An old professor of education had a monochromatic theme. It was his only theme and the topic of his every lecture. It was this:
All he ever did was lecture; he never assigned any problems.
Your problems all relate to the computer subroutine wavemovie() (Lynn, Gonzalez, JFC, Hale, Li, Karrenbach). It makes a movie of a sum of monochromatic waves. As it stands it will produce a movie (three-dimensional matrix) of waves propagating through a focus. The whole process from compilation through computation to finally viewing the film loop takes about a minute. A sample frame of the movie is in Figure 3.
Figure 3 First frame of movie generated by wavemovie().
It shows a snapshot of the (x,z)-plane. Collapsing spherical waves enter from the top, go through a focus and then expand again. Notice that the wavefield is small but not zero in the region of geometrical shadow. In the shadow region you see waves that appear to be circles emanating from point sources at the top corners. Notice that the amplitudes of expanding spherical waves drop off with distance and collapsing spherical waves grow towards the focus. We will study the program that made this figure and see many features of waves and mathematics.