2017 H2 Superposition Lecture Notes (Teachers)

9646 H2 PHYSICS Lecture Notes Nanyang Junior College 1 Chapter 11 SUPERPOSITION Content Stationary waves Diffraction Interference Two-source interference patterns Diffraction grating 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 if two-source interference fringes are to be observed. (i) recall and use the equation λ =a x/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 sin θ = 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). AND previously in the WAVES (chapter 10) syllabus: Candidates should be able to (n) Determine the wavelength of sound using stationary waves.

9646 H2 PHYSICS Lecture Notes Nanyang Junior College 2 1 PRINCIPLE OF SUPERPOSITION Previously in Waves, we were introduced to the wave model. In this chapter, we are studying the effect when more than one wave exist s simultaneously. Like how each pebble creates its own ripple in the wate r, what happens when two pebbles are thrown into the pond and the ripples spread out and overlap? When two waves meet, the resultant displacement is the vector sum of the displacements due to each individual wave. The figure below shows two wave pulses travelling in opposite directions. When the two waves meet, the resultant displacement of the rope is always equal to the sum of the displacements produced by each pulse. After the waves separate, they behave as if they had never met. Consider two transverse waves A and B emitted by two sources meeting and interfering at the point X. y t X Wave A Wave B Source A Source B

9646 H2 PHYSICS Lecture Notes Nanyang Junior College 3 Graph A shows the displacement -time graph of a particle at X when only source A is switched on. Graph B shows the displacement -time graph of the same particle at X when only source B is switched on. (a) Sketch the displacement-time graph of the particle at X when both source A and source B are switched on simultaneously. (b) The amplitude of the wave from Source B is now doubled and Source B is in antiphase to Source A. Sketch the displacement-time graph of the particle at X due to both waves. Graph A Graph B t t 0 0 t y y y y 0 y 0 0 Phase Difference between A & B, AB= 0 y0 2y0 Graph A t t y y 0 0 0 y t Total Phase Difference between A & B, AB=  Graph B 2y 0 y0

9646 H2 PHYSICS Lecture Notes Nanyang Junior College 4 (c) Source B is now out of phase with Source A. Sketch the displacement -time graph of the particle at X due to both waves. How about superposing two waves of different frequencies? 1.1 COHERENCE Consider the Sources C and D below with a frequency of 100 Hz and 75 Hz respectively. Would it be meaningful to discuss their phase difference? Sources are said to be coherent if they have constant phase difference. (i.e. the phase difference of the sources does not change with time)  This is only possible when the two sources have the same frequency, the same wavelength and the same wave speed.  In particular for light, it is impossible to achieve coher ence if two separate sources are used. Therefore to overcome this problem, we use a single source and a pair of double slits to split the light into two identical sources. Refer to Section 4.3.2 and Appendix for further discussion. We will study mainly how the principle of superposition is applied to formation of stationary waves, interference of waves and diffraction of light. y0 Graph A Graph B t t y0 y y y t 0 0 0 t t Resultant wave Source C Source D t t In phase In phase Antiphase t Phase Difference between A & B, AB= /2 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4

9646 H2 PHYSICS Lecture Notes Nanyang Junior College 5 2 STATIONARY WAVES 2.1 FORMATION OF STATIONARY WAVES explaine d using graphical method Stationary waves result from the superposition of two waves of equal amplitude and frequency travelling with the same speed but in opposite directions. Consider two identical waves moving in opposite directions. The waveform of the rightward -moving wave (solid line) after every T4 1 is drawn, do the same for the leftward -moving wave (dotted line). You will see a stationary wave set up after applying the principle of superposition. Displacement, y Position, x t = 0 Displacement, y Position, x t = ½ T Displacement, y Position, x t = T Individual Waves Resultant Stationary Wave Position, x Position, x Position, x Position, x Position, x Displacement, y Position, x t = ¾ T Displacement, y Position, x t = ¼ T A -A A -A A -A A -A A -A

9646 H2 PHYSICS Lecture Notes Nanyang Junior College 6 2.2 CHARACTERISTICS OF STATIONARY WAVES  As the wave pattern does not appear to move in either direction along the wave and the positions of the crests and troughs of the wave are fixed with time, the resultant waveform is known as a stationary or standing wave.  We represent stationary waves on a string as follows:  Points marked by “ N” in the diagram above are called displacement nodes. The particles at these points do not oscillate because the component waves travelling in opposite direction superpose destructively (destructive interference) at these points where total = odd integer multiples of  radians. The positions of nodes do not change with time.  Points marked by “ A” in th e diagram above are called displacement a ntinodes. The particles at these points oscillate with maximum amplitude because the component waves travelling in opposite direction superpose constructively (constructive interference) at these points where total = zero or even integer multiples of  radians. The positions of antinodes do not change with time.  The stationary wave is divided by nodes and antinodes into equal segments (or “loops”), and each segment is of the same length. The distance between 2 adjacent nodes, or 2 adjacent antinodes is half a wavelength.  All particles (except nodes) in the stationary wave are oscillating with simple harmonic motion at the same frequency, but not at the same am