation along a screen or ripple tank.

Keep source frequency and geometry stable, identify a central maximum and measure fringe spacing or nodal lines.

Two microwave horns fed by one oscillator produce repeated high and low signal positions as a detector is moved across the pattern.

A single source with reflections can make a standing pattern, but it is not automatically the same as a two-source interference geometry.

To see steady two-source fringes, sources should have the same frequency and a constant phase relationship; the geometry must also allow path differences to vary.

Similar amplitudes make maxima and minima distinct, and a screen far enough away can make fringe spacing easier to resolve.

Independent lamps usually wash out visible fringes because their phase difference changes randomly, whereas one laser split into two paths can remain coherent.

Equal source distance alone does not establish coherence; coherence is about phase stability over the observation time.

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.

For a grating with slit spacing d, constructive maxima occur when d sinθ=nλ, where n is an integer order.

Use the grating line density to find d, measure θ from the central maximum and check that |sinθ|≤1.

A 600 lines mm⁻¹ grating has d=1.67 μm; first order at θ=22° gives a wavelength near 6.2×10⁻⁷ m.

n labels bright orders, not the number of slits; the central maximum is n=0.

Use a grating of known line spacing, observe a bright maximum of order n, measure θ and calculate λ=d sinθ/n.

Measure symmetric +n and −n angles where possible, average them, and convert line density to metres per line before substituting.

A laser’s first-order angle on a calibrated grating gives its wavelength; repeating for another colour compares wavelengths directly.

Do not use the angle from the grating normal incorrectly, and do not confuse line density with slit spacing.