Deformed Waves Can Produce Images
In Physics 7C so far we have introduced waves and discussed their interactions. Our model of waves has been so useful because it enables us to apply the same basic ideas to a wide variety of phenomena; namely waves on ropes, sound waves, light waves and other types of waves. In this chapter we will be dealing with waves that travel from one medium to another. In such a case two things can happen: part of the wave can bounce back into the original medium which we refer to as reflection, and part of the wave can travel into the next medium, a transmission. When a wave travels into a new medium the wave is typically "bent" or deformed in some way, a phenomenon we call refraction.
We can combine these effects of reflecting and bending waves to make the waves appear as if they are being created at different locations than they actually are. If the waves in question are light waves then this means that we see images at places distinct from where the objects themselves are. While most of our examples will involve light waves, it is important to realize that all types of waves will reflect and refract as they pass from one medium to another. (For completeness we mention that there are two other methods by which the path of light can be altered: absorption and scattering. We will not discuss these further.)
Representing a Wave with Wavefronts
So far we've pictured waves using only an oscillating sine functions. For one-dimensional waves this was adequate, but for two- or higher dimensional waves this representation becomes difficult, and so we introduce the idea of wavefronts and rays. Let us start by thinking of dropping a stone in water and letting the ripples propagate outward. Some time later, the ripples appear as shown on the figure below on the left. In this figure, parts of the wave are obscured by other waves and it is generally difficult to draw and visualize interactions between waves like this.
These limitations make it a difficult to use representation, so we adopt a more useful representation that omits a few details. One such representation is the wavefront representation in which we choose to only draw one part of the waves. The figure above and on the right is a wavefront representation of the picture on the left where we've chosen only to show the peaks of the waves. Because we know the wave is oscillating up and down, and traveling outward, you should have a reasonable idea of what the wave is doing just by looking at the picture of the wavefronts.
Occasionally we shall draw wavefronts for the peaks and troughs in different colors so that we can superpose them, recalling that peak-and-peak or trough-and-trough interference is constructive, while peak-and-trough interference is destructive.
If the medium that the wave is propagating through is isotropic (i.e. behaves the same in all locations) the wave will spread out at the same speed in all directions and the wavefronts will be either concentric circles for 2-D waves or concentric spheres for 3-D waves. As the wavefronts travel further from the source, they start to appear less curved. Far enough away, the wavefront doesn't appear curved at all, like being on the edge of a giant circle. When we can approximate waves as being flat we call them plane waves, because the wavefront of a 3-D wave (like sound) resembles a plane at far distances. Plane waves are convenient because we can approximate that the wave travels in one direction, which enables us to describe it using the harmonic wave equation we developed.
Representing a Wave with Rays
We can simplify representing a wave in a different way; consider only the direction that a particular piece of the wave is traveling. We can join these directions and trace out a path of a particular piece of a wave. These paths are called rays, and are always perpendicular to the wavefronts. Examples of rays are shown in the figure below as arrows.
Notice that we can draw whichever rays are convenient to use; in the above example, we drew many rays going in all directions. On the right, we concentrated the rays in the part where we discussed the wavefronts looking flat so we can discuss one more important quality of rays: when you are a far distance from the source of the wave, the rays are approximately parallel. The extra rays do not indicate that there is more energy in that part of the wave, or that this section is particularly important. We are simply drawing those rays because they bring attention to the phenomenon we wish to discuss. Throughout this section on optics we will select our rays to illustrate the points we wish to make.
Two (plane) waves with the same wavelength are coming in from far away. One is travelling right and the other is travelling from the bottom of the page to the top, as shown in the picture below. Find the interference pattern created by the two waves when they cross.
The picture above showed the wavefronts for the peaks only. To complete the picture, we include both wavefronts from the peaks (in red) and the troughs (in blue). We know if two of the same color cross (i.e. peak-and-peak, or trough-and-trough) we get constructive interference, while if a red and blue wavefront cross (i.e. peak-and-trough) we get destructive interference. This is shown below.f
On the left we have drawn the individual wavefronts crossing, and on the right we have put C for constructive interference and 0 for destructive interference. Both of these are easier to draw and visualize than a picture of the wave that keeps all the wave's information (like the one below).