HCI 10 Superposition Lecture Notes

Hwa Chong Institution (College) H2 Physics C1 2023 1 Chapter 10 SUPERPOSITION Interference pattern produced by two sources placed at different distances apart and of different wavelengths. Taken from www.arthistoryclub.com Playlist of Lecture Example Solutions and Others available at https://youtube.com/playlist?list=PL_b5cjrUKDlbeRspxU7s_UswuIC_U1VMJ

Hwa Chong Institution (College) H2 Physics C1 2023 2 Syllabus 9749 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 waves 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 equations  = a x / D for double slit interference. (j) recall and use the equation sin 𝜃 = 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  = n  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.)

Hwa Chong Institution (College) H2 Physics C1 2023 3 Contents Syllabus 9749…………………………………………………………………………………………….2 Content 2 Learning Outcomes 2 10.1 – Introduction ……………………………………………………………………………………. 4 10.2 – Principle of Superposition ………………………………………………………………….. 4 10.3 – Interference ……………………………………………………………………………………. 5 10.4 – Interference of Two Wave Sources ………………………………………………………... 7 10.4.1 – Interference of Waves from Two Sources that are in Phase 7 10.4.2 – Accounting for Phase Difference at the Sources 11 Tutorial 10A: Superposition and Interference …………………………………………………... 13 10.5 – Interference of Light Waves: Young’s Double Slit Experiment ……………………… 17 10.6 – Diffraction Grating …………………………………………………………………………… 23 Tutorial 10B: Young’s Double Slits and Diffraction Grating …………………………………. 28 10.7 – Diffraction of Waves …………………………………………………………………………. 36 10.7.1 – Diffraction of water waves 36 10.7.2 – Single Slit Diffraction Pattern 37 10.7.3 – Resolving Power and Rayleigh’s Criterion 40 Tutorial 10C: Single Slit and Rayleigh’s Criterion ……………………………………………… 43 10.8 – Stationary Waves …………………………………………………………………………….. 47 10.8.1 – Characteristics of Stationary Wave 48 10.8.2 – Stationary Waves in Strings 50 10.8.3 – Standing Waves in Air Column 53 10.8.4 – What about the pressure? 58 10.8.5 – Investigating the speed of sound using stationary waves 60 Tutorial 10D: Stationary Waves ……………………………………………………………………. 61 Appendices Appendix A: Huygen’s Principle 66 Appendix B: Single Slit Diffraction Pattern 68 Appendix C: Ripple Tank 69 Appendix D: Spectrometer 70

Hwa Chong Institution (College) H2 Physics C1 2023 4 10.1 Introduction In your secondary schools, you have seen that all waves exhibit reflection and refraction. For the A-levels, we extend our understanding of waves by looking at two other phenomena that are exhibited by waves: interference and diffraction. We will learn about Young's double slit experiment, the measurement of wavelengths of light and sound, and the formation of musical notes in strings and woodwind instruments through the formation of stationary waves. 10.2 Principle of Superposition Imagine throwing two pebbles into a still pond. Each pebble makes its own ripples and as the ripples spread out, they kind of overlap. What determines the resultant “ripple” that we see? The displacement of the resultant wave is determined by the Principle of Superposition: Principle of Superposition: The principle of superposition states that when two or more waves of the same kind overlap, the resultant displacement at any point at any instant is given by the vector sum of the individual displacements that each individual wave would cause at that point at that instant. yresultant = y1 + y2 + y3 + ....+ yN where yresultant is the displacement of the resultant wave and y1, y2, ..., yN are the individual displacements of the waves at the point of consideration. The statement of the Principle of Superposition is a popular exam question. Be careful to use the word “displacement” and not “amplitude” in your statement. Fig. 10.2.1 and 10.2.2 are pictorial representations of the superposition of two wave pulses, y1 and y2, travelling in opposite directions. When the waves begin to overlap, the resultant displacement of the waveform on the string at every point at any time is given by y1 + y2. Fig. 10.2.1. Constructive interference of two wave pulses. Fig. 10.2.2. Destructive interference of two wave pulses.

Hwa Chong Institution (College) H2 Physics C1 2023 5 Two pulses travelling on a stretched string in opposite directions pass through each other. When the pulses overlap the net displacement of the string is equal to the sum of the displacements produced by each pulse. Note that when the two pulses separate, they will continue to move in their original directions as if nothing has happened. From this illustration, we also see that: After two travelling waves have passed through each other, the pulse shapes of each wave are unchanged, as if the two pulses had never met. 10.3 Interference This combination of separate waves in the same region of space at the same time to produce a resultant wave is called interference: Interference Interference is the superposing or overlapping of two or more waves to give a resultant wave whose displacement is given by the Principle of Superposition, which states that displacement of the resultant wave at any point is the vector sum of the displacements of the individual waves at that point. Constructive interference Constructive interference occurs at a point when two waves meet in phase at that point. The resultant amplitude of the oscillation at that point is therefore a maximum.  Phase difference between the two waves at that point = 0, 2, 4, …  Resultant amplitude AR = A1 + A2 Constructive Interference of Transverse Waves Fig. 10.3.1. Two transverse waves in phase undergo constructive interference at every point to produce a wave of maximum amplitude. Constructive Interference of Longitudinal Waves Fig. 10.3.2. Two longitudinal waves in phase undergo constructive interference at every point to produce a wave of increased intensity. In this case, since the amplitudes of the 2 longitudinal waves are the same, the resultant amplitude will be double that of each individual wave. Destructive Interference: Destructive interference occurs at a point when two waves meet in antiphase (phase difference of 180o). The resultant amplitude of the oscillation at that point is therefore a minimum.  Phase difference between the two waves at that point = , 3, 5,…  Resultant ampli