November 1997 Presentation Topic (continued)

The particular question we are interested is the following: Given a wavefront,
or the shape of a wave,
at a certain time *T*, how do we predict the wavefront at later times *t*>*T*?
Huygens' principle states that every point on any wavefront may be considered
as a new source of waves, and all the points on any wavefront may be used
alternatively with the source to predict any later wavefront. For simplicity
we will only consider wave propagation in two dimensions. In this case we can
find the wavefront at a later time by drawing a circle of radius *t*-*T*
centered at each point of the wavefront at time *T*. The new fronts are the
curves which are tangent to the all these circles.

The simplest application of Huygens' principle is when the initial front is
just a
point. In this case a unique circular wave is formed from that point. This can
be seen, for example, when a stone is dropped on a water surface; see
figure 1. In this case only one wavefront is formed. The second easiest
example is the wavefront corresponding to a straight line. We can imagine the
waves that are formed when a rod is dropped on a water surface. Two linear
wavefronts are going to form, propagating away and in opposite directions,
from the rod. This behavior
is illustrated in figure 2. We also illustrate the wavefront evolution
in other cases in figures 3 and 4. These pictures
correspond to evolutions for
short times, i.e., when *t*-*T* is small. As we can imagine, the behavior of these
wavefronts for large times, except for the first and second ones, can be
rather complicated. Our goal is to discuss the behavior of the fronts in
figure 3 for large times.

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