9.31 Figure 9.30 shows an equiconvex lens (of refractive index 1.50) in contact with a liquid layer on top of a plane mirror. A small needle with its tip on the principal axis is moved along the axis until its inverted image is found at the position of the needle. The distance of the needle from the lens is measured to be 45.0cm. The liquid is removed and the experiment is repeated. The new distance is measured to be 30.0cm. What is the refractive index of the liquid? FIGURE 9.30 253 Reprint 2025-26 Physics Notes Reprint 2025-26 Wave Optics Chapter Ten WAVE OPTICS 10.1 INTRODUCTION In 1637 Descartes gave the corpuscular model of light and derived Snell’s law. It explained the laws of reflection and refraction of light at an interface. The corpuscular model predicted that if the ray of light (on refraction) bends towards the normal then the speed of light would be greater in the second medium. This corpuscular model of light was further developed by Isaac Newton in his famous book entitled OPTICKS and because of the tremendous popularity of this book, the corpuscular model is very often attributed to Newton. In 1678, the Dutch physicist Christiaan Huygens put forward the wave theory of light – it is this wave model of light that we will discuss in this chapter. As we will see, the wave model could satisfactorily explain the phenomena of reflection and refraction; however, it predicted that on refraction if the wave bends towards the normal then the speed of light would be less in the second medium. This is in contradiction to the prediction made by using the corpuscular model of light. It was much later confirmed by experiments where it was shown that the speed of light in water is less than the speed in air confirming the prediction of the wave model; Foucault carried out this experiment in 1850. The wave theory was not readily accepted primarily because of Newton’s authority and also because light could travel through vacuum255 Reprint 2025-26 Physics and it was felt that a wave would always require a medium to propagate from one point to the other. However, when Thomas Young performed his famous interference experiment in 1801, it was firmly established that light is indeed a wave phenomenon. The wavelength of visible light was measured and found to be extremely small; for example, the wavelength of yellow light is about 0.6 mm. Because of the smallness of the wavelength of visible light (in comparison to the dimensions of typical mirrors and lenses), light can be assumed to approximately travel in straight lines. This is the field of geometrical optics, which we had discussed in the previous chapter. Indeed, the branch of optics in which one completely neglects the finiteness of the wavelength is called geometrical optics and a ray is defined as the path of energy propagation in the limit of wavelength tending to zero. After the interference experiment of Young in 1801, for the next 40 years or so, many experiments were carried out involving the interference and diffraction of lightwaves; these experiments could only be satisfactorily explained by assuming a wave model of light. Thus, around the middle of the nineteenth century, the wave theory seemed to be very well established. The only major difficulty was that since it was thought that a wave required a medium for its propagation, how could light waves propagate through vacuum. This was explained when Maxwell put forward his famous electromagnetic theory of light. Maxwell had developed a set of equations describing the laws of electricity and magnetism and using these equations he derived what is known as the wave equation from which he predicted the existence of electromagnetic waves*. From the wave equation, Maxwell could calculate the speed of electromagnetic waves in free space and he found that the theoretical value was very close to the measured value of speed of light. From this, he propounded that light must be an electromagnetic wave. Thus, according to Maxwell, light waves are associated with changing electric and magnetic fields; changing electric field produces a time and space varying magnetic field and a changing magnetic field produces a time and space varying electric field. The changing electric and magnetic fields result in the propagation of electromagnetic waves (or light waves) even in vacuum. In this chapter we will first discuss the original formulation of the Huygens principle and derive the laws of reflection and refraction. In Sections 10.4 and 10.5, we will discuss the phenomenon of interference which is based on the principle of superposition. In Section 10.6 we will discuss the phenomenon of diffraction which is based on Huygens- Fresnel principle. Finally in Section 10.7 we will discuss the phenomenon of polarisation which is based on the fact that the light waves are transverse electromagnetic waves. * Maxwell had predicted the existence of electromagnetic waves around 1855; it was much later (around 1890) that Heinrich Hertz produced radiowaves in the laboratory. J.C. Bose and G. Marconi made practical applications of the Hertzian 256 waves Reprint 2025-26 Wave Optics

10.5 INTERFERENCE OF LIGHT WAVES AND YOUNG’S EXPERIMENT We will now discuss interference using light waves. If we use two sodium lamps illuminating two pinholes (Fig. 10.11) we will not observe any interference fringes. This is because of the fact that the light wave emitted from an ordinary source (like a sodium lamp) undergoes abrupt phase changes in times of the order of 10–10 seconds. Thus the light waves coming out from two independent sources of light will not have any fixed phase relationship and would be incoherent, when this FIGURE 10.11 If two sodium happens, as discussed in the previous section, the lamps illuminate two pinholes intensities on the screen will add up. S1 and S2, the intensities will add The British physicist Thomas Young used an up and no interference fringes will ingenious technique to “lock” the phases of the waves be observed on the screen. emanating from S1 and S2. He made two pinholes S1 and S2 (very close to each other) on an opaque screen [Fig. 10.12(a)]. These were illuminated by another pinholes that was in turn, lit by a bright source. Light waves spread out from S and fall on both S1 and S2. S1 and S2 then behave like two coherent sources because light waves coming out from S1 and S2 are derived from the same original source and any abrupt phase change in S will manifest in exactly similar phase changes in the light coming out from S1 and S2. Thus, the two sources S1 and S2 will be locked in phase; i.e., they will be coherent like the two vibrating needle in our water wave example [Fig. 10.8(a)]. The spherical waves emanating from S1 and S2 will produce interference fringes on the screen GG¢, as shown in Fig. 10.12(b). The positions of maximum and minimum intensities can be calculated by using the analysis given in Section 10.4. (a) (b) FIGURE 10.12 Young’s arrangement to produce interference pattern. 265 Reprint 2025-26 Physics We will have constructive interference resulting in a bright xd region when = nl. That is, D n λD x = xn = ; n = 0, ± 1, ± 2, ... (10.13) d On the other hand, we will have destructive xd 1 interference resulting in a dark region when = (n+ ) l D 2 that is  1 D1829) ) ; n  0,  1,  2 (10.14) x = xn = (n+– 2 d Thomas Young Thus dark and bright bands appear on the screen, (1773 – 1829) English as shown in Fig. 10.13. Such bands are called fringes. physicist, physician and Equations (10.13) and (10.14) show that dark and(1773 Egyptologist. Young worked bright fringes are equally spaced. on a wide variety of scientific problems, ranging from the structure of the eye and the mechanism ofYOUNG vision to the decipherment of the Rosetta stone. He revived the wave theory of light and recognised that interference phenomenaTHOMAS provide proof of the wave properties of light. FIGURE 10.13 Computer generated fringe pattern produced by two point source S1 and S2 on the screen GG¢ (Fig. 10.12); d = 0.025 mm, D = 5 cm and l = 5 × 10–5 cm.) (Adopted from OPTICS by A. Ghatak, Tata McGraw Hill Publishing Co. Ltd., New Delhi, 2000.) 10.6 DIFFRACTION If we look clearly at the shadow cast by an opaque object, close to the region of geometrical shadow, there are alternate dark and bright regions just like in interference. This happens due to the phenomenon of diffraction. Diffraction is a general characteristic exhibited by all types of waves, be it sound waves, light waves, water waves or matter waves. Since the wavelength of light is much smaller than the dimensions of most 266 obstacles; we do not encounter diffraction effects of light in everyday Reprint 2025-26 Wave Optics observations. However, the finite resolution of our eye or of optical instruments such as telescopes or microscopes is limited due to the phenomenon of diffraction. Indeed the colours that you see when a CD is viewed is due to diffraction effects. We will now discuss the phenomenon of diffraction. 10.6.1 The single slit In the discussion of Young’s experiment, we stated that a single narrow slit acts as a new source from which light spreads out. Even before Young, early experimenters – including Newton – had noticed that light spreads out from narrow holes and slits. It seems to turn around corners and enter regions where we would expect a shadow. These effects, known as diffraction, can only be properly understood using wave ideas. After all, you are hardly surprised to hear sound waves from someone talking around a corner! When the double slit in Young’s experiment is replaced by a single narrow slit (illuminated by a monochromatic source), a broad pattern with a central bright region is seen. On both sides, there are alternate dark and bright regions, the intensity becoming weaker away from the centre (Fig. 10.15). To understand this, go to Fig. 10.14, which shows a parallel beam of light falling normally on a single slit LN of width a. The diffracted light goes on to meet FIGURE 10.14 The geometry of path a screen. The midpoint of the slit is M. differences for diffraction by a single slit. A straight line through M perpendicular to the slit plane meets the screen at C. We want the intensity at any point P on the screen. As before, straight lines joining P to the different points L,M,N, etc., can be treated as parallel, making an angle q with the normal MC. The basic idea is to divide the slit into much smaller parts, and add their contributions at P with the proper phase differences. We are treating different parts of the wavefront at the slit as secondary sources. Because the incoming wavefront is parallel to the plane of the slit, these sources are in phase. It is observed that the intensity has a central maximum at q = 0 and other secondary maxima at q l (n+1/2) l/a, which go on becoming weaker and weaker with increasing n. The minima (zero intensity) are at q l nl/a, n = ±1, ±2, ±3, .... FIGURE 10.15 Intensity The photograph and intensity pattern corresponding distribution and photograph of to it is shown in Fig. 10.15. fringes due to diffraction There has been prolonged discussion about at single slit. difference between intereference and diffraction among 267 Reprint 2025-26 Physics scientists since the discovery of these phenomena. In this context, it is interesting to note what Richard Feynman* has said in his famous Feynman Lectures on Physics: No one has ever been able to define the difference between interference and diffraction satisfactorily. It is just a question of usage, and there is no specific, important physical difference between them. The best we can do is, roughly speaking, is to say that when there are only a few sources, say two interfering sources, then the result is usually called interference, but if there is a large number of them, it seems that the word diffraction is more often used. In the double-slit experiment, we must note that the pattern on the screen is actually a superposition of single-slit diffraction from each slit or hole, and the double-slit interference pattern. 10.6.210.6.210.6.210.6.210.6.2 SeeingSeeingSeeingSeeingSeeing thethethethethe singlesinglesinglesinglesingle slitslitslitslitslit diffractiondiffractiondiffractiondiffractiondiffraction patternpatternpatternpatternpattern It is surprisingly easy to see the single-slit diffraction pattern for oneself. The equipment needed can be found in most homes –– two razor blades and one clear glass electric bulb preferably with a straight filament. One has to hold the two blades so that the edges are parallel and have a narrow slit in between. This is easily done with the thumb and forefingers (Fig. 10.16). Keep the slit parallel to the filament, right in front of the eye. Use spectacles if you normally do. With slight adjustment of the width of the slit and the parallelism of the edges, the pattern should be seen with its bright and dark bands. Since the position of all the bands (except the central one) depends on wavelength, they will show some colours. Using a filter for red or blue will make the fringes clearer. With both filters available, the wider fringes for red compared to blue FIGUREFIGUREFIGUREFIGUREFIGURE 10.1610.1610.1610.1610.16 can be seen. Holding two blades to In this experiment, the filament plays the role of the first slit S in form a single slit. A bulb filament viewed Fig. 10.15. The lens of the eye focuses the pattern on the screen (the through this shows retina of the eye). clear diffraction With some effort, one can cut a double slit in an aluminium foil with bands. a blade. The bulb filament can be viewed as before to repeat Young’s experiment. In daytime, there is another suitable bright source subtending a small angle at the eye. This is the reflection of the Sun in any shiny convex surface (e.g., a cycle bell). Do not try direct sunlight – it can damage the eye and will not give fringes anyway as the Sun subtends an angle of (1/2)°. In interference and diffraction, light energy is redistributed. If it reduces in one region, producing a dark fringe, it increases in another region, producing a bright fringe. There is no gain or loss of energy, which is consistent with the principle of conservation of energy. * Richard Feynman was one of the recipients of the 1965 Nobel Prize in Physics 268 for his fundamental work in quantum electrodynamics. Reprint 2025-26 Wave Optics

10.2 HUYGENS PRINCIPLE We would first define a wavefront: when we drop a small stone on a calm pool of water, waves spread out from the point of impact. Every point on the surface starts oscillating with time. At any instant, a photograph of the surface would show circular rings on which the disturbance is maximum. Clearly, all points on such a circle are oscillating in phase because they are at the same distance from the source. Such a locus of points, which oscillate in phase is called a wavefront; thus a wavefront is defined as a surface of constant FIGURE 10.1 (a) Aphase. The speed with which the wavefront moves outwards from the diverging spherical source is called the speed of the wave. The energy of the wave travels wave emanating from in a direction perpendicular to the wavefront. a point source. The If we have a point source emitting waves uniformly in all directions, wavefronts are then the locus of points which have the same amplitude and vibrate in spherical. the same phase are spheres and we have what is known as a spherical wave as shown in Fig. 10.1(a). At a large distance from the source, a small portion of the sphere can be considered as a plane and we have what is known as a plane wave [Fig. 10.1(b)]. Now, if we know the shape of the wavefront at t = 0, then Huygens principle allows us to determine the shape of the wavefront at a later time t. Thus, Huygens principle is essentially a geometrical construction, which given the shape of the wafefront at any time allows us to determine the shape of the wavefront at a later time. Let us consider a diverging FIGURE 10.1 (b) At a wave and let F1F2 represent a portion of the spherical wavefront at t = 0 large distance from (Fig. 10.2). Now, according to Huygens principle, each point of the the source, a small wavefront is the source of a secondary disturbance and the wavelets portion of the emanating from these points spread out in all directions with the speed spherical wave can of the wave. These wavelets emanating from the wavefront are usually be approximated by a plane wave.referred to as secondary wavelets and if we draw a common tangent to all these spheres, we obtain the new position of the wavefront at a later time. FIGURE 10.2 F1F2 represents the spherical wavefront (with O as centre) at t = 0. The envelope of the secondary wavelets emanating from F1F2 produces the forward moving wavefront G1G2. The backwave D1D2 does not exist. 257 Reprint 2025-26 Physics Thus, if we wish to determine the shape of the wavefront at t = t, we draw spheres of radius vt from each point on the spherical wavefront where v represents the speed of the waves in the medium. If we now draw a common tangent to all these spheres, we obtain the new position of the wavefront at t = t. The new wavefront shown as G1G2 in Fig. 10.2 is again spherical with point O as the centre. The above model has one shortcoming: we also have a backwave which is shown as D1D2 in Fig. 10.2. Huygens argued that the amplitude of the secondary wavelets is maximum in the forward direction and zero in the backward direction; by making this adhoc assumption, Huygens could explain the absence of the backwave. However, this adhoc assumption is not satisfactory and the absence of the backwave is really justified from more rigorous wave theory. In a similar manner, we can use Huygens principle to determine the shape of the wavefront for a plane wave propagating through a medium (Fig. 10.3). FIGURE 10.3 Huygens geometrical construction for a 10.3 REFRACTION AND REFLECTION OF plane wave PLANE WAVES USING HUYGENS PRINCIPLE propagating to the right. F1 F2 is the 10.3.1 Refraction of a plane wave plane wavefront at t = 0 and G1G2 is the We will now use Huygens principle to derive the laws of refraction. Let PP¢ wavefront at a later represent the surface separating medium 1 and medium 2, as shown in time t. The lines A1A2, Fig. 10.4. Let v1 and v2 represent the speed of light in medium 1 and B1B2 … etc., are medium 2, respectively. We assume a plane wavefront AB propagating in normal to both F1F2 the direction A¢A incident on the interface at an angle i as shown in the and G1G2 and figure. Let t be the time taken by the wavefront to travel the distance BC. represent rays. Thus, BC = v1 t FIGURE 10.4 A plane wave AB is incident at an angle i on the surface PP¢ separating medium 1 and medium 2. The plane wave undergoes refraction and CE represents the refracted wavefront. The figure corresponds to v2 < v1 so that the refracted waves bends towards the 258 normal. Reprint 2025-26 Wave Optics In order to determine the shape of the refracted wavefront, we draw a sphere of radius v2t from the point A in the second medium (the speed of the wave in the second medium is v2). Let CE represent a tangent plane drawn from the point C on to the sphere. Then, AE = v2 t and CE would represent the refracted wavefront. If we now consider the triangles ABC and AEC, we readily obtain BC v1τ sin i = = (10.1) AC AC and AE v 2τ sin r = = (10.2) AC AC ChristiaanChristiaanChristiaanChristiaanChristiaan HuygensHuygensHuygensHuygensHuygens where i and r are the angles of incidence and refraction, (1629(1629(1629(1629(1629 ––––– 1695)1695)1695)1695)1695) Dutch respectively. Thus we obtain physicist, astronomer, mathematician and the sin i v1 founder of the wave = sin r v 2 (10.3) theory of light. His book, Treatise on light, makes From the above equation, we get the important result CHRISTIAAN fascinating reading even that if r < i (i.e., if the ray bends toward the normal), the today. He brilliantly speed of the light wave in the second medium (v2) will be explained the double less then the speed of the light wave in the first medium refraction shown by the (v1). This prediction is opposite to the prediction from mineral calcite in this the corpuscular model of light and as later experiments work in addition to HUYGENSshowed, the prediction of the wave theory is correct. Now, reflection and refraction. if c represents the speed of light in vacuum, then, He was the first to analyse circular and c simple harmonic motion (1629 n1 = – v1 (10.4) and designed and built improved clocks and and telescopes. He discovered c the true geometry of 1695) n2 = v 2 (10.5) Saturn’s rings. are known as the refractive indices of medium 1 and medium 2, respectively. In terms of the refractive indices, Eq. (10.3) can be written as n1 sin i = n2 sin r (10.6) This is the Snell’s law of refraction. Further, if l1 and l 2 denote the wavelengths of light in medium 1 and medium 2, respectively and if the distance BC is equal to l1 then the distance AE will be equal to l2 (because if the crest from B has reached C in time t, then the crest from A should have also reached E in time t ); thus, λ1 BC v1 = = λ2 AE v 2 or v1 v 2 = (10.7) 259 λ1 λ2 Reprint 2025-26 Physics The above equation implies that when a wave gets refracted into a denser medium (v1 > v2) the wavelength and the speed of propagation decrease but the frequency n (= v/l) remains the same. 10.3.2 Refraction at a rarer medium We now consider refraction of a plane wave at a rarer medium, i.e., v2 > v1. Proceeding in an exactly similar manner we can construct a refracted wavefront as shown in Fig. 10.5. The angle of refraction will now be greater than angle of incidence; however, we will still have effect n1 sin i = n2 sin r . We define an angle ic by the following equation n 2 sin i c = (10.8) Doppler n1 and Thus, if i = ic then sin r = 1 and r = 90°. Obviously, for i > ic, there can not be any refracted wave. The angle ic is known as the critical angle and for all angles of incidence greater than the critical angle, we will not have any refracted wave and the wave will undergo what is known as total internal reflection. The phenomenon of total internal reflection and its resonance applications was discussed in Section 9.4. refraction, diffraction, interference, of FIGURE 10.5 Refraction of a plane wave incident on a rarer medium for which v2 > v1. The plane wave bends away from the normal. Demonstration http://www.falstad.com/ripple/ 10.3.3 Reflection of a plane wave by a plane surface We next consider a plane wave AB incident at an angle i on a reflecting surface MN. If v represents the speed of the wave in the medium and if t represents the time taken by the wavefront to advance from the point B to C then the distance BC = vt In order to construct the reflected wavefront we draw a sphere of radius vt from the point A as shown in Fig. 10.6. Let CE represent the tangent plane drawn from the point C to this sphere. Obviously 260 AE = BC = vt Reprint 2025-26 Wave Optics FIGURE 10.6 Reflection of a plane wave AB by the reflecting surface MN. AB and CE represent incident and reflected wavefronts. If we now consider the triangles EAC and BAC we will find that they are congruent and therefore, the angles i and r (as shown in Fig. 10.6) would be equal. This is the law of reflection. Once we have the laws of reflection and refraction, the behaviour of prisms, lenses, and mirrors can be understood. These phenomena were discussed in detail in Chapter 9 on the basis of rectilinear propagation of light. Here we just describe the behaviour of the wavefronts as they undergo reflection or refraction. In Fig. 10.7(a) we consider a plane wave passing through a thin prism. Clearly, since the speed of light waves is less in glass, the lower portion of the incoming wavefront (which travels through the greatest thickness of glass) will get delayed resulting in a tilt in the emerging wavefront as shown in the figure. In Fig. 10.7(b) we consider a plane wave incident on a thin convex lens; the central part of the incident plane wave traverses the thickest portion of the lens and is delayed the most. The emerging wavefront has a depression at the centre and therefore the wavefront becomes spherical and converges to the point F which is known as the focus. In Fig. 10.7(c) a plane wave is incident on a concave mirror and on reflection we have a spherical wave converging to the focal point F. In a similar manner, we can understand refraction and reflection by concave lenses and convex mirrors. From the above discussion it follows that the total time taken from a point on the object to the corresponding point on the image is the same measured along any ray. For example, when a convex lens focusses light to form a real image, although the ray going through the centre traverses a shorter path, but because of the slower speed in glass, the time taken is the same as for rays travelling near the edge of the lens. FIGURE 10.7 Refraction of a plane wave by (a) a thin prism, (b) a convex lens. 261 (c) Reflection of a plane wave by a concave mirror. Reprint 2025-26 Physics Example 10.1 (a) When monochromatic light is incident on a surface separating two media, the reflected and refracted light both have the same frequency as the incident frequency. Explain why? (b) When light travels from a rarer to a denser medium, the speed decreases. Does the reduction in speed imply a reduction in the energy carried by the light wave? (c) In the wave picture of light, intensity of light is determined by the square of the amplitude of the wave. What determines the intensity of light in the photon picture of light. Solution (a) Reflection and refraction arise through interaction of incident light with the atomic constituents of matter. Atoms may be viewed as oscillators, which take up the frequency of the external agency (light) causing forced oscillations. The frequency of light emitted by a charged oscillator equals its frequency of oscillation. Thus, the frequency of scattered light equals the frequency of incident light. (b) No. Energy carried by a wave depends on the amplitude of the 10.1 wave, not on the speed of wave propagation. (c) For a given frequency, intensity of light in the photon picture is determined by the number of photons crossing an unit area per EXAMPLE unit time. 10.4 COHERENT AND INCOHERENT ADDITION OF WAVES 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 14 of your Class XI textbook. Indeed the entire field of interference is based on the superposition (a) (b) principle according to which at a particular point in the medium, the resultant FIGURE 10.8 (a) Two needles oscillating in displacement produced by a number ofphase in water represent two coherent sources. (b) The pattern of displacement of water waves is the vector sum of the displace- molecules at an instant on the surface of water ments produced by each of the waves. showing nodal N (no displacement) and Consider two needles S1 and S2 moving antinodal A (maximum displacement) lines. periodically up and down in an identical fashion in a trough of water [Fig. 10.8(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 10.8(b) shows the position of crests (solid circles) and troughs (dashed circles) at a given instant of time. Consider a point P for which 262 S1 P = S2 P Reprint 2025-26 Wave Optics Since the distances S1 P and S2 P are equal, waves from S1 and S2 will take the same time to travel to the point P and waves that emanate from S1 and S2 in phase will also arrive, at the point P, in phase. Thus, if the displacement produced by the source S1 at the point P is given by y1 = a cos wt then, the displacement produced by the source S2 (at the point P) will also be given by y2 = a cos wt Thus, the resultant of displacement at P would be given by y = y1 + y2 = 2 a cos wt Since the intensity is proportional to the square of the amplitude, the resultant intensity will be given by I = 4 I0 where I0 represents the intensity produced by each one of the individual sources; I0 is proportional to a2. In fact at any point on the perpendicular bisector of S1S2, the intensity will be 4I0. The two sources are said to FIGURE 10.9interfere constructively and we have what is referred to as constructive (a) Constructive interference. We next consider a point Q [Fig. 10.9(a)] interference at a for which point Q for which the S2Q –S1Q = 2l path difference is 2l. (b) Destructive The waves emanating from S1 will arrive exactly two cycles earlier interference at a than the waves from S2 and will again be in phase [Fig. 10.9(a)]. Thus, if point R for which the the displacement produced by S1 is given by path difference is 2.5 l. y1 = a cos wt then the displacement produced by S2 will be given by y2 = a cos (wt – 4p) = a cos wt where we have used the fact that a path difference of 2l corresponds to a phase difference of 4p. The two displacements are once again in phase and the intensity will again be 4 I0 giving rise to constructive interference. In the above analysis we have assumed that the distances S1Q and S2Q are much greater than d (which represents the distance between S1 and S2) so that although S1Q and S2Q are not equal, the amplitudes of the displacement produced by each wave are very nearly the same. We next consider a point R [Fig. 10.9(b)] for which S2R – S1R = –2.5l The waves emanating from S1 will arrive exactly two and a half cycles later than the waves from S2 [Fig. 10.10(b)]. Thus if the displacement FIGURE 10.10 Locus produced by S1 is given by of points for which y1 = a cos wt S1P – S2P is equal to zero, ±l, ± 2l, ± 3l. then the displacement produced by S2 will be given by y2 = a cos (wt + 5p) = – a cos wt 263 Reprint 2025-26 Physics where we have used the fact that a path difference of 2.5l corresponds to a phase difference of 5p. The two displacements are now out of phase and the two displacements will cancel out to give zero intensity. This is referred to as destructive interference. To summarise: If we have two coherent sources S1 and S2 vibrating in phase, then for an arbitrary point P whenever the path difference, S1P ~ S2P = nl (n = 0, 1, 2, 3,...) (10.9) we will have constructive interference and the resultant intensity will be 4I0; the sign ~ between S1P and S2 P represents the difference between S1P and S2 P. On the other hand, if the point P is such that the path difference, 1 S1P ~ S2P = (n+ ) l (n = 0, 1, 2, 3, ...) (10.10) 2 we will have destructive interference and the resultant intensity will be zero. Now, for any other arbitrary point G (Fig. 10.10) let the phase difference between the two displacements be f. Thus, if the displacement produced by S1 is given by y1 = a cos wt then, the displacement produced by S2 would be interference y2 = a cos (wt + f) wave and the resultant displacement will be given by on y = y1 + y2 = a [cos wt + cos (wt +f)] = 2 a cos (f/2) cos (wt + f/2) experiments Tank The amplitude of the resultant displacement is 2a cos (f/2) and therefore the intensity at that point will be I = 4 I0 cos2 (f/2) (10.11) Ripple http://phet.colorado.edu/en/simulation/legacy/wave-interference If f = 0, ± 2 p, ± 4 p,… which corresponds to the condition given by Eq. (10.9) we will have constructive interference leading to maximum intensity. On the other hand, if f = ± p, ± 3p, ± 5p … [which corresponds to the condition given by Eq. (10.10)] we will have destructive interference leading to zero intensity. Now if the two sources are coherent (i.e., if the two needles are going up and down regularly) then the phase difference f at any point will not change with time and we will have a stable interference pattern; i.e., the positions of maxima and minima will not change with time. However, if the two needles do not maintain a constant phase difference, then the interference pattern will also change with time and, if the phase difference changes very rapidly with time, the positions of maxima and minima will also vary rapidly with time and we will see a “time-averaged” intensity distribution. When this happens, we will observe an average intensity that will be given by I = 2 I0 (10.12) 264 at all points. Reprint 2025-26 Wave Optics When the phase difference between the two vibrating sources changes rapidly with time, we say that the two sources are incoherent and when this happens the intensities just add up. This is indeed what happens when two separate light sources illuminate a wall.