Wavefronts, Refraction, and Snell’s Law
1. Wavefronts
If you drop a rock into a pool of water, circular ripples will spread out from the place where the rock hit the water. (See Fig 1 to right.) These circular ripples are waves, and they can be drawn on paper as a set of concentric circles. Each circle represents what is called a wavefront. A wavefront is just all the points along a traveling wave that have the same phase (which means they are at the same relative location along the wave—e.g., all points that are lined up along a crest, or all points that are lined up along a trough). It is easiest to imagine that each line is a crest of a 2D wave. Note that each circular ripple is expanding outward from the point at which the rock hit the water. Wavefronts are always perpendicular to the direction in which the wave is traveling.
Of course, not all waves are circular. A plane wave is one in which wavefronts are parallel straight lines. (See Fig 2 to right.) Again, the direction in which the wave propagates is perpendicular to each wavefront. Thus, in the figure to the right, the wave could be traveling either to the right or to the left. (Assume right, as indicated by the arrow.) The distance between two adjacent lines is one wavelength.
2. Refraction
The speed at which a wave travels depends on what kind of medium the wave is traveling in. For example, light travels faster in air than it does in water. When a wave travels from one medium into another, some strange things happen. We can use the method of drawing waves described above to see what strange things happen.
First, the wavelength changes. Look at Fig 3 below. A wave is moving from medium 1 to medium 2, and the speed at which the wave travels in medium 2 is faster than in medium 1. When a wavefront crosses over the boundary into medium 2, it speeds up while the wavefronts behind it in medium 1 are still moving at the original slower speed. As a result, each wavefront gets farther ahead of the wavefront behind it when it crosses the boundary, and the wavelength increases as the wave crosses the boundary. This is what happens when a wave crosses from a slower medium to a faster medium.
The opposite effect occurs when a wave crosses from a faster medium to a slower medium. (See Fig 4 below.) Each wavefront slows down just after it crosses the boundary, while the wavefront behind it is still traveling at the original, faster speed. Thus the wavefronts get closer together after crossing the boundary, and the wavelength is shortened.
Note that the frequency of the wave remains the same in both mediums. (The frequency at which the incoming wavefronts hit the boundary in medium 1 is the same as the frequency at which the outgoing wavefronts leave the boundary in medium 2.) It is only the speed and the wavelength that change. From the equation c = fλ, we get that c/λ = f = constant. Thus if speed increases, wavelength decreases proportionally; and if speed decreases, wavelength increases (so that frequency remains constant).
The second interesting thing that happens when a wave crosses from one medium to another is that the direction of propagation changes (unless the wavefronts are parallel to the boundary, as in the examples above). This reason for this is similar to the reason the wavelength changes. Consider a wave crossing a boundary at an angle, as shown in the figure below. When the wavefronts are not parallel to the boundary, one side of the wavefront crosses the boundary first. That part of the wavefront is then traveling faster than the part that is still in the first medium. As a result, the wavefront gets bent. Once all of the wavefronts have crossed the boundary, their direction is different from what it was before. As a result, the wave is traveling in a different direction.
In order to describe this effect mathematically, we need to define an angle at which the wave hits the boundary between the two mediums, called angle of incidence, and an angle at which the wave leaves the boundary, called the angle of refraction. These angles are measured from the normal to the boundary.
The general rule in qualitative terms is:
1. When a wave crosses from a slower medium to a faster medium, its wavelength gets longer and its direction of propagation turns away from the normal (angle of refraction is greater than angle of incidence).
2. When a wave crosses from a faster medium to a slower medium, its wavelength gets shorter and its direction of propagation turns toward the normal (angle of refraction is less than angle of incidence).
The mathematical equation that describes this behavior is called Snell’s Law:
Here θi is the angle of incidence and θr is the angle of refraction; vi is the speed of the wave in medium 1 (the medium in which the wave is incident on the boundary), and vr is the speed of the wave in medium 2. We can define the relative index of refraction nrel between the two materials to be the ratio of the speeds at which the wave travels in the two mediums:
Then Snell’s Law can be written as . For engineering purposes, it is often useful to determine the relative index of refraction between two materials. With some waves it is easier to measure angles of incidence and refraction than it is to measure wave speeds.
In optics (i.e., when dealing with light), we define the absolute index of refraction for a material to be
, where c is the speed at which light travels in a vacuum, and v is the speed at which light travels in the material. Using absolute indices of refraction, Snell’s Law can be written as
,
where n1 is the absolute index of refraction of medium 1 and n2 is the absolute index of refraction of medium 2. Angle θ1 is the angle between the direction of propagation and the normal of the boundary in medium 1, and angle θ2 is the angle between the direction of propagation and the normal in medium 2.
Practice Problems
1. A material’s index of refraction is actually different for waves of different frequencies. In general, higher frequency waves are refracted more strongly than lower frequency waves. Since different colors of light have different frequencies, they are refracted by different amounts when they cross boundaries between two mediums. This is how rainbows are formed: White light is composed of all colors (frequencies) of visible light. When white light from the sun is refracted by drops of water in the atmosphere, the different-colored components of the light are refracted by different amounts. For this problem, consider a thin ray of white light that is incident from air onto a block of glass at an angle of 60°. The highest frequency component of the white ray is blue light, for which the glass’s index of refraction is 3.2. The lowest frequency component of the ray is red light, for which the index of refraction is 1.8. Use Snell’s Law to find the angle over which the white ray of light will be refracted. (Use 1.0 for the index of refraction of air.)
Δθr = ______
2. A ray of yellow light has wavelength 580 nm in air and 460 nm in an unknown material. Find the absolute index of refraction of the material. (Again, the index of refraction of air is 1.0.) Here’s an interesting question: Does the color of light depend on its wavelength or on its frequency? (In space or air, this question doesn’t matter since each frequency corresponds to only one wavelength. But when light of a particular wavelength goes from one medium to another, its wavelength changes while its frequency remains the same. Has its color changed?)
n = ______
3. A ray of light starts in glass and hits a boundary between the glass and air at just the right angle so that its angle of refraction is 90°. If the index of refraction of the glass is 2.0, what must the angle of incidence be? (This is called the “critical angle”.) What will happen if the angle of incidence is greater than this angle?
θi = ______
______
4. Find the critical angle for light in a material with (absolute) index of refraction 1.8 bounded by another material with (absolute) index of refraction 1.3.
θc = ______