. C OHERENT AND I NCOHERENT A DDITION OF W AVES In this section we will discuss the interference pattern produced by the superposition of two waves. You may recall that we had discussed the superposition principle in Chapter of your Class XI textbook. Indeed the entire field of interference is based on the superposition principle according to which at a particular point in the medium, the resultant displacement produced by a number of waves is the vector sum of the displacements produced by each of the waves .
Consider two needles S and S moving periodically up and down in an identical fashion in a trough of water [Fig. . (a)]. They produce two water waves, and at a particular point, the phase difference between the displacements produced by each of the waves does not change with time; when this happens the two sources are said to be coherent .
Figure . (b) shows the position of crests (solid circles) and troughs (dashed circles) at a given instant of time. Consider a point P for which S P = S P Since the distances S P and S P are equal, waves from S and S will take the same time to travel to the point P and waves that emanate from S and S in phase will also arrive, at the point P, in phase. Thus, if the displacement produced by the source S at the point P is given by y = a cos , t then, the displacement produced by the source S (at the point P) will also be given by y = a cos , t Thus, the resultant of displacement at P would be given by y = y + y = a cos , t Since the intensity is the proportional to the square of the amplitude, the resultant intensity will be given by I = I where I represents the intensity produced by each one of the individual sources; I is proportional to a .
In fact at any point on the perpendicular bisector of S S , the intensity will be I . The two sources are said to (a) (b) FIGURE . (a) Two needles oscillating in phase in water represent two coherent sources. (b) The pattern of displacement of water molecules at an instant on the surface of water showing nodal N (no displacement) and antinodal A (maximum displacement) lines.