In this module we use exponential functions to study the way that colors change underwater. Our work begins at Fermat's Reef. Click here to open a new window with the Fermat's Reef simulation. Note that Fermat's Reef is a big world and you may need to enlarge the new window to see the whole thing. Arrange these two windows on your desktop so you can move easily back-and-forth between them. One way to do this is arrange them so that they overlap and part of each one is visible even when the other one is active. That way you can make the inactive window active by clicking on its exposed part. A link on the new Fermat's Reef page leads to instructions on how to use this simulation.
In this module we will use the fourth icon -- the white square. This icon represents a white square. We can use it to investigate how colors change underwater. Water absorbs different colors at different rates. Experiment by lowering the white square into the water to see what happens. Notice how its color changes as it gets deeper and deeper. As you lower the white square notice how the red, green, and blue components of its color change at different rates.
Lower the white square to a depth of ten feet. Press the Record Data Point button to record this reading. Notice that only 31 percent of the red light from the surface has reached this depth. More of the green light (88 percent) and still more of the blue light (96 per cent) has reached this depth. Now press the Clear Data button to clear the data recording window.
Now record a series of measurements at depths of 0, 1, 2, 3, ... 20 feet by moving the white square to a depth of zero feet and pressing the Record Data Point button and then repeatedly lowering the white square one foot at a time and pressing the same button. When you are done making the readings, highlight the data in the usual way by clicking and dragging the mouse. With the data highlighted, copy it by pressing CTRL-c (Windows) or COMMAND-c (Macintosh). It is a good idea to copy the data several times. For some reason the first time doesn't always work. Next open your spreadsheet. In the spreadsheet click the square in the upper left corner and press CTRL-v (Windows) or COMMAND-v (Macintosh) to paste your data into the spreadsheet. Now you can work in the spreadsheet with the data that you just collected.
Using your spreadsheet see if you can fit an exponential function to the data for the color red. Then try the same thing for the color green and then for the color blue. Does it make sense that an exponential function should describe how light is absorbed by water?
Now suppose that we are interested in black-and-white images rather than color images. You can see from a little experimentation at Fermat's Reef that by the time you reach a depth of 30 or 40 feet the colors are pretty washed out in any case. For black and white images we are interested in the average intensity of light of all colors. Using your spreadsheet compute the average light intensity at each depth. The first few rows of this spreadsheet are shown below.
Do you think that the average intensity can be described by an exponential function? Why or why not? Check your answer by trying to use your spreadsheet to fit an exponential function tom this data.
In other visits to Fermat's Reef you may have experimented with the process of correcting the colors for photographs taken near sunset or sunrise. We want to do the same thing now for photographs taken underwater. While boating off the coast of North Carolina a well-known mathematician and grandfather dropped the picture below into the water. He dove into the water immediately to try to save the picture but it was guarded by a school of fish, They did let him get close enough to the photograph to make the picture below. As you can see, the colors are distorted because it came to rest at a depth of eight feet[Footnote]. You can correct the color balance and save the picture by multiplying the red component, the green component, and the blue component of this picture by appropriate numbers. Fill in the appropriate numbers in the form below and press the Correct It!! button.
Footnote: More precisely the total distance from the surface of the water to the photograph and then to the camera was eight feet, so the light traveled through eight feet of water.