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Opus in profectus

Refraction

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Practice

practice problem 1

Waves travel in all directions in the open ocean, but they always approach the land nearly perpendicular to the shore. Why does this happen?

solution

Ocean waves normally form when the wind grips the surface of the water and tries to drag it along. The friction at the interface gives the water a little tug and piles it up into a wave. Short burst of wind make little ripples and strong, steady winds make larger waves or swells. Regardless of size, waves generated by this means generally propagate in the direction that the wind is blowing. Since the wind can and does blow in every direction, waves can and do travel in any direction when they are formed.

Overhead cartoon representation of ocean waves approaching the shore

The speed of an ocean wave is affected by the depth of the water through which it is propagating. As the sea floor approaches the shore it rises and depth decreases. The shallower the water, the slower the wave speed. (The relationship is a complex one, but near shore speed is approximately proportional to the square root of depth.) Waves entering a medium with slower wave speed are refracted towards the normal. Where the sea floor rises suddenly, the refraction is abrupt. Where it rises gradually, the refraction is gradual. The closer a wave gets, the more perpendicular its propagation. The result is that most waves near the shore will eventually wind up heading nearly perpendicular to the shore no matter what direction they were traveling in initially.

practice problem 2

A ray of light is traveling from air to crown glass. The angle that this ray makes with the surface of the glass is 30°. Determine each of the following angles…
  1. the angle of incidence
  2. the angle of reflection
  3. the angle of refraction
  4. the angle between the reflected and refracted rays
  5. the angle between the incident and refracted rays
  6. the angle between the incident and reflected rays

solution

Start with a sketch. Let's make the surface horizontal, because we gotta pick something. Label one side air. Label the other side crown glass. Add an incident ray 30° above the surface. Add a normal to the surface at the point where the ray strikes the surface, since angles in geometric optics are measured from the normal. Sketch a reflected ray that looks symmetric to the incident ray, since it's obeying the law of reflection. Sketch in a transmitted ray that's refracted toward the normal, since the ray is entering a medium whe the speed of light is slower.

The set up

  1. The angle of incidence is measured from the normal, not the surface. Use the compliment of 30°

    90° − 30° = 60°

  2. The angle of reflection equals the angle of incidence. That's the law of reflection.

    60° = 60°

  3. Use Snell's law to compute the angle of refraction.

    n1 sin θ1 =  n2 sin θ2
    (1.00)(sin 60°) =  1.52 sin θ2
    θ2 =  35°

  4. Subtract the two angles on the right side of the normal from 180° to determine the angle between the reflected and refracted rays.

    180° − 60° − 35° = 85°

  5. Add the two angles adjacent to the right angle in the lower left corner to determine the angle between the incident and refracted rays.

    30° + 90° + 35° = 155°

  6. Double the angle of incidence to determine the angle between the incident and reflected rays.

    60° + 60° = 120°

The solution

practice problem 3

The diagram below shows a ray of light (A) incident upon an air-water interface. Several possible rays (B–K) are also shown.

Rays of light going into and out of a point on an interface with a protractor overlay

  1. Using the protractor in the diagram, measure the angle of incidence.
  2. Using the law of reflection, determine the angle of reflection, then select the lettered ray that best represents the reflected ray.
  3. Using Snell's law, determine the angle of refraction, then select the lettered ray that best represents the transmitted ray.

solution

  1. The angle of incidence is the angle between a line normal to the interface and the incident ray. Using the protractor supplied we get 53°. You can get this number by just counting off the tick marks, or by doing a little math. The incident ray is at 143° and the normal to the interface is at 90°.

    143° − 90° = 53°

  2. According to the law of reflection, the angle of reflection equals the angle of incidence. So once again the answer is 53°. Ray C best represents this direction.

  3. Snell's law is usually written like this.

    n1 sin θ1 = n2 sin θ2

    For this problem…

    n1 =  1.00 (air)
    θ1 =  53°
    n2 =  1.33 (water)
    θ2 =  ?

    Substituting these numbers…

    (1.00)(sin 53°) = 1.33 sin θ2

    gives us this answer…

    θ2 = 37°

    Ray G best represents this direction.

Rays of light going into and out of a point on an interface with a protractor overlay

practice problem 4

The earliest recorded refraction experiments with actual measurements was performed sometime in the 2nd century by the Greco-Egyptian mathematician-astronomer Claudius Ptolemy (85–165). Ptolemy could have deduced Snell's law of refraction from his data, but some of the needed mathematical concepts hadn't been invented yet (like the sine function). Use the values in the tables below and your favorite application and device to perform a linear regression analysis (a line of best fit) that can be used to determine the relative indexes of refraction at each interface. How well do these indexes compare to their modern day values? How well does Ptolemy's data illustrate Snell's law?
air-water
angle of
incidence (°)		 angle of
refraction (°)
10 08.0
20 15.5
30 22.5
40 29.0
50 35.0
60 40.5
70 45.5
80 50.0
air-glass
angle of
incidence (°)		 angle of
refraction (°)
10 07.0
20 13.5
30 19.5
40 25.0
50 30.0
60 34.5
70 38.5
80 42.0
water-glass
angle of
incidence (°)		 angle of
refraction (°)
10 09.5
20 18.5
30 27.0
40 35.0
50 42.5
60 49.5
70 56.0
80 62.0
Source: L'Ottica di Claudio Tolomeo da Eugenio Ammiraglio di Sicilia Scrittore del Secolo XII Ridotta in Latino sovra la traduzione Araba di un testo Greco imperfetto ora per la prima volta conforme a un codice della Biblioteca Ambrosiana per deliberazione della R. Accademia delle Scienze di Torino pubblicata da Gilberto Govi Socio della stessa Accademia (The Optics of Claudius Ptolemy from Admiral Eugene of Sicily, writer of the 12th century. Abridged in Latin from the Arabic translation of an imperfect Greek text. Now for the first time, conforming to a code of the Ambrosian Library by resolution of the Royal Academy of Sciences of Turin. Published by Gilberto Govi, member of the same Academy)

solution

Start with Snell's law…

n1 sin θ1 = n2 sin θ2

Rearrange it a bit so that it looks like the equation for a straight line…

sin θ1 =  n2  sin θ2  + 0  
n1  
y =  m  x  + b  
 

Now, if we set up three scatter plots so that…

sin θ2 , the angle of refraction, goes on the horizontal (x) axis and
sin θ1 , the angle of incidence, goes on the vertical (y) axis

Then the data should fit a set of straight lines with…

n2/n1 , the relative index of refraction, as the slope (m) and
0  as the y-intercept (b)

And that's what we get.

Scatter plot with line of best fit
Scatter plot with line of best fit
Scatter plot with line of best fit

The table below summarizes the statistical calculations.

Statistical summary
air-water air-glass water-glass
nptolemy 1.323 1.517 1.152
nmodern 1.333 1.523 1.143
Δ −0.72% −0.41% +0.79%
b −0.006 −0.006 −0.019
r2 0.998 0.998 0.999
  • The entries in the first row (nptolemy) are the relative indexes of refraction computed from the slope (m) of the line that best fits to the sines of Ptolemy's data.
  • The second row (nmodern) has the currently accepted values for these indexes. I assumed that Ptolemy's "glass" was crown glass. It's the most common form of glass now in the 21st century, and it probably was the most common form in the 2nd century as well. I also assumed that the index for air was 1.000 just to make life easier.
  • The third row (Δ) is the deviation between Ptolemy's values and modern values stated as a percent. These are all less than 1%, which means Ptolemy's values (had he computed them) are basically the same as modern values. I would be happy with results like these if I used 2nd century scientific equipment.
  • The coefficients of determination (r2) in the fourth row are all very close to 1, meaning that the sines of these data are very well described by a linear relationship.
  • The y-intercepts (b) in the fifth row are all very nearly zero, which means that the sines are directly proportional to one another.

In summary: These data seem to show an experiment that agrees with Snell's law.

Or do they? Let's look at the residuals — the difference between the observed value and the value that the model predicts for that observation or, in more visual terms, the distance from the data points and the line of best fit. For the relationship to be truly linear, there should be no pattern in those distances. The data should be randomly scattered in a band on either side of the line of best fit resulting in a null residual plot. In the case of Ptolemy's data, that isn't what we get. There is a definite pattern in the residuals, a pattern that is the same for all three data sets, a pattern that is almost perfectly pattern-like.

A plot of residual values with values tracing out a sideways 'S' shape
A plot of residual values with values tracing out a sideways 'S' shape
A plot of residual values with values tracing out a sideways 'S' shape

So what's going on here? The angles of incidence are separated by 10° between observations. Nothing surprising about that. That was his independent variable, the one he controlled. Now take a look at the angles of refraction. They are separated by a number that decrease by half a degree with each observation.

θ1(°) θ2(°) Δ1(°) Δ2(°)
10 08½    
20 15½ 7½  
30 22½ 7½ ½
40 29½ 6½ ½
50 35½ 6½ ½
60 40½ 5½ ½
70 45½ 5½ ½
80 50½ 4½ ½
θ1(°) θ2(°) Δ1(°) Δ2(°)
10 07½    
20 13½ 6½  
30 19½ 6½ ½
40 25½ 5½ ½
50 30½ 5½ ½
60 34½ 4½ ½
70 38½ 4½ ½
80 42½ 3½ ½
θ1(°) θ2(°) Δ1(°) Δ2(°)
10 09½    
20 18½ 9½  
30 27½ 8½ ½
40 35½ 8½ ½
50 42½ 7½ ½
60 49½ 7½ ½
70 56½ 6½ ½
80 62½ 6½ ½

The data are mathematically ideal. A scatter plot of the angles instead of their sines results in a parabola every time. A quadratic curve of best fit strikes each data point precisely. The residuals are all exactly 0. The coefficients of determination are all exactly 1. This shows that the data do not agree with Snell's law, but rather that they agree with some other kind of law.

A scatter plot that fits perfectly to a parabola
A scatter plot that fits perfectly to a parabola
A scatter plot that fits perfectly to a parabola

The fact that Ptolemy's data looks like it illustrates Snell's law probably means it was based on actual controlled observation and careful measurement. The fact that the data fits perfectly to a mathematical model that isn't Snell's law probably means that the carefully collected data were adjusted to fit that model. These data are both real and fake at the same time.