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November 1997 Presentation Topic (continued)

# Problems

Question 1

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

Question 2

Let be a straight line. Find for t>0.

Question 3

Let S be the parabola defined by Sketch the graphs of for t>0. Figure 3 shows 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 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 has a singularity.

Question 5

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

Question 6

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

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