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November 1997 Presentation Topic (continued)


Question 1

Let  (IMAGE) be a smooth curve. Assuming uniqueness of the fronts, show that the wave fronts  (IMAGE) defined by S at time t>0 are obtained in the following way: For  (IMAGE) , let  (IMAGE) be the unit normal vectors to S at the point (x,y). Then


Question 2

Let  (IMAGE) be a straight line. Find  (IMAGE) for t>0.

Question 3

Let S be the parabola defined by  (IMAGE) Sketch the graphs of  (IMAGE) for t>0. Figure 3 shows  (IMAGE) for small values of t.

Question 4

Let S be the parabola defined in problem 3. Prove that the front that corresponds to the unit normal with negative y-coordinate is a smooth curve in  (IMAGE) for all times. Also prove that the front corresponding to the normal vector with positive y-coordinate is smooth for t small, but develops singularities for t large. Find the first value of t for which  (IMAGE) has a singularity.

Question 5

Find the set of points on the plane  (IMAGE) which is the union of the sets of singular points of  (IMAGE) t>0.

Question 6

After this discussion do you know why the cusp formed in the tea cup is brighter?


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