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 t