Superposition Notes

 2014 Yeow Kok Han Page 1 SSuuppeerrppoossiittiioonn 11 SSuuppeerrppoossiittiioonn When 2 particles collide, they bounce off each other or sometimes stick together. However, when waves meet they interact in a very unique way called the principle of superposition: Fig. 1.1 shows how 2 wave pulses on a string interact. Notice how they pass right through each other and thereafter continue their journey as if nothing has happened. Such behaviour distinguishes waves from particles. Waves from two sources do not meet by travelli ng in opposite directions only. Also, instead of an interaction of pulses, the waves can be continuous. Fig. 1.2 shows 2 sources S 1 & S 2 continuously emitting waves meeting at points P 1, P 2 & P 3 among many other possible points. For the rest of this topic, we will ke ep things simple by considering the sources and meetings points to be all on the same plane. When two waves of the same kind meet at a point, the resultant displacement at the point is a vector sum of their individual displacements. Fig. 1.1a Fig. 1.1b When two waves of the same kind meet at a point, the resultant displacement at the point is a vector sum of their individual displacements. S1 S2 P1 P2 P3 Fig. 1.2

 2014 Yeow Kok Han Page 2 SSuuppeerrppoossiittiioonn ooff WWaatteerr WWaavveess SS11 && SS22 IInn PPhhaassee In Fig. 1.3 , 1.4 & 1.5, sources S 1 & S2 oscillate in phase and continuously radiate water waves of the same frequency . P is a selected point where we consider how the waves meet and superpose. To simplify things, we assume that as the waves travel from S 1 & S2 to P, the respective amplitudes A1 and A2 do not decrease. | S2P  S1P | is called the path difference (PD) which is the difference in distance travel led by the waves from S 1 and S2. We shall see that PD is one factor in determining the result of superposition. In Fig. 1.3, P is a point equidistant from the sources. As the waves originated in phase and travelled the same distance to meet at P, they wi ll meet in phase i.e. the trough of one wave will always meet the trough of the other and similarly for crests. The result of superposition is an oscillation at P with amplitude = A1 + A2. In Fig. 1. 4, waves originated from S1 and S2 in phase but waves from S2 always travelled an extra ½ to reach P. The different path lengths cause the waves to always meet at P in anti-phase(crest meet trough), resulting in cancellation of displacements by the principle of superposition. The amplitude at P is = | A1  A2 | which equals zero if the individual amplitudes are equal. In Fig. 1.5, S1 and S2 are in phase again but the PD is now  compared to ½ in Fig. 1.4. Now the result of superposition is an oscillation at P with amplitude = A1 + A2. Whenever the ampli tudes add, we say that the waves from S 1 and S2 interfere constructively and whenever the amplitudes subtract, we say they interfere destructively. There is no special term for situations in between constructive and destructive interference. To generalise: PD, path difference is a factor determining if waves from 2 sources meet in phase or otherwise. When 2 wave trains meet in phase continuously, the resulting amplitude is the sum of the individual wave amplitudes and the situation is called constructive interference. When 2 wave trains meet in anti-phase continuously, the resulting amplitude is the difference between the individual wave amplitudes and the situation is called destructive interference. When 2 sources are in phase, the conditions for constructive or destructive interference are shown in the box on the left. S1 S2 PD = S2P  S1P = 2½  2 = ½ Destructive Interference P S1 S2 P PD = S2P  S1P = 2  2 = 0 Constructive Interference Fig. 1.3 Fig. 1.4 S1 S2 PD = S2P  S1P = 3½  2½ =  Constructive Interference P Fig. 1.5 A1 A2 Ap Ap Ap = A1 + A2 Ap = | A1  A2 | Ap Ap = A1 + A2 PD PD For sources S1 and S2 in phase, when PD = n, waves meet in phase  constructive interference PD = (n + ½ ), waves meet in anti-phase  destructive interference where n is integer.

 2014 Yeow Kok Han Page 3 SS11 && SS22 IInn AAnnttii--PPhhaassee The difference between Fig. 1.6, 1.7 & 1.8 compared to Fig. 1.3, 1.4 & 1.5 respectively is that the sources S1 & S2 now oscillate in anti -phase instead. We shall see that the initial phase difference when the waves are produced is the other factor that will also determine the result of superposition. In Fig. 1.6, t he initial phase difference between the waves generated at S 1 and S 2 is maintained as the waves travelled to meet at P because the paths travelled have the same distance. Hence the waves meet in anti - phase, resulting in cancellation of displacements. Resulting amplitude Ap = | A1  A2 |. In Fig. 1. 7, waves from S 1 and S 2 have initial phase difference 180  or  rad. Recall in the topic Waves that  corresponds to a phase difference of 360  or 2  rad. The waves from S 2 travelled an extra distance of ½ to meet waves from S1 thus introducing an extra 180 phase difference and allowing the waves to meet in phase. Resulting amplitude Ap = A1 + A2. In Fig. 1.8 , the PD is now  compared to ½ in Fig. 1.7. Due to the PD, t he extra phase difference introduced is 360  or one cycle, which in effect is equivalent to a phase difference of 0. Taking into account the initial phase difference of 180  due to the sources, the waves would meet in anti-phase. Resulting amplitude Ap = | A1  A2 |. Again, to generalise: Initial phase difference between 2 wave sources is another factor determining if waves from 2 sources meet in phase or otherwise. When 2 sources are in anti -phase, the conditions for constructive or destructive interference are shown in the box on the left. For sources S1 and S2 in anti-phase, when PD = n, waves meet in anti-phase  destructive interference PD = (n + ½ ), waves meet in phase  constructive interference where n is integer. Ap = | A1  A2 | S1 P PD = S2P  S1P = 2½ - 2 = ½ Constructive Interference S1 S2 P PD = S2P  S1P = 2 - 2 = 0 Destructive Interference PD = S2P  S1P = 3½  2½ =  Destructive Interference S1 S2 P Fig. 1.6 Fig. 1.7 Fig. 1.8 S2 Ap Ap = | A1  A2 | A1 A2 PD PD Ap = A1 + A2 Ap Ap

 2014 Yeow Kok Han Page 4 RRiippppllee TTaannkk A ripple tank can be used to create circular or plane waves depending on set - up. Fig. 1.9 shows plane waves being created. To see the interference of circular waves from two point sources S 1 and S2, the paddle will be suspended above the water s urface and 2 vertical rods attached to the paddle. As the paddle oscillates, the rods will dip in and out of the water surface, creating circular ripples. Fig. 1.10 and 1.11 show how the water waves from 2 in phase sources produce an interference pattern made up of lines of constructive and destructive interference. For the points where crest meets trough , crest meets crest or trough meets trough, the path difference can be worked out by counting the difference in the number of rings away from each source. When waves of the same  from 2 sources meet, an interference pattern made up of lines of constructive and destructive interference is formed.  crest trough crest meets crest S1 S2 2 1½  0 ½ I-x plot along