MOST SHADOWS ARE EASILY understood, but those that are cast on the bottom of a pool of water sometimes have strange features. For example, when a leaf floats on water, its shadow is normal if the water is one or two centimeters deep, but in somewhat deeper water the shadow has a peculiar bright border. Equally unusual are the fleeting shadows left on the bottom of a shallow pool when an object is drawn through the water quickly and then removed: nothing opaque is left in the water, and yet dark disks with bright borders play on the bottom. A pencil can also produce an odd shadow. Stick the pencil partly into the water and then tilt it in various directions. For many orientations the shadow consists of two sausage-like shapes separated by a band of light.

The puzzling aspects of these shadows are due to the fact that light is refracted at the air-water boundary. That is, its direction of travel changes because its speed changes. In a vacuum light travels at 3 x 10⁸ meters per second, the ultimate speed in the universe. In air it travels slightly slower because it occasionally interacts with h air molecules along its path. In water, where it is slowed by interactions it with much more densely packed molecules, it travels at only three-fourths its speed in a vacuum.

Textbooks commonly treat light as being a wave with a series of straight wavefronts; the direction of travel is represented by a ray perpendicular to the wavefronts. When the light is refracted through a boundary, the initial ray is called the incident ray and the final ray is called the refracted ray. The orientation of the incident ray is measured with respect to a line, called the normal, that is perpendicular to the boundary. If the boundary is curved, the normal is perpendicular to a tangent to the boundary at the point where the ray crosses. The angle between the normal and the incident ray is called the angle of incidence.

If light travels from the air into a flat pool of water with the incident ray aligned with the normal, each wavefront slows uniformly as it crosses the boundary, and so the light's direction of travel does not change. For any other orientation of the incident ray, each wavefront crosses the boundary gradually. The part that first enters the water slows down before the rest of the wavefront does. The nonuniform reduction in speed produces a kink that moves along the wavefront and alters the wavefront's direction of travel: the light is refracted. The larger the angle of incidence is, the more the light is refracted.

When the water surface is flat, a bundle of parallel rays is refracted equally and the rays illuminate the bottom of the pool uniformly. When the water surface is curved, however, the refraction is not equal. The rays have different angles of incidence, because the normals are oriented differently where different rays cross into the water. The nonuniform refraction gives rise to nonuniform illumination on the bottom of the pool.

A concave surface diverges light rays, thereby decreasing the illumination in the region of the bottom where the rays would otherwise have gone. A convex surface converges light rays; the region where the rays cross, and where the light is therefore brighter, is called a caustic. Depending on the shape of the water surface, the caustic can be a point, a line or a three-dimensional "surface." If the bottom of the pool happens to intercept the caustic, you see a bright point or line on the bottom. If the bottom intercepts the rays either before or after they cross, it is illuminated somewhat, but not as brightly as it would be at the caustic.

Small waves provide curved surfaces that throw fleeting, complex patterns of light on the bottom of the pool, but the motion is too rapid to follow. A better way to study refraction through a curved water surface is to float a small object whose edges--will draw the water surface into a-concave or convex shape. Where an edge is slightly higher than the normal water level, water is pulled up to make a concave surface; where the edge is lower, water is pulled down to make a convex surface.

In 1983 Michael V. Berry and J. V. Hajnal of the University of Bristol described how the curved water surfaces surrounding a floating object influence the object's shadow. To repeat their experiments, partially fill a white container with water. Then gently lower a razor blade (not the kind that has a heavy reinforced edge) onto the water. If you are careful to keep the blade flat and not to break the surface, the blade will float. The water under the blade is depressed below the normal water level by the weight of the blade, and so the water surface along the edge of the blade is convex.

Illuminate the blade and water with an overhead lamp at least a meter away and examine the shadow cast by the blade on the bottom of the container. If the depth of the water is less than three centimeters or so, the shadow is normal: a dark rendition of the blade. If the water is deeper, the shadow has a bright border. Berry and Hajnal describe a way to change quickly from one type of shadow to the other. Add water to a depth of five centimeters and then put a sheet of white paper on the bottom of the container; the blade's shadow on the paper has a bright border. Raise the paper until it is one or two centimeters below the blade; the shadow becomes normal. I prefer to change the depth of the shadow by adding or removing water in the container. Pouring water into the container usually does not disturb the blade; I remove water by sucking it out through a drinking straw.

Along the curved water surface next to the blade's edge, rays are refracted so that they converge, but their point of focus depends on where they traverse the surface [see top illustration at right]. The surface curvature is greatest near the blade, and so rays entering the water there converge earlier than rays entering the water farther from the edge. Rays traversing the nearby flat surface do not converge at all.

The ray that enters the water right at the edge of the blade is known as the shadow ray; if there were no water, that ray would always mark the border of the shadow. Consider another ray that is slightly farther from the razor's edge than the shadow ray is. The curved water surface focuses that ray and the shadow ray to form a caustic at a certain depth below the normal water level. Call that depth the critical depth. (In the illustration the bottom of the container is shown at three depths, the middle one of which is the critical depth.)

Suppose you begin with about two centimeters of water in the container. The depth is less than the critical value (which in my experiments is about three centimeters), and so the bottom of the container intercepts the shadow ray and its neighboring ray before they cross. The blade's shadow is normal and the shadow ray marks its border. Slowly add water to the container while you monitor the shadow. As the water reaches the critical depth, the border of the shadow develops a caustic because it is at the focus of the shadow ray and its neighbor.

Add more water to the container. A caustic remains at the shadow's border, but it is not formed by the shadow ray and its neighbor. The caustic is now due to two closely spaced rays that pass through- the water surface somewhat farther from the blade's edge. Since the surface curvature is less there than it is at the edge, the rays focus at a greater depth than the shadow ray and its neighbor. The shadow ray no longer skims the border of the shadow but falls in the illuminated region on the bottom of the container somewhat away from the shadow. The width of the shadow is no longer set by the shadow ray, as in a normal shadow, but by the caustic that encroaches on the shadow. If you continue to add water, the location of the pair of rays responsible for the caustic at the border of the shadow gradually shifts away from the blade's edge. The distance to the bottom of the container is increasing; the caustic is at the convergence of rays that pass through the water surface where it is less curved.

The critical-depth value depends in theory on the weight of the blade. A heavier blade should depress the water more,

increasing the curvature of the water surface next to the blade. The shadow ray and its neighbor should cross earlier-at a shallower critical depth. In an experiment designed to check the theory I drained the water to a depth of 2.5 centimeters and floated a single blade as a control. The blade produced a normal shadow. Then I stacked three blades and lowered them onto the water. Although the intact stack floated for only a few seconds, I was able to study its evanescent shadow. The border of the shadow was bright. The weight of the stack depressed the water level more than the single blade did, decreasing the critical depth so that a caustic illuminated the border of the shadow. The h caustic was wider than the caustics I had seen with a single blade. Once k; water seeped into the stack, the lower blades sank, leaving the top blade floating. The shadow immediately became normal.

Berry and Hajnal pointed out that a variety of floating objects, such as leaves and insects, can create shadows with bright edges if the water is deep enough and if the objects depress the water surface. When floating objects elevate the water surface so that the surface is concave, the shadows have normal borders because the light rays diverge rather than forming a caustic. The shadows are also different in size. A floating razor blade produces a shadow that is larger than itself. If you lift the blade so that it makes the water surface concave, the shadow is smaller than the blade because the diverging rays pass under the blade, shrinking the shadow [see illustration above right].

Berry and Hajnal also showed how shadows with caustic borders can be produced by the curved sides of a vortex in water. They created the vortex by spinning a magnet placed under the container, which in turn rotated a bar magnet in the water, on the bottom of the container. When light from an overhead source passed through the vortex, refraction formed a wide bright ring that surrounded a dark interior. The inside and outside edges of the ring were caustics.

A few months ago M. H. Sterling, Michael A. Gorman, P. J. Widmann, S. C. Coffman and Robert M. Kiehn of the University of Houston and John A Strozier, Jr., of the State University Empire State College described how similar vortex shadows can be made without elaborate equipment. You can repeat their observations the next time you take a bath. After letting the water settle, briskly move an object horizontally through the top layer and then remove it. For a few seconds you will see dark circles, each with a bright border, playing on the bottom of the tub. The border is actually a narrow version of the bright ring seen by Berry and Hajnal. Try a variety of objects to find the one that best generates vortexes. Sterling and his co-workers used a circular paddle. As it moved, water circulated from the front of the paddle around the edge to fill in the space left at the back of the paddle. The visible swirls remaining on the water surface were connected by a vortex tube below the surface.

The group's investigation was initiated by an observation Kiehn made at an outdoor swimming pool. When he came out of the water, he left two dark disks on the bottom of the pool, each surrounded by a narrow bright ring. The shadows lasted for as long as 10 minutes. Kiehn reasoned that the shadows were caused by the refraction of sunlight through vortexes he had left in the water. But exactly what kind of vortex shape was responsible?

Members of the group considered two types of shape, one concave (parabolic) and the other convex (hyperbolic). If the vortex were entirely parabolic, the light rays would diverge; the region below the vortex might be dim enough to appear dark, but the divergence could not create a bright ring. If the vortex were entirely hyperbolic, the rays would focus to form a bright ring, but the water surfaces at the center of the vortex would meet at a sharp angle-a situation that is physically unlikely. The group found that the best model for the vortex is a blend of the two shapes: the vortex has a parabolic core that is surrounded by a hyperbolic surface.

The core creates the dark disk of the shadow. The bright ring that limits the size of the disk is due to closely spaced rays that pass through the hyperbolic surface and focus on the bottom. The rays that pass through the hyperbolic surface closer to the core converge too early to contribute to the ring and end up spread over the illuminated region outside the ring. The rays that pass through the surface farther from the core do not have a chance to converge and also end up outside the ring.

Which rays are responsible for the caustic depends on the depth of the water. If the vortex glides over water of decreasing depth, the responsibility for the caustic shifts to the rays closer to the core; the dark disk shrinks to some minimum size and then disappears if the water becomes even shallower. If the vortex moves over deepening water, the disk widens. The ring may grow wide enough so that you can distinguish its caustic edges.

I found I could make a similar shadow on the bottom of a container of water by blowing on the water through a straw. The resulting dimple on the surface usually casts a circular shadow with a bright ring, but when I blow almost horizontally, the shadow develops intriguing distortions.

The divergence of light rays by a concave water surface is responsible for the odd appearance of the shadow of a pencil inserted into water. The effect was reported in 1967 by Cyrus Adler of New York, who discovered it while idly playing with a pencil during a bath. He noticed a curious divided shadow of the pencil, cast on the bottom of the tub by light from the single lamp in the room. He called it "the shadow-sausage effect" and correctly explained the illuminated gap between the sausages as being caused by the concave water surface along the sides of the pencil.

Intrigued by Adler's observation, I toyed with a pencil in a container of water. To follow my play, fill a container with water to a depth of about six centimeters, place it directly below a light and insert an unsharpened pencil vertically into the water. When you first insert the pencil, you may see a large shadow with a bright edge, because the pencil initially depresses the water surface. Either wait until the water climbs the shaft or pull the pencil up slightly so that the water surface around the shaft is concave. Consider the rays that pass through the water surface on one side of the pencil. They diverge in various directions and some are blocked by the submerged segment of the pencil. The ray that just skirts the end of the pencil is the shadow ray. The rays on the opposite side of the pencil undergo similar divergence and also have a shadow ray.

When you insert the pencil several centimeters into the water, the two shadow rays do not cross before they reach the bottom of the container, you will see a shadow of the pencil that is narrower than the pencil. Gradually lift the pencil. The shadow narrows and then disappears, replaced by a h bright spot. The shadow rays now cross before they are intercepted by the bottom of the container. The bright spot is caused by light rays from opposite sides of the pencil that overlap when they reach the bottom of the container. Lift the pencil barely above the normal flat level of the water. If the liquid bridge clings to the end of the pencil, the bright spot is maintained. If you lift the pencil too far, the bridge fails and a normal shadow of the pencil immediately appears.

Next insert the pencil at an angle of 45 degrees and again either wait for h the water to climb the shaft or pull the pencil slightly upward. The water surface is again concave near the pencil, but now the extent of the curvature varies around the sides of the shaft [see illustration above]. It is greatest where the pencil forms a 45-degree angle with the water, smallest at the obtuse angle on the opposite side of the pencil and intermediate at intermediate points around the pencil.

Both the dry and the submerged parts of the pencil cast normal shadows, but the short segment of the pencil surrounded by a curved water surface does not. Consider the shadow rays that pass through the two regions of intermediate curvature on opposite sides of the pencil. They converge toward the area on the bottom that lies between the shadows cast by the dry and the submerged parts of the pencil. If the water is shallow enough, the shadow rays are intercepted by the bottom before they cross; you will see a narrow shadow connecting the two wider shadows. If the water is deeper, however, the shadow rays cross and eliminate the connecting shadow. You see an illuminated gap between the two shadows, as is shown in the illustration.

Adler noted that the gap had a complex structure of bright and dark regions; those near the shadows were brighter than the interior of the gap, which was grayish. I saw two dark lines radiating from the shadows' tips-as if "anticaustics" were being produced.

I poured a thick layer of corn oil over the water in my container and inserted a pencil through both liquids at a 45-degree angle. Now the pencil's shadow had three parts separated by illuminated gaps. One gap was from the refraction at the air-water interface, the other from the refraction at the water-oil interface.

You might investigate how other objects cast shadows on the bottom in shallow water. For example, a floating hair creates a string of different shadows. Some are dark and others have bright borders. Can you tell from the shadow which segments of a hair are fully submerged and which lie above the normal water level?

Bibliography

SHADOW-SAUSAGE EFFECT. Cyrus Adler in American Journal of Physics, Vol. 35, No. 8, pages 774-776; August, 1967.

THE SHADOWS OF FLOATING OBJECTS AND DISSIPATING VORTICES. M. V. Berry and J. V. Hajnal in Optica Acta, Vol. 30, No. 1, pages 23-40; January, 1983.

WHY ARE THESE DISKS DARK? THE OPTICS OF RANKINE VORTICES. M. H. Sterling, M. Gorman, P. J. Widmann, S. C Coffman, J. Strozier and R M. Kiehn in Physics of Fluids, Vol. 30, No. 11, pages 3624-3626; November, 1987.

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