Huygens secondary point source

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 2.

 

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Figure 2 Waves going through a gap in a barrier have semicircular wavefronts (if the wavelength is long compared to the gap size).

A plane wave incident on the barrier from the open ocean will send a wave through the gap in the barrier. It is an observed fact that the wavefront in the harbor becomes a circle with the gap as its center. The difference between this beam of water waves and a light beam through a window is in the ratio of wavelength to hole 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 3 shows hyperbolas increasing in density from left to right.

 

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Figure 3 A barrier with many holes (top). Waves, (x , t)-space, seen beyond the barrier (bottom).

All those waves at nonvertical angles must somehow combine with one another to extinguish all evidence of anything but the plane wave.

A Cartesian coordinate system has been superimposed on the ocean surface with x going along the beach and z 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 x and t. From this data they can make inferences about the existence of gaps in the barrier out in the (x , z)-plane. Figure 4a shows the arrival time at the beach of a wave from the ocean through a gap.

 

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Figure 4 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 x-axis is compressed from Figure 2. (Gonzalez)

The earliest arrival occurs nearest the gap. What mathematical expression determines the shape of the arrival curve seen in the (x , t)-plane?

The waves are expanding circles. An equation for a circle expanding with velocity v about a point ( x₃ , z₃ ) is  

$${ ( x - x_3 ) }^2\ \ +\ \ {( z - z_3 )}^2\ \ =\ \ v^2 \, t^2$$ (1)

Considering t to be a constant, i.e. taking a snapshot, equation (1) is that of a circle. Considering z to be a constant, it is an equation in the (x , t)-plane for a hyperbola. Considered in the (t , x , z)-volume, equation (1) is that of a cone. Slices at various values of t show circles of various sizes. Slices of various values of z show various hyperbolas. Figure 4 shows four hyperbolas. The first is the observation made at the beach z₀ = 0. The second is a hypothetical set of observations at some distance z₁ out in the water. The third set of observations is at z₂, an even greater distance from the beach. The fourth set of observations is at z₃, nearly all the way out to the barrier, where the hyperbola has degenerated to a point. All these hyperbolas are from a family of hyperbolas, each with the same asymptote. The asymptote refers to a wave that turns nearly 90$^\circ$ at the gap and is found moving nearly parallel to the shore at the speed dx/dt of a water wave. (For this water wave analogy it is presumed--incorrectly--that the speed of water waves is a constant independent of water depth).

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.