RI Chap 11 Superposition Lecture Notes

11 SUPERPOSITION H2 Physics 9478 Content Page 11.1 Principle of Superposition 2 11.2 Standing Waves 3 11.3 Diffraction Through a Finite-size Gap 14 11.4 Interference of Two or More Coherent Sources 15 11.5 Single-Slit Diffraction 30 11.6 Appendix 39 Learning Outcomes Candidates should be able to: (a) explain and use the principle of superposition in simple applications (b) show an understanding of experiments which demonstrate standing (stationary) waves using microwaves, stretched strings and air columns. (c) explain the formation of a standing (stationary) wave using a graphical method, and identify nodes and antinodes, differentiating between pressure and displacement nodes and antinodes for sound waves. (d) determine the wavelength of sound using standing (stationary) waves. (e) show an understanding of the terms interference, coherence, phase difference and path difference. (f) show an understanding of phenomena which demonstrate two- source interference using water waves, sound waves, light waves and microwaves. (g) show an understanding of the conditions required for two- source interference fringes to be observed. (h) recall and use the equation ax D λ= to solve problems for double-slit interference, where a is the slit separation and x is the fringe separation. (i) recall and use the equation sinan θλ= to solve problems involving the principal maxima of a diffraction grating, where a is the slit separation. (j) describe the use of a diffraction grating to determine the wavelength of light (knowledge of the structure and use of a spectrometer is not required). (k) show an understanding of phenomena which demonstrate diffraction through a single slit or aperture, or across an edge, such as the diffraction of water waves in a ripple tank with both a wide gap and a narrow gap, or the diffraction of sound waves from loudspeakers or around corners. (l) recall and use the equation sinb θλ= to solve problems involving the positions of the first minima for diffraction through a single slit of width b. (m) recall and use the Rayleigh criterion bθλ≈ for the resolving power of a single aperture.

Page| 2 11.1 Principle of Superposition  Introduction A wave is a disturbance that travels through a medium or vacuum. For mechanical waves, like sound and water waves, the disturbance refers to the displacement of the particles of the media from their equilibrium position. For electromagnetic waves, the dist urbance refers to the varying electric and magnetic field. What happens when two waves of the same type (e.g., sound waves from two sources) meet at a point in space? Fig. 11.1 illustrates two different sets of two waves meeting at an instant of time. Fig. 11.1 Note that displacement of the resultant wave at any position is the vector sum of the displacements due to the two waves. -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0 1 2 3 4 5 6 7 8 9 10 Resultant Wave Wave 2 Wave 1 -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0 1 2 3 4 5 6 7 8 9 10 Resultant Wave Wave 2 Wave 1 Principle of Superposition The principle of superposition states that when two or more waves of the same type meet at a point in space, the resultant displacement at that point is equal to the vector sum of the displacements of the individual waves at that point.

RAFFLES INSTITUTION YEAR 56 PHYSICS DEPARTMENT Page| 3 11.2 Standing Waves  Formation of a Standing Wave Note: Since the waves have the same frequency and speed, they also have the same wavelength. A standing wave is also known as a stationary wave. Fig. 11.2 shows a set of graphs representing two progressive waves of equal amplitude and frequency travelling in opposite directions. The superposition of the two waves is observed along the line of propagation, giving rise to a resultant waveform. At 0t = , superposition of the waves gives rise to a resultant wave which has twice the amplitude of either progressive wave At 1 8tT= , the waves have moved 8λ in opposite directions. Resultant wave has lower maximum amplitude. At 1 4tT= , the waves have moved 28 4λλ = in opposite directions. The resultant amplitude is zero everywhere. At 3 8tT= , the waves have moved 38λ in opposite directions. Resultant wave has lower maximum amplitude. At 1 2tT= , the resultant wave has twice the amplitude of either progressive wave once again. Fig. 11.2 t = 0 t = T/8 t = 2T/8 = T/4 t = 3T/8 t = 4T/8 = T/2 Standing Waves A standing wave is formed by the superposition of two progressive waves of the same type, amplitude, frequency and speed, travelling along the same path but in opposite directions.

Page| 4 This continues to 3 4tT= , and resultant displacement is zero everywhere. Finally, at exactly one period, when tT= , the resultant wave has twice the amplitude of either progressive wave once again. The graphical representation of a standing wave is shown in Fig. 11.3. Fig. 11.3 Each loop is described as an envelope where the resultant displacement varies rapidly and is outlined by curves that represents the amplitudes of the wave at each position. The solid and dashed lines represent both extremes when the constituent waves are in phase.  Properties of a Standing Wave 1. The wave profile does not propagate. As such, the resultant wave is known as a standing wave or stationary wave. 2. The particles of the wave oscillate (except those at the nodes) about their respective equilibrium positions with the same frequency, but different amplitudes. The frequency is the same as that of the two component waves. 3. An antinode is a point in a standing wave where the amplitude is the maximum. The component waves always arrive in phase at the antinodes. 4. A node is a point in a standing wave where the amplitude is zero. The component waves always arrive anti-phase at the nodes. 5. Between two adjacent nodes, all particles oscillate in phase, i.e., they reach their respective maxima, minima and equilibrium positions at the same instant. Note that these particles do not have the same amplitude. 6. Distance between two adjacent nodes (or antinodes) is ½λ. Particles in neighbouring segments vibrate 180° (or π rad) out of phase with each other. Node A node is a point in a standing wave where the amplitude is zero. Antinode An antinode is a point in a standing wave where the amplitude is the maximum.

RAFFLES INSTITUTION YEAR 56 PHYSICS DEPARTMENT Page| 5  Comparisons of Standing vs Progressive Waves Standing Wave Progressive Wave Wave profile The wave profile does not move. The wave profile moves in the direction of propagation with the velocity of the wave. Energy No net energy is transported. Energy is transported in the direction of propagation. Amplitude Amplitude varies from zero at the nodes to maximum at the antinodes. Every particle along the wave oscillates with the same amplitude. Frequency All particles except the nodes oscillate in simple harmonic motion with the same frequency as the progressive wave. All particles oscillate in simple harmonic motion with the same frequency as the progressive wave. Phase All particles between adjacent nodes oscillate in phase. Particles between adjacent segments oscillate in anti- phase. Particles within one wavelength oscillate with different phases. Wavelength Distance between adjacent nodes or adjacent antinodes is 1 2 λ . Distance between two adjacent particles oscillating in phase is λ.

Page| 6 Example 1 [J90/1/13] Progressive waves of frequency 300 Hz are superposed to produce a system of stationary waves in which adjacent nodes are 1.5 m apart. Calculate the speed of the progressive waves. [Solution] Since wavelength of progressive waves is twice the internodal distance, 2 1.5 3.0 mλ ×== Speed 1300 3.0 900 m sv f λ −== = ×