Larry J. Ruff
FOCUS ON RAYS
It's summer. You are sitting outside, enjoying the sights, sounds, and hot sun. You begin to wonder why the sun's rays seem to be focused on you. As you sit up straight and look around, the bright reflection off a nearby metal sculpture blinds you, the adjacent greenhouse glares at you, and you catch a flash from a high-flying jet. Now you're bothered enough to shift to a different location, but oddly enough you still seem to be at a focal point. Now you realize that you are really sweating, so you decide to go inside where you can more safely contemplate the dazzling effects of focusing. While cooling down, why not write some fun educational software or problem sets and send it to us at EduQuakes!
One of my favorite seismological programs is Seismic Waves, written by Alan Jones at SUNY Binghamton and made available for download through his Web site (http://www.geol.binghamton.edu/faculty/jones/jones.html). This program shows the wavefronts for various seismic phases as they propagate through the mantle and core, and you can see some of the caustics and other curious wavefront behaviors. Seismologists have always used ray tracing to study seismic phases and produce travel time curves, but it is not as common to produce wavefront tracing plots.
Most ray tracing is done with the "shooting method", where a certain number of rays with some angular spread shoot out from the source and end up back at the Earth's surface somewhere. We typically show just the final picture where all rays have been traced (e.g., the ray tracing plots shown for the PS/SP phases in the online EduPhase section of EduQuakes). To get the complete picture, we need to add time along the ray paths. For example, the final ray tracing may show that two ray paths cross and thus imply that there were a focal point and high wave amplitudes at that place, but what if there is a temporal mismatch as the waves went through that point at different times? For a sharp transient source, clearly there is a reduction in the wave amplitude. For a continuous wave source, such as the sun, the answer is more complicated, as the level of constructive and destructive interference depends on wave period and temporal mismatch. For a transient source, the best way to see the complete picture is to follow the wavefronts as they propagate along. This can be done in a single ray tracing picture by connecting iso-times across the ray paths to show the wavefront at a several values of travel time. Seismic Waves takes this one step further with an animation of the wavefront position with time.
The Earth provides a rich environment for the study of curious wave effects due to the presence of low-velocity zones and high-velocity zones, and simply from the spherical geometry. Even if our Earth had constant velocities within each major region, we would still get some interesting seismic phases due to the focusing/defocusing effects of spherical boundaries. For educational purposes, students can easily appreciate wave-focusing effects with straight ray paths interacting with curved boundaries--the wonderful world of optics. Indeed, if we focus on light waves rather than seismic waves, students have many practical examples around them: rain drops, a magnifying glass, camera lenses, microscopes, corner-cube reflectors, telescopes (or satellite dishes), and solar ovens.
To understand the light refraction effects through different materials such as air, water, and glass, students need just one rule about ray paths: Snell's Law. However, for the remainder of this column, I will keep things even simpler by focusing just on waves in air reflected off some surface. Then we need just the one basic rule extracted from Snell's Law that connects the angles of the incident and reflected rays. We can get interesting behavior when the reflecting surface is curved such that the local tangent plane to the surface changes its orientation. To show the focusing effect, we need some way to display the fact that wave amplitude increases as ray paths converge--plotting the ray paths together with the wavefronts gives a qualitative indication of this effect. Of course, one can "feel" this wave amplitude effect by conducting real experiments with the sun and an adjustable reflecting surface--but make sure that you have thick sunglasses and tough skin!
A safer way to conduct these experiments is to use the computer, which also offers the advantage of slowing down time so that we can follow the progress of a single wavefront. A good programming project for seismologists and enthusiastic students is to make a user-friendly interactive program for some basic problems in wavefront tracing. Recall that Seismic Waves already does the problem for waves propagating through the Earth. Thus my emphasis here is on the development of wavefront tracing through simple structures where the user can modify the structure. I have scanned the educational software listings but not found such a basic program. The graphics for the 3-D problem are quite a challenge, but the 2-D problem is quite simple. To get things going for this software project, I have scripted a very simple 2-D wavefront tracer available for download (via a link in the online EduQuakes column).
The main virtue of my contribution is that it is easy to use, since it is a HyperCard stack. It can be downloaded by a click from your Web browser onto a Macintosh, and it will run without HyperCard since it has been converted to a stand-alone application. The stack contains a few built-in shapes of the reflecting surface, but other scripters can add more modules if they have HyperCard on their machines (contact me for a download of the source stack). The built-in modules include the 2-D parabolic reflector (of course!) and a few other shapes.
Most people know that a parabolic reflector is the ideal shape for concentrating an incoming plane wave to a point (in 3-D). There are some good questions and problems for students that lie just beyond the parabolic reflector. What if you wanted to concentrate the incoming waves onto a plane surface, e.g., heating of a plate for a water heater--what is the best shape? Or what if you want to design a solar oven to cook a sausage, or a potato--what is the best shape? Recently I read an article where some of the new large reflecting telescopes are using mirrors that do not have the classic parabolic shape. One of the designs uses a mirror with a spherical shape. Can you think of the advantages and disadvantages of a spherical versus a parabolic shape? Finally, can a secondary mirror be used to correct for some of the deficiencies of a spherical mirror? Speaking of spherical mirrors, are there any well observed seismic phases that are reflected off the spherical mirror formed by the underside of the Earth's surface?
Teaching waves to students is a challenging endeavor, and most of us employ light waves and optics to help explain some aspects of wave propagation and focusing. This is one topic where software can also help to develop intuition about waves. I hope that this invitation will prompt you to come in out of the hot sun and focus on producing more and better wave-propagation educational software!
SRL encourages guest columnists to contribute to "EduQuakes." Please contact Larry Ruff with your ideas. His e-mail address is firstname.lastname@example.org.
Posted: 23 September 1999