At the point where two waves cross, the total displacement is the vector sum of the individual displacements due to each wave at that point. This is the principle of superposition. If these two waves are either both transverse or both longitudinal, interference effects may be observed. It is not necessary for the two waves to have the same frequencies or amplitudes for the above statements to be true, although these are the waves considered in this chapter.
Consider two transverse waves of the same frequency and amplitude travelling in opposite directions superimposed on one another. Interference takes place between the two waves and a standing or stationary wave is produced. The standing wave is shown in Figure 18.1
The wave does not progress to the left or right and certain parts of the wave called nodes, labelled N in the diagram, do not oscillate. Those positions on the wave that undergo maximum disturbance are called antinodes, labelled
A. The distance between adjacent nodes or adjacent antinodes is , where is the wavelength. Standing waves may be set up in a string, for example, when a wave is reflected at the end of the string and is superimposed on the incoming wave. Under these circumstances standing waves are produced only for certain frequencies. Also, the nodes may not be perfect because the reflected wave may have a slightly reduced amplitude.
Two sound (longitudinal) waves of the same amplitude and frequency travelling in opposite directions and superimposed on each other also produce a standing wave. In this case there are displacement nodes where the medium does not oscillate and displacement antinodes where the displacement is a maximum.
The interference effects mentioned above are not always restricted to the line between the two sources of waves. Two dimensional interference patterns are produced on the surface of water in a ripple tank, for example. In this case, two dippers, usually oscillating in phase and with the same frequency, produce circular ripples on the surface of the water and interference takes place where the circular ripples overlap. The resulting interference pattern is shown in Figure 18.2. The sources of the waves are S1 and S2.
When sea waves are incident on a barrier that is parallel to them a disturbance is observed beyond the barrier in that region where it might be thought that the water would remain undisturbed. This is because waves may spread round obstacles into regions which would be in shadow if the energy travelled exactly in straight lines. This phenomenon is called diffraction. All waves whether transverse or longitudinal exhibit this property. If light, for example, is incident on a narrow slit, diffraction takes place. The diffraction pattern on the screen placed beyond the slit is not perfectly sharp. The intensity of the image varies as shown in Figure 18.3.
A consequence of diffraction is that if light from two sources that are close together pass through a slit or small circular aperture, the diffraction patterns of the two sources may overlap to such an extent that they appear to be one source. If they are to be distinguished as two separate sources, the angular separation, 8, in radians, of the two sources, must be greater than , where b is the wavelength of the light and b is the width of the slit (see Figure 18.4).
For a circular aperture the condition becomes:
If light falls on two narrow parallel slits with a small separation, light passes through both slits and because of diffraction there is an overlapping of the light and interference takes place. This is shown in Figure 18.5. The interference effects are similar to those described for water ripples above.
Suppose the light from the two slits meet at a point on a distant screen. Since the distance between the slits is much less that the slit to screen distance the two light beams will be very nearly parallel. See Figure 18.6. If the path difference is n , where n is an integer and is the wavelength there will be constructive interference and a maximum intensity occurs on the screen.
But from Figure 18.6, the path difference is BC, that is, d sin 8. Thus for a maximum intensity on the screen, n D d sin 8, that is:
The intensity of the interference pattern on the screen at various distances from the polar axis is shown in Figure 18.7. The pattern is modified by the type of diffraction pattern produced by a single slit.
A diffraction grating is similar to the two-slit arrangement, but with a very large number of slits. Very sharp values of maximum intensity are produced in this case. If the slit separation is d and light is incident along the normal to the grating, the condition for a maximum is:
If white light is incident on a diffraction grating a continuous spectrum is produced because the angle at which the first order emerges from the grating depends on the wavelength. Thus the diffraction grating may be used to determine the wavelengths present in a source of light.
Atoms in a crystal diffract X-rays that are incident upon them and information may be gained about crystal structure from the analysis of the diffraction pat- tern obtained. When X-rays strike atoms in a crystal, each atom scatters the X-rays in all directions. However, in certain directions constructive interference takes place. In Figure 18.8 a lattice of atoms is shown, in which X-rays strike atoms and are scattered. The X-rays emerging in a particular direction are considered.
Three planes of atoms are shown. The X-rays ‘reflected’ from the top and middle planes (and any other pair of adjacent planes) will be in phase if their path difference is n , where n is an integer and is the wavelength of the X-rays.