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1© Manhattan Press (H.K.) Ltd. Constructive and destructive interference Mathematical approach Mathematical approach 9.8 Interference of water waves

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2 © Manhattan Press (H.K.) Ltd. 9.8 Interference of water waves (SB p. 65) Interference of water waves Interference is the effect produced by the superposition of waves from two coherent sources passing through the same region.

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3 © Manhattan Press (H.K.) Ltd. 9.8 Interference of water waves (SB p. 65) Interference of water waves Two wave sources are said to be coherent if: - the phase difference between the sources is constant, and - two waves should have same frequency, and - two waves should have comparable amplitudes for interference to occur. The interference pattern produced in a ripple tank using two sources of circular waves which are in phase with each other.

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4 © Manhattan Press (H.K.) Ltd. 9.8 Interference of water waves (SB p. 66) Constructive & destructive interference The two sources S 1 and S 2 are in phase and coherent. Therefore, the wavelengths of waves from S 1 and S 2 are the same, say λ.

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5 © Manhattan Press (H.K.) Ltd. 9.8 Interference of water waves (SB p. 66) Interference of water waves At A 1, produce double crest, constructive interference occurs, antinode At A 2, produce double trough, constructive interference occurs, antinode For all the points along the line joining A 1 and A 2, the distances from S 1 and S 2 are equal. It is antinodal line which are joining all the antinodes.

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6 © Manhattan Press (H.K.) Ltd. 9.8 Interference of water waves (SB p. 66) Constructive interference For constructive interference to occur, n = 0, 1, 2, 3, … Path difference = n

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7 © Manhattan Press (H.K.) Ltd. 9.8 Interference of water waves (SB p. 66) At X 1, superposition of a crest and a trough produces zero amplitude, destructive interference occurs, node At X 2, superposition of a crest and a trough produce zero amplitude, destructive interference occurs, node The line joining all the nodes is called a nodal line. Destructive interference

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8 © Manhattan Press (H.K.) Ltd. 9.8 Interference of water waves (SB p. 67) Destructive interference For destructive interference to occur, n = 0, 1, 2, 3, … For the point X 1, n = 1, For the point Y 1, n = 2, Path difference = (n ) Go to Example 9 Example 9

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9 © Manhattan Press (H.K.) Ltd. 9.8 Interference of water waves (SB p. 68) Mathematical approach By the principle of superposition, the resultant displacement y due to the two waves is: This can be simplified using the relationship: where y 0 is the amplitude of the resultant displacement.

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10 © Manhattan Press (H.K.) Ltd. 9.8 Interference of water waves (SB p. 68) Constructive interference Constructive interference occurs when the waves have no phase difference, i.e., = 0. Thus, the resultant amplitude is given by Maximum amplitude is obtained for = 2 , 4 , …etc

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11 © Manhattan Press (H.K.) Ltd. 9.8 Interference of water waves (SB p. 69) Destructive interference Destructive interference occurs when the waves are out of phase, i.e., = . The resultant amplitude is Minimum amplitude is obtained for = 3 , 5 , …etc

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12 © Manhattan Press (H.K.) Ltd. 9.8 Interference of water waves (SB p. 69) Phasor diagram A phasor diagram is used to obtain the amplitude at the other points in the medium where the phase difference is not an integral multiple of . A phasor is a rotating vector used to represent a sinusoidally varying quantity, e.g. the displacement y at a point in a wave.

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13 © Manhattan Press (H.K.) Ltd. 9.8 Interference of water waves (SB p. 69) Suppose the displacement y 1 at a point P due to a wave motion can be represented by the equation As time t increases, the tip of the phasor rotates counter-clockwise around the circle with constant angular speed ω. Its projection on the y- axis, y 1 which is equal to y 0 sinωt represents the displacement at the point P at any time t. Phasor diagram

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14 © Manhattan Press (H.K.) Ltd. 9.8 Interference of water waves (SB p. 69) Suppose the displacement at the point P due to another wave is represented by Then by the Principle of Superposition of Waves, the resultant of displacement y at the point P due to the two waves is given by Phasor diagram

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15 © Manhattan Press (H.K.) Ltd. 9.8 Interference of water waves (SB p. 70) The resultant displacement y can be obtained using a phasor diagram by finding the vector sum of y 1 and y 2. The amplitude of y is given by the length Y of the rotating vector OA. Phasor diagram

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16 © Manhattan Press (H.K.) Ltd. 9.8 Interference of water waves (SB p. 69) To find the length Y of the resultant displacement, it is easier to draw the phasor diagram for y 1 and y 2 at time t = 0 as shown. Phasor diagram OB represents the phasor for y 1. AB represents the phasor for y 2. OA of length Y represents the phasor for y = y 1 + y 2. The value of Y can be calculated using the cosine rule since OB = BA = y 0 and angle OBA = 180 o . Go to Example 10 Example 10

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17 © Manhattan Press (H.K.) Ltd. End

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18 © Manhattan Press (H.K.) Ltd. 9.8 Interference of water waves (SB p. 67) Q : Q : Two wave generators S 1 and S 2 placed 4 m apart in a water tank, produce water waves of wavelength 1 m. P is a point 3 m from S 1 as shown in the figure. Solution Assuming each generator produces a wave at P which has an amplitude A. When the generators are operating together and in phase, what is the resultant amplitude at P?

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19 © Manhattan Press (H.K.) Ltd. Solution: Solution: Distance of P from S 2 = = 5 m Path difference = PS 2 PS 1 = (5 3) m = 2 m = 2 Since the path difference is 2λ, constructive interference occurs. Thus, the resultant amplitude at P is A + A = 2A. Return to Text 9.8 Interference of water waves (SB p. 67)

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20 © Manhattan Press (H.K.) Ltd. Q : Q : (a) Write down a progressive wave equation. With the aid of suitable diagrams, explain the meanings of the quantities appearing in your equation. (b) Sketch two similar sinusoidal waves each of amplitude A and with phase differences such that, when superposed, the waves would produce (i) constructive interference, (ii) destructive interference. 9.8 Interference of water waves (SB p. 70)

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21 © Manhattan Press (H.K.) Ltd. Q : Q : (c) If two waves have exactly a phase difference of 60° and are superposed, find (i) by means of a phasor diagram, the amplitude of the resultant wave in terms of A, (ii) the ratio of the power carried by the resultant wave to the total power carried by the two component waves considered separately. Does the result obtained contradict the principle of conservation of energy? Solution 9.8 Interference of water waves (SB p. 70)

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22 © Manhattan Press (H.K.) Ltd. Solution: Solution: (a) The progressive wave equation is: y = a sin( ) where a = amplitude, y = displacement of a particle at a distance x from O at time t, λ= wavelength, = angular frequency of wave. 9.8 Interference of water waves (SB p. 71)

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23 © Manhattan Press (H.K.) Ltd. Solution (cont’d): Solution (cont’d): (b) (i) Constructive interference (ii) Destructive interference Phase difference = radians 9.8 Interference of water waves (SB p. 71)

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24 © Manhattan Press (H.K.) Ltd. Solution (cont’d): Solution (cont’d): (c) (i) If phase difference = 60°, the amplitude of the resultant wave is given by the length of PR in the phasor diagram below. In the triangle PQR, using the cosine rule, PR 2 = PQ 2 + QR 2 – 2(PQ)(QR) cos120° = A 2 + A 2 – 2A 2 cos120° = 3A 2 PR = A Therefore, the amplitude of the resultant wave is A. 9.8 Interference of water waves (SB p. 72)

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25 © Manhattan Press (H.K.) Ltd. Solution (cont’d): Solution (cont’d): (ii) Power carried by wave Intensity (Amplitude) 2 The result does not contradict the Principle of Conservation of Energy because where constructive interference occurs, the power carried by the resultant wave is greater than the sum of the power carried by the two waves. The extra power comes from areas where destructive interference occurs. Return to Text 9.8 Interference of water waves (SB p. 72)

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