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SIMMER

November 1997 Presentation Topic (continued)

Problems

Question 1

Let S in R^2 be a smooth curve. Assuming uniqueness of the fronts, show that the wave fronts S_( +/- t) defined by S at time t>0 are obtained in the following way: For (x,y) in S, let +/- N(x,y) be the unit normal vectors to S at the point (x,y). Then

S_( +/- t)= { (x,y) +/- t N(x,y): (x,y) in S}.
Question 2

Let S in R^2 be a straight line. Find S_( +/- t) for t>0.

Question 3

Let S be the parabola defined by S = { (x,y) in R^2 : y=x^2 }. Sketch the graphs of S_( +/- t) for t>0. Figure 3 shows S_( +/- t) 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 R^2 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 S_t has a singularity.

Question 5

Find the set of points on the plane R^2 which is the union of the sets of singular points of S_( +/- t), t>0.

Question 6

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