Snell on the Beach

Carole (C) was sitting on the beach watching her little brother Ivan (I) play in the water. It was only knee-deep, and she was not worried about him at all. He always enjoyed clowning around, grimacing and gesticulating. When he suddenly cried "Help!" Carole just waved at him, but when his head disappeared under water and stayed there, she jumped up and ran to help. Fred (F) was impressed to see her veer toward Henry (H), almost stepping on Donna (D), instead of going straight for Ivan. But Carole was a certified life guard and knew that she would get there faster by this little detour, and Fred knew why this was so. Since she ran twice as fast on the sand as through the splashing water, she had learned that the cosine of the angle FDC should be twice a big as that of HDI. The general rule was that these cosines should be in the same ratio as the respective speeds. The instuctor had said that this was just a fact, called Snell's Law, and that you could not understand it unless you knew Calculus.

The attentive reader might wonder what Grant (G) and Judy (J) are doing in the picture, and the answer is: very little. For now, let us just say that Grant was ogling Judy through his binoculars. This whole story could be told entirely without them, except that Judy stubbornly insisted on being part of this excursion, and as she went so did Grant. We shall say more about them at the end of this tale.

Fred's real name was Fernando, and at school they called him "Ferd the nerd" because he liked mathematics. When Carole had told him about Snell's Law, his first reaction was to pull out a piece of paper and scribble. "It's easy, Carole", he said, "... but, hm, yes, I am using derivatives. There's got to be another way: that Dutchman Snell died in 1626 -- long before Newton was even born." She was used to that kind of stunt from him, but this much detail astonished her. "How come you remember that?" she wanted to know. "Ten years after Shakespeare", was his answer. The next day, he was back with two neat drawings, the first of which is shown here on the left. That was his style: only rarely did he use algebra with Carole because he knew it did not convince her. She had taken an "interdisciplinary" course, where she learned to speak about derivatives and integrals but not to work with them. Fred spent hours with her using reams of graph paper, until one day she exclaimed: "You mean, the steeper the curve, the faster it's moving away from the x-axis? And that steepness, is the derivative? Wow!"

Their conversation about the new diagram was rather private, so we'll paraphrase it. If you go back to the beach picture and imagine D moving from F to H, you'll easily see that CD would be increasing while DI would be decreasing. As you slide your pencil-tip from left to right along the horzontal line in the new drawing, its distance from the upper green area is increasing (like CD), while its distance from the lower gray area is decreasing (like DI). As this drawing is not to scale anyway, we might as well regard the lower distance as representing not DI itself but mDI, where m is a positive numerical factor reflecting the ratios of the speeds on sand and through water. Then CD+mDI would be repesented by the vertical distance between the green and gray regions, right?
That distance is minimal at the place where they would just barely touch if you slid them together vertically. "Yes," said Carole, " the red points would be kissing." "And the chaperone could separate them with one straight thrust of her cane", Fred added. "They'd share the same tangent line, you mean, "Carole sighed, "(you are so romantic, Fred) -- if you don't wish to slide them together, let us say, it's where their boundaries have the same steepness."

Back to the beach scene. As D' moves toward D, the triangle FD'C gradually morphs into FDC, and we wish to compare the growth of the hypotenuse (CD' to CD) with that of the side (FD' to FD). If we mark D" on CD so that CD" always equals CD' in length (i.e., the green triangle remains isosceles throughout), the changes in hypotenuse and side are represented by D"D and D'D, respectively. Now, as D' slides toward D (in synchronisation with its twin D"), the green base angles approach 90 degrees, and the quotient D"D / D'D approaches -- hold your breath -- the cosine of the included angle FDC.

So, now you have all the ingredients to roll your own proof of Snell's Law. But how Grant and Judy fit into this ? Well, it so happens that in issue Number 7 some graphics expert thought it would be pretty if the whole picture were inside a circle, with Donna in the centre and Ivan on the rim (where Judy is now). With CD = DI of equal length, CD+mDI was constant, and the argument was sunk. But who needs a theory when you have facts?

EXERCISE: If v and w are Carole's speed on land and water, respectively, what is the total time it takes her to reach Ivan? What is the meaning of the factor m ? How does it show up in the segment FG ?