EJC Physics H212a Superposition 2023 1. Notes (full)

9749(202 3) H2 Physics H212 Superposition – Notes Page 1 of 28 H2 Topic 12a – Superposition Anti-reflection (AR) coating commonly found on lenses, and Active Noise Cancellation (ANC) commonly found on higher -end audio devices are just two amongst many applications of interference, which is a phenomenon based on the principle of superposition. Content • Principle of superposition • Stationary waves • Diffraction • Two-source interference • Single slit and multiple slit diffraction Learning Outcomes Candidates should be able to: (a) explain and use the principle of superposition in simple applications (b) show an understanding of the terms interference, coherence, phase difference and path difference (c) show an understanding of experiments which demonstrate stationary waves using microwaves, stretched strings and air columns (d) explain the formation of a stationary wave using a graphical method, and identify nodes and antinodes (e) explain the meaning of the term diffraction (f) show an understanding of experiments which demonstrate diffraction including the diffraction of water waves in a ripple tank with both a wide gap and a narrow gap (g) show an understanding of experiments which demonstrate two-source interference using water waves, sound waves, light and microwaves (h) show an understanding of the conditions required for two-source interference fringes to be observed (i) recall and solve problems using the equation λ = ax / D for double-slit interference using light (j) recall and use the equation sinθ = λ / b to locate the position of the first minima for single slit diffraction (k) recall and use the Rayleigh criterion θ ≈ λ / b for the resolving power of a single aperture (l) recall and use the equation d sinθ = nλ to locate the positions of the principal maxima produced by a diffraction grating (m) describe the use of a diffraction grating to determine the wavelength of light (the structure and use of a spectrometer are not required) incoming noise 2nd sound introduced silence

9749(202 3) H2 Physics H212 Superposition – Notes Page 2 of 28 12.0 Introduction Unlike particles, two (or more) waves can exist at the same point in space, at the same time. In this topic, we will look at the interaction between two or more waves when they meet and overlap. 12.1 Principle of Superposition Be careful to use “displacement” rather than “amplitude” when defining superposition. In a wave, the displacement of each particle or segment of the wave varies from zero to the maximum possible value, and therefore the resultant displacement when multiple waves meet will change with time Example 1 On the same axes, use the principle of superposition to sketch the resultant variation of displacement with time when two waves A and B meet and overlap at a single particle. Note: the principle of superposition applies at each point along the wave, at the same point in time, to give the resultant waveform. displacement 0 time A B The Principle of Superposition states that when two or more waves meet and overlap, the resultant displacement is the vector sum of the displacement of each individual wave A series of snapshots that show two pulses travelling in opposite directions along a stretched string. Where the two pulses meet and overlap, we see the resultant wave, not the individual waves. The waves’ travel are not altered after they overlap.

9749(202 3) H2 Physics H212 Superposition – Notes Page 3 of 28 12.2 Interference It is inevitable to invoke the definition of superposition when describing interference. The added consideration is that interference describes a spatial pattern where there are large displacements and minimal displacements . Since intensity is directly proportional to the square of amplitude, correspondingly there are regions of high intensity (maxima) and regions of minimal intensity (minima). We term these regions of constructive interference and destructive interference. By referring to 2 waves meeting and overlapping: phase difference of waves at that point type of interference resultant displacement is … ( )0 2 4 2, ,...,, nππ π even integers of π / meet in-phase constructive larger than the largest individual displacement ( )ππ π + 3 21, ,..., n odd integers of π / meet anti-phase destructive smaller than the largest individual displacement *not necessarily zero resultant (left) A ripple tank. 2 dippers are connected to the same oscillating bar and just touch the water surface when at rest. Each dipper is a source of circular ripples, which overlap to give an interference pattern. A B (bottom) The circular lines are the wavefronts of crests, which means the mid-point between 2 concentric wavefronts are troughs. A is a position of constructive interference; there is destructive interference at B. An interference pattern is a pattern of maxima and minima which results from 2 or more waves meeting and overlapping such that the resultant displacement is the vector sum of the displacement of each individual wave. *depending on the context, will need to bring in additional details provided in question to explain the phase difference of the 2 waves meeting -> type of interference -> maxima / minima formed Two pulses can travel on a stretched string in opposite directions and undergo destructive interference when they meet and overlap. Thereafter the pulses continue without having been permanently affected.

9749(202 3) H2 Physics H212 Superposition – Notes Page 4 of 28 Example 2 Two wave sources S1 and S2, which are synchronised to each other, provide wavefronts as shown. The amplitude of the resultant of the waves is observed to be a maximum along the direction MM. On the same diagram, draw a line to show (i) a second direction along which the amplitude is a maximum (label this line AA), (ii) a direction in which the amplitude is a minimum (label this line NN). Note: the interference pattern comprises the MM, AA, and NN lines; not the wavefronts. The minimum amplitudes are when crests meet troughs. The two-source set up in Example 2 can be created using sound waves, microwaves and light: S1 • S2 • M M A A A A N N N N signal generator speakers screen double slit microwave source metal sheets microwave probe & meter

9749(202 3) H2 Physics H212 Superposition – Notes Page 5 of 28 Example 3 Circular water waves may be produced by oscillating dippers at points P and Q as shown. The waves from P alone have the same amplitude at point R as the waves from Q alone. The dippers oscillate in phase with a period of 1.5 s to produce waves of speed 4.0 cm s-1. Explain why the water particles are at rest at point R. Waves from P and Q, meet and overlap at point R in anti-phase, and undergo destructive interference. Since the waves individually have the same amplitude at point R, the vector sum of displacements from waves from P and Q is zero. To explain the resultant intensity at a particular position in an interference pattern: (i) identify the wave sources and the position where the waves meet and overlap (ii) make reference to either path difference s∆ or time lag t∆ (iii) work out the phase difference φ∆ that the waves meet and overlap via 2 st T φ λπ ∆∆∆= = (iv) determine if there is constructive or destructive interference For point (iii), it matters if the sources are in-phase or not. See next example as an illustration. P • Q • • R wavefront 48 cm 33 cm Solution: λ P λ λ λ λ λ λ λ 48 cm λ λ λ λ λ Q meet at R 33 cm zero resultant displacement time

9749(202 3) H2 Physics H212 Superposition – Notes Page 6 of 28 Example 4 Microwaves of the same amplitude a