TJC 11 Superposition

SUPERPOSITION 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 diffraction, interference, coherence, phase difference and path difference (f) show an understanding of phenomena which demonstrate two-source interference using water waves, sound waves, light 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 to solve problems for double-slit interference, where a is the slit separation and x is the fringe seperation (i) recall and use the equation asin = n to solve problems involving the principal maxima of a diffraction grating, where a is the slit seperation (j) describe the use of a diffraction grating to determine the wavelength of light (knowledge of the structure and use of a spectrometer are not required). (k) (a) 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 bsin =  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   /b for the resolving power of a single aperture, where b is the width of the aperture * Covered in Unit 10: Waves Temasek Junior College

2 1 Principle of superposition LO (a) Principle of superposition states that when two or more waves of the same kind meet at a point, the resultant displacement of the waves at any point is the vector sum of the displacement due to each wave acting independently. 1.1 Important terms LO (e) (a) Phase difference Phase difference is an angular measure of the fraction of a cycle two particles in a wave or two waves are out of step. • When the phase difference between two waves is zero, or an integer multiple of a cycle, the waves are said to be in phase. • When the phase difference between two waves is an odd integer multiple of half cycle, the waves are said to be anti-phase. (b) Coherence Two sources are said to be coherent if waves from each source have a constant phase difference between them (and therefore they must have the same frequency). Two coherent waves need not be in phase. For example, (c) Interference Interference refers to the results of the superposition of two or more waves of the same kind. The displacement of the resultant wave at any point is given by the principle of superposition. • Constructive interference occurs when two or more waves meet in phase and superpose to produce a resultant wave of maximum amplitude (where the amplitudes add up). • Destructive interference occurs when two or more waves meet anti-phase (or 180° out of phase) and superpose to produce a resultant wave of zero or minimum amplitude (where the amplitudes cancel). Fig (a) two waves from coherent sources Fig (b) two waves from incoherent sources

3 (d) Path difference Path length of a wave is the distance travelled by a wave. Path difference is the difference in the two path lengths. i.e. Path difference is the difference in the distance travelled by two waves from their respective coherent wave sources to a point. Path difference is often expressed in terms of a number of wavelengths. For example, consider waves originating from two coherent sources that oscillate in phase, • if the path difference is zero or an integer number of wavelengths, the waves from the two sources would meet in phase and interfere constructively. i.e. Condition for constructive interference is path difference = n, where n = 0,1,2,3,... • if the path difference is an odd integer number of half wavelengths, the waves from the two sources would meet anti-phase and interfere destructively. i.e. Condition for destructive interference is path difference = (n + ½), where n = 0,1,2,3,... Note: 1. Since path difference of one wavelength  corresponds to a phase difference  of 2 radians (or 360), we can write the proportion 2 L  = Thus, path difference L of n is equivalent to phase difference  of (n)2 rad, and path difference L of (n + ½) is equivalent to phase difference  of (n + ½)2 rad, where n = 0,1,2,3,... 2. For two coherent sources that oscillate anti-phase, the conditions above above would be reversed.

4 2 Diffraction LO (e) Diffraction is the spreading of waves around an obstacle or through a gap, into its geometrical shadow. Geometrical shadow refers to region which the waves would not have reached if they had travelled in a straight line. 2.1 Diffraction experiment with a ripple tank LO (k) Diffraction of waves can be demonstrated using a ripple tank as shown. Plane or circular water waves can be generated using a plane dipper or a rod dipper respectively. Light from a lamp shines upon the water from above and illuminates a white screen placed directly below the tank. When plane water waves produced by the ripple tank pass through obstacle, gaps (or slits) of varying widths, the following diffraction effect could be observed. Fig (a) shows diffraction of water waves around an obstacle. In Fig (b), the gap width is comparable to wavelength, diffraction is significant as the waves emerge from the gap with circular wavefronts, as if spreading out from a point source. The waves spread noticeably around the gap into the geometrical shadow. In Fig (c), the gap width is much larger than the wavelength, diffraction is insignificant. The spreading of the waves around the edges of the gap becomes less. shallow water tray plane dipper rod dipper motion of dipper Fig (a) obstacle Fig (b) small gap Fig (c) wide gap

5 Note: 1. Diffraction does NOT change the wavelength of the wave; i.e. the space between adjacent wavefronts remains constant. 2. Diffraction can be explained with Huygen's principle, which states that every point on a wavefront can be considered as a source of tiny wavelets that spread out in the forward direction at the speed of the wave itself. The new wavefront is the envelope of all the wavelets as illustrated in the diagram. 3. Two-source interference 3.1 Conditions for observable interference pattern LO (g) For interference pattern to be observable, 1. The waves must be of the same kind and superpose at a point. 2. The waves must be coherent. i.e. the waves have a constant phase difference between them. (Otherwise the pattern will change with time due to changing phase difference and quick shifting patterns are not observable). 3. The waves must have about the same amplitude. (Otherwise the pattern has poor contrast due to incomplete cancellation of wave amplitudes at points of destructive interference). 4. For transverse waves, they must be unpolarised or have the same plane of polarisation. (Interference can only occur for wave oscillations in the same pl