Superposition — OCR A-Level Physics

The principle of superposition states that when two or more waves meet at a point, the resultant displacement at that point equals the vector sum of the individual displacements of each wave. Displacement is a vector, so direction matters: crests meeting crests add to give a larger displacement, while a crest meeting a trough can partially or completely cancel. For example, two identical waves each of amplitude 2.0 cm arriving in phase produce a resultant amplitude of 4.0 cm; arriving in antiphase they produce zero resultant displacement. The principle applies to all wave types, including sound, light and microwaves, and underlies interference patterns. It applies at every instant, so the resultant displacement changes as the waves move.

Superposition experiments use two coherent sources so that a stable interference pattern can be observed and measured. With sound, two loudspeakers driven by the same signal generator produce loud and quiet regions detected by a microphone moved along a line; the distance between adjacent maxima relates to wavelength. With light, a laser passing through a double slit produces bright and dark fringes on a screen, and fringe spacing is measured with a ruler. With microwaves, a transmitter and a pair of slits or a double source produce maxima and minima detected by a probe receiver. In each case the procedure involves aligning sources, keeping distances fixed, moving the detector or screen systematically, and recording positions of maxima and minima to determine wavelength or to demonstrate superposition.

Graphical methods show superposition by plotting displacement against position or time for each wave separately, then adding the displacements point by point to obtain the resultant graph. For two waves of equal amplitude and wavelength, plotting them on the same axes makes it clear where crests align to give a doubled displacement and where a crest meets a trough to give zero. For waves of different amplitudes, the resultant graph still follows from adding ordinates at each position. The method works for any wave shapes, including pulses, and helps predict the resultant displacement without needing a formula. Care is needed to add displacements with their signs, since displacement above the axis is positive and below is negative.

Interference is the superposition of waves from two or more sources, producing regions of maximum and minimum resultant displacement. Coherence means waves have a constant phase difference and the same frequency, which is needed for a stable interference pattern. Path difference is the difference in distance travelled by two waves from their sources to a point; when it equals a whole number of wavelengths the waves arrive in phase and interfere constructively, and when it equals an odd number of half wavelengths they arrive in antiphase and interfere destructively. Phase difference describes how far one wave is ahead of another in its cycle, measured in degrees or radians; a phase difference of 360° or 2π rad corresponds to one whole cycle.

When two waves meet, their displacements add. Constructive interference gives maximum amplitude when the waves arrive in phase: phase difference is zero or a whole number of cycles (0, 2π, 4π, ...), and path difference is zero or a whole number of wavelengths (0, λ, 2λ, ...). Destructive interference gives minimum amplitude when the waves arrive in antiphase: phase difference is an odd number of π (π, 3π, 5π, ...), and path difference is an odd number of half-wavelengths (λ/2, 3λ/2, 5λ/2, ...). Phase difference and path difference are linked by Δφ = 2πΔx/λ, where Δx is path difference and λ is wavelength. For example, if λ = 0.60 m and Δx = 0.90 m, then Δx = 1.5λ, so Δφ = 3π rad and the interference is destructive.

Two-source interference can be demonstrated with sound and microwaves. For sound, two loudspeakers driven by the same signal act as coherent sources; a microphone moved along a line parallel to the speakers detects alternating loud and quiet regions. For microwaves, a single transmitter feeds two horn aerials to produce coherent waves; a receiver moved along a line detects maxima and minima. In both cases, maxima occur where path difference is a whole number of wavelengths (0, λ, 2λ, ...) and minima where path difference is an odd number of half-wavelengths (λ/2, 3λ/2, ...). The fringe spacing or separation of maxima depends on wavelength, source separation and distance to the detector. Using microwaves of wavelength about 3 cm gives measurable spacings on a laboratory bench.

Young's double-slit experiment demonstrates interference of visible light. Monochromatic light passes through a single slit to produce coherent waves, which then illuminate two closely spaced slits. The light from the two slits overlaps on a screen, producing a pattern of bright and dark fringes. Bright fringes occur where path difference is a whole number of wavelengths (nλ), and dark fringes where path difference is an odd number of half-wavelengths ((n + ½)λ). The fringe spacing x is given by x = λD/a, where λ is wavelength, D is distance from slits to screen, and a is slit separation. For example, with λ = 6.0 × 10⁻⁷ m, D = 2.0 m and a = 0.50 mm, x = (6.0 × 10⁻⁷ × 2.0) / (0.50 × 10⁻³) = 2.4 × 10⁻³ m = 2.4 mm.

This guided reading covers the techniques and procedures used to determine the wavelength of light using a double-slit and a diffraction grating, as required by section 4.4.3(g). You will learn how to set up a laser or a light source with a single slit to illuminate a double-slit, measuring slit separation a, screen distance D, and fringe spacing x to find wavelength using λ = ax/D. You will also explore using a diffraction grating, measuring the angle θ of the nth order maximum to calculate wavelength using nλ = d sin θ. The guidance explains how to record measurements accurately, such as measuring across multiple fringes to find x, and how to evaluate uncertainties in these optical experiments.

This relationship links the wavelength λ of any wave to the geometry of a two-source interference pattern. Here a is the separation of the two coherent sources, x is the fringe spacing measured between corresponding points on adjacent maxima, and D is the perpendicular distance from the source plane to the screen. The condition a ≪ D means the source separation is much smaller than the screen distance, so the rays reaching a point on the screen are nearly parallel and the small-angle approximation holds. Rearranged, λ = ax/D. For example, with a = 0.50 mm, x = 2.4 mm and D = 1.20 m, λ = (0.50 × 10⁻³ m × 2.4 × 10⁻³ m) / 1.20 m = 1.0 × 10⁻⁶ m. The equation applies to all waves, including sound and microwaves, provided the geometry condition is met.

This statement introduces the practical techniques for measuring the wavelength of light by producing an interference pattern and analysing its geometry. A monochromatic source, such as a laser or a filtered lamp, illuminates either a double slit or a diffraction grating. The resulting pattern of bright and dark fringes is projected onto a screen, and measurements of fringe spacing or fringe positions are combined with the known slit or grating spacing and the screen distance. The wavelength is then calculated from the appropriate equation. The procedure requires care: the source must be coherent, the screen distance measured perpendicular to the source plane, and fringe spacings found by averaging over many fringes to reduce uncertainty. The two named techniques, double-slit and diffraction grating, are treated in the following statements.

In the double-slit technique, coherent light passes through two narrow slits separated by a small distance a and falls on a screen a distance D away. Bright fringes occur where the path difference from the two slits is a whole number of wavelengths, and dark fringes where it is an odd number of half wavelengths. The fringe spacing x is measured between corresponding points on adjacent bright fringes, ideally by measuring across several fringes and dividing by the number of intervals. The wavelength follows from λ = ax/D. For example, if a = 0.25 mm, D = 2.00 m and ten fringe intervals span 48.0 mm, then x = 4.80 mm and λ = (0.25 × 10⁻³ m × 4.80 × 10⁻³ m) / 2.00 m = 6.0 × 10⁻⁷ m. The slits must be narrow enough to diffract light and close enough that a ≪ D.

A diffraction grating has many equally spaced parallel slits, with the grating spacing d equal to the reciprocal of the number of lines per metre. When monochromatic light passes through the grating, sharp maxima occur at angles θ given by d sin θ = nλ, where n is the order of the maximum. The zeroth-order maximum is at θ = 0°, and higher orders appear symmetrically on both sides. To determine wavelength, the angle θ for a chosen order is measured, often by finding the positions of the maxima on a screen or using a spectrometer, and λ = d sin θ / n. For example, with 300 lines per millimetre, d = 1 / (300 × 10³ m⁻¹) = 3.33 × 10⁻⁶ m; if the first-order maximum is at 10.0°, then λ = 3.33 × 10⁻⁶ m × sin 10.0° = 5.79 × 10⁻⁷ m. The many slits make the maxima very sharp, so angles can be measured precisely.