November 1997 Presentation Topic (continued)

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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