NYJC EJC 2026 Superposition Tutorial

9814 H3 Physics (2026) Superposition Tutorial 1 H3 Superposition Tutorial 1 (a) Fig. 1.1 shows the equilibrium and actual positions at an instant in time of a series of particles forming part of a stationary sound wave. Particle M is at its maximum displacement. (i) On Fig. 1.2, sketch the variation of the displacement with distance, taking right as positive. Label your sketch with “P”. [2] (ii) On Fig. 1.2, sketch the variation of displacement with distance 1 4 period later. Label your sketch with “Q”. [1] (b) The standing wave in (a) is formed in a pipe that is closed at one end and open at the other. Sound waves from a coherent source travel parallel to the axis of the pipe. They reflect off the wall of the pipe, subsequently meet and interfere with sound waves from the same source, as shown in Fig. 1.3. Note that there is a -phase change upon reflection off the wall. (i) State the condition for the formation of a displacement antinode. (ii) Hence, determine the shortest distance x (as a multiple of the wavelength, ) from the wall of the pipe at which a displacement antinode is formed. equilibrium positions actual positions Fig. 1.1 Fig. 1.2 displacement distance M sound waves from a coherent source wave reflected off the wall incident wave wall of pipe Fig. 1.3 x

9814 H3 Physics (2026) Superposition Tutorial 2 2 (a) Fig. 2.1 shows two point-sources of waves, S1 and S2. P is a point equidistant from the two sources. Fig. 2.1 The sources radiate waves of the same wavelength. The amplitudes at P due to S1 and S2 are A1 and A2, respectively. At P, the time variation of the disturbance due to the wave from S1 may be represented by 11 cosy A t = . (i) If, at P, there is a phase angle  between the waves from the two sources, write down an expression for the disturbance y2 due to the wave from S2. (ii) By a graphical method, or otherwise, show that Ar, the amplitude of the resultant displacement at P, is given by 2 2 2 1 2 1 2 2 cosrA A A A A = + + [4] (b) Fig. 2.2 shows two vertical slits Q1 and Q2 in an opaque screen, a distance d apart. Fig. 2.2 A vertical strip-light source producing monochromatic light of wavelength 590 nm is set up at L, exactly one metre from Q1 on the normal to the screen, so that both slits are illuminated. A detector is placed at D, exactly equidistant from the slits on the other side of the screen. Readings of the light intensity at D are taken as follows: I1 with the slit Q2 covered but Q1 open; I2 with Q1 covered but Q2 open; and Ir with both slits open. It is found that Ir is exactly equal to the sum of I1 and I2. (i) By reference to the relationship given in (a)(ii) above, deduce the separation d of the slits. [7] (ii) Is this the only possible value for d? Explain. [2] [N92/Q4] S1 S2 P L Q1 Q2 D d 1.00 m

9814 H3 Physics (2026) Superposition Tutorial 3 3 In a double-slit experiment for the measurement of the wavelength of light from a laser, the two slits are 1.00 mm apart, and are 3.00 m in front of a screen placed parallel to the plane of the slits. (a) The distance of the fifth bright fringe from the central bright fringe is found to be 9.50 mm. what is the wavelength of the laser light? [4] The experimenter finds that the measurements of the separation of the bright fringes from the central bright fringe are subject to an uncertainty u which depends on the order n of the fringe (the number of the fringes away from the central bright fringe). This uncertainty is given by 20.010 0.0004un=+ where u is measured in mm, and n takes the values of 1, 2, 3 … according to the order of the fringe. (b) Suggest two practical reasons why the uncertainty might be expected to be larger for larger values of n. [2] (c) Because of the uncertainty u in the fringe separation, the value of the wavelength you calculated in (i) is also subject to an uncertainty. Find this uncertainty. Express the wavelength, with the associated uncertainty, to an appropriate number of significant figures. [4] (d) To obtain the least uncertainty in the values of the wavelength, on what order of fringe should the measurements be taken? [4] [N93/Q5(c)] 4 (a) In Fig. 4.1, S1 and S2 are two equal point sources a distance d apart. Fig. 4.1 The sources emit waves of wavelength  in phase. P is a point on the line passing through the midpoint of S 1S2 and making an angle  with the centre line. The distances of P from S 1 and S 2 are r1 and r2 respectively. The equation for the displacement y1 of the wave arriving at P from S1 is 11 2sinoy y r t   =−  equation 4.1 P S1 S2 d centre line 

9814 H3 Physics (2026) Superposition Tutorial 4 (i) Write down an equation, similar to equation 4.1, for the displacement y2 of the waves arriving at P from S2. [1] (ii) Hence, or otherwise, write down an expression for the phase difference  between the waves arriving at P from S1 and S2. [1] (iii) When P is a very great disatnce from the sources, S1P is approximately parallel to S2P. Making this approximation, find an expression for  in terms of , d and . [2] (iv) Hence, obtain the condition for P to be a point at which the amplitude of the resultant disturbance is a maximum. [3] (b) A stereo system in a large hall has two identical speakers, S 1 and S2, 1.2 m apart. The amplitude of the output of each speaker is proportional to the voltage across its terminals. The voltage input to each speaker is adjusted by means of a balance control. The arrangement is illustrated in Fig. 4.2. Fig. 4.2 Initilally, the speakers are emitting signals of frequency 1000 Hz which are in phase. The balance control is set so that there is a voltage of 6.0 V r.m.s. across each speaker. An observer stands on the centre line at the point A, 15 m away from the point half-way between the speakers, and hears a loud sound of intensity Imax. As he moves along the line at right angles ot the centre line, he observes that the intensity of the sound first falls to zero at the point B, a distance y from A. The speed of sound in the air in the hall is 330 m s-1. (i) Estimate the distance y. State any assumptions you make. [4] (ii) At what other frequencies of operation of the speakers would the point B also be a position of zero intensity? [2] (iii) With the speakers emitting the original signal of frequency 1000 Hz, the balance control is now adjusted so that the voltages across S1 and S2 are 3.0 V r.m.s. and 9.0 V r.m.s. respectively. 1. In terms of Imax, find the new intensities at points A and B. 2. Show that, although the average of the voltages across the speakers is the same as before (i.e. 6.0 V r.m.s.), the adjustment of the balance control has caused a change in the total power output of the system. Discuss this result with reference to the principle of conservation of energy. [5] [N98/Q9]

9814 H3 Physics (2026) Superposition Tutorial 5 5 The mercury spectrum contains two intense yellow lines with wavelengths 577.0 nm and 579.1 nm respectively. Light from a mercury lamp is incident normally on a diffraction grating ruled with 4300 lines per centimetre. To observe the yellow lin