Waves on the ocean have wavelengths comparable to those of waves in seismic prospecting (15-500 meters), but ocean waves move slowly enough to be seen. Imagine a long harbor barrier parallel to the beach with a small entrance in the barrier for the passage of ships. This is shown in Figure 7.1.
Figure 1. Waves going through a gap in a barrier have semicircular wavefronts (if the wavelengt h is long compared to the gap size).
Linearity is a property of all low-amplitude waves (not those foamy, breaking waves near the shore). This means that two gaps in the harbor barrier make two semicircular wavefronts. Where the circles cross, the wave heights combine by simple linear addition. It is interesting to think of a barrier with many holes. In the limiting case of very many holes, the barrier disappears, being nothing but one gap alongside another. Semicircular wavefronts combine to make only the incident plane wave. Hyperbolas do the same. Figure 7.2 shows hyperbolas increasing in density from left to right.
Figure 2. A barrier with many holes (top). Waves, -space, seen beyond the barrier (bottom).
A Cartesian coordinate system has been superimposed on the ocean surface with going along the beach and measuring the distance from shore. For the analogy with reflection seismology, people are confined to the beach (the earth's surface) where they make measurements of wave height as a function of and . From this data they can make inferences about the existence of gaps in the barrier out in the -plane. The first frame of Figure 7.3 shows the arrival time at the beach of a wave from the ocean through a gap.
Figure 3. The left frame shows the hyperbolic wave arrival time seen at the beach. Frames to the right show arrivals at increasing distances out in the water. The -axis is compressed from Figure 7.1.
The waves are expanding circles.
An equation for a circle expanding with velocity about
a point is
If the original incident wave was a positive pulse, the Huygens secondary source must consist of both positive and negative polarities to enable the destructive interference of all but the plane wave. So the Huygens waveform has a phase shift. In the next section, mathematical expressions will be found for the Huygens secondary source. Another phenomenon, well known to boaters, is that the largest amplitude of the Huygens semicircle is in the direction pointing straight toward shore. The amplitude drops to zero for waves moving parallel to the shore. In optics this amplitude drop-off with angle is called the obliquity factor.