STARS TWINKLE, MIRAGES APPEAR IN desert country and paved highways shimmer under the summer sun because air that rises above warm objects is less dense than the surrounding air. Such differences in density alter the refractive property of air and thereby bend light rays from their normal paths. The effect can be used to photograph normally invisible phenomena such as convection currents in air as well as the flow of air around speeding objects, including bullets and the wings of model airplanes. Recently Robert Walan, then a senior at the Carnegie Institute of Technology, built an apparatus for photographing air density variations that not only records convection currents and flow patterns but also represents zones of different pressure within the patterns in different colors.

"I built the apparatus in order to analyze the flow of air inside a miniature wind tunnel," Walan writes. "It is essentially a system for making what are called Schlieren photographs, but it differs from conventional Schlieren systems in that it records refraction effects in selected colors rather than in monochrome.

"In the conventional Schlieren system light from a small source is focused on a knife edge adjusted in such a way that the edge intercepts part of the rays. The unobstructed rays that proceed beyond the knife edge are focused by a second lens (or a parabolic mirror) to graze in turn a second knife edge parallel to the first. An opaque object placed in the cone of light between the first knife edge and the second lens, a region known as the test section, will appear in silhouette at the image plane of the lens beyond the second knife edge, where it can be projected on a viewing screen or a photographic film [see illustration at left below]

"When the air in the test section is uniformly dense, all parts of the screen are lighted equally except the portion shadowed by the

object; the intensity of the light that reaches the screen is determined by the amount of light intercepted by the knife edges. When a pressure gradient exists in the air of the test area, however, some of the light rays are bent. Some rays that would normally fall on the screen will be intercepted by the second knife edge; the screen will appear darker than normal in the region where these rays formerly impinged. Conversely, some rays that would normally be intercepted by the knife edge will be refracted away from the edge and will now reach the screen, adding to the background illumination and making the screen in certain areas brighter than average. As a result of both effects, variations of density in the test area appear on the screen as patterns of light and shade. If the temperature of the object in the test section differs from that of the surrounding air, convection currents form and their image can be seen rising from the silhouette.

"The system is attractive as an analytical device because its action depends solely on light. It has no moving parts and therefore is free of inertia effects, and there is no mechanical linkage to the test object-that might affect the free movement of air. The results are easy to interpret and can be expressed quantitatively by measuring the images with an inexpensive light meter. One problem, however, is that the system is extremely sensitive to the quality of the optical parts, since defects in the lenses or parabolic mirrors can also bend light rays. (Indeed, a similar apparatus, the Foucault knife edge, is used routinely by optical workers to check the quality of both lenses and mirrors.) Hence the apparatus must be equipped with lenses that are carefully corrected, particularly with respect to spherical and chromatic aberration. Small lenses of the required quality are comparatively inexpensive. The diameter of the test section is determined however, by the diameter of the second lens; most experiments require a test section at least three inches in diameter The cost of corrected lenses of this diameter was beyond me, so I substituted parabolic mirrors, which can be made inexpensively by the methods suggested in Amateur Telescope Making: Book One edited by Albert G. Ingalls (Scientific American, Inc., 1962).

"When a single parabolic mirror is substituted for the second lens in a simple Schlieren system, the arrangement introduces a particularly undesirable form of distortion known as coma. Light from the source must fall on the mirror at rather a large angle in relation to the optical axis or the image will be reflected to a position so close to the source that cannot be examined. Coma, which increases in proportion to the angle

reflection, makes a point source appears as a small comet shape. Fortunately Schlieren systems can be constructed in a variety of ways. In one popular arrangement light from the source falls on a parabolic mirror from one side and is reflected to a second parabolic mirror from which the rays proceed to a screen on the opposite side. Coma is canceled in the course of the double reflection from opposite sides. The light between the two mirrors is transmitted as parallel rays: a collimated beam that produces maximum detail in the image. The test section lies between the mirrors, and its length, which depends on the separation of the mirrors, can be made as long as one wishes. By adding two small condensing lenses and a direct-vision prism and substituting a pair of adjustable slits for the knife edges, I converted the arrangement into a color system, as illustrated by the accompanying schematic diagram [Figure 1].

"A line source of white light (such as the straight, narrow filament of an automobile spotlight bulb) is placed at the exact focus of a small achromatic condensing lens. The collimated rays that merge from the lens traverse the prism and are dispersed into their spectral colors. Then they are focused by a second condensing lens onto a slit adjusted to transmit a sharply defined beam of colored light. A parabolic mirror located exactly one focal length from the slit receives the converging rays and reflects them as a collimated beam to a second parabolic mirror, which in turn focuses the image of the source on a second slit assembly located on the opposite side of the apparatus, as illustrated. The second slit is adjusted to intercept all but one color of the beam. An objective lens beyond the second slit functions as a camera and projects an image of the test section on a screen or color film.

"When the density of the air in the test section (the region between the two parabolic mirrors) is uniform, the screen is uniformly illuminated and is monochromatic. When variations of density in local regions of the test section refract the rays, other colors, which would normally be intercepted by the edges of the slits, are transmitted to the screen, replacing the background color. The image then appears in polychrome, with higher densities depicted by a color nearer one end of the spectrum and low densities by a color on the opposite side.

"I usually adjust the second slit to pass the orange-yellow part of the spectrum. This creates a yellowish background when the air of the test section is of uniform density. The first slit is then adjusted to pass the red-orange-yellow-green segment of the spectrum. Compressions appear in red and expansions in green, as illustrated by the accompanying color photographs [see above]. The sensitivity of the system, which depends on the position and focal length of the mirrors and the width of the slits, is broad enough to accommodate disturbances set up in the test chamber by objects as different in temperature as a hot soldering iron and a finger.

"The optical system of a color Schlieren apparatus is not difficult to design. The experimenter arbitrarily selects the desired diameter of the test section. I decided on a width of three inches. This determines the diameter of the parabolic mirrors. To minimize the effects of optical aberrations such as coma and astigmatism (the tendency of mirrors to focus lines in one orientation more sharply than in others) a focal length must be selected that allows the entrance and exit cones of rays to make an angle of not more than five degrees in relation to the parallel rays of the test section. The focal length of my mirrors was 11 times the diameter of the mirrors, a focal ratio, or 'speed,' of f/ll. A higher focal ratio may be selected, but a lower one invites construction difficulties. (At f/11, incidentally, the curvature of a 'parabolic' mirror is essentially a section of a sphere a fairly easy surface for amateurs to grind and polish.)

"The geometry of the mirrors determines the minimum over-all length of the system, which is twice the focal length of the mirrors. The lens and slit components can be located at any desired angle with respect to the test section by deflecting the entrance and exit cones to the side by means of plane, front-surfaced mirrors. (The insertion of each additional surface in the optical path causes approximately a 5 per cent loss of light.) When a camera lens is not present, the second mirror forms a real image in its focal plane. When a camera lens is added, the proper location of the screen and film plane is determined experimentally.

"For maximum performance the system must be carefully aligned so that all components of the optical train, including the slits, are symmetrical in relation to the optical axis. The components must be rigidly mounted on vibrationless supports. I machined the mounting fixtures from brass and slotted the bases of most fittings for hold-down screws so that they can be shifted about half an inch in either direction along the optical axis. They can also be adjusted vertically. All components can be rotated vertically and moved transversely with respect to the light beam.

"The system is assembled in two units, one comprising the light source, direct-vision prism and associated condensing lenses, the first slit, a deflecting plane mirror and one parabolic mirror. The second unit includes the remaining parabolic mirror, a pair of deflecting mirrors, the second slit, the camera lens and screen. The test section is lengthened by simply sliding the units apart. The base supports of some units are mounted in brass guide plates that confine their transverse movement to one plane, a feature that simplifies the adjustment of the system. After aligning the system initially I made a full-scale drawing of the position of each component directly on the mounting board. The drawing saved a lot of time during subsequent experiments because components could be moved and quickly returned to approximately their proper positions. Final adjustments had to be made, of course, by trial and error.

"Ideally the source of light should be bright, white and in the form of a straight line to match the geometry of the slits. My source is an automobile spotlight bulb designed for operation at 12 volts. When exposing Kodacolor film, I operate the lamp on alternating current at twice its rated voltage. Power is supplied by a 110-volt line through a variable transformer. I made my slits of single-edge safety-razor blades, spraying the sides with white lacquer to improve their visibility.

"Although most of my experiments have been made in color, the apparatus can be converted for use as a conventional black-and-white system merely by removing the prism assembly. A direct-vision type of prism is preferred because of its optical properties, but any weak prism in crown glass may be substituted if some loss in performance can be tolerated. Such prisms are inexpensive and can be procured, along with condensing lenses and glass blanks for the parabolic mirrors, from the Edmund Scientific Co, in Barrington, N.J."

"The Foucault pendulum is a fascinating demonstration of the fact that the earth rotates in relation to the stars, but it is not a very good timekeeper. In principle the device is merely a weight suspended by a long wire and free to swing in a vertical plane at any orientation. If such a pendulum were to be set swinging directly over one of the earth's poles, the plane of the swing would appear to make a complete rotation in 24 hours with respect to the earth's surface. Since no force acts on the pendulum to alter the a direction of its swing, the apparent rotation of the plane of vibration can be explained only on the assumption that the earth rotates. Moreover, when the experiment is repeated at the earth's equator, the plane of vibration remains fixed in relation to the surface. At intermediate latitudes it has been shown that the rate at which the plane of vibration appears to rotate is equal to the time of the earth's rotation (24 hours a day) divided by the trigonometric sine of the latitude at which the experiment is conducted. In the case of short pendulums a correction factor must be applied: the f computed rate is multiplied by the factor (1 - 2A²/8L²), in which A is the amplitude of the swing and L the length of the pendulum, both quantities expressed in centimeters. At the latitude of New York City (40 degrees 45 minutes North) the ideal Foucault pendulum would appear to complete one revolution in 36 hours 50 minutes.

The performance of actual pendulums, however, usually comes only within about 15 per cent of this figure, as mentioned previously in this department [see "The Amateur Scientist", SCIENTIFIC AMERICAN, June, 1958]. None of the Foucault pendulums the editor of this department has examined (including the one on display at the Griffith Observatory in Los Angeles and the splendid installation that adorns the entrance hall of the United Nations General Assembly building in New York) betters the 15 per cent error. To B. B. Bingham of Athol, Mass., a deviation from clock time of this magnitude seemed unreasonably large. After some three years of experimentation he has reduced it substantially by constructing a miniaturized version of the Foucault pendulum. Bingham's apparatus keeps time with an average error of only 2 percent on most days and is considerably more accurate on some days.

The design is unusually attractive for home installation because the pendulum is less than seven feet long and is equipped with a transistorized drive. Once started, it runs indefinitely without attention.

The iron bob, cut from a piece of mild steel shafting and fully annealed, is four inches wide and two inches thick. The center is drilled and threaded to take a collar that makes a close fit with a pin vise. The pin vise grips the bottom end of a length of No. 24 music wire that serves as the pendulum arm. The upper end of the wire is threaded through a fixed washer, called the Charron-ring assembly, and a three-jawed hollow chuck. It then passes over a suspension pulley attached to the ceiling and fastens to the end of a threaded rod that passes through a fixed bracket and engages a thumb nut, as shown in the illustration in Figure 8. Adjusting the nut alters the length of the wire between the jaws of the chuck and the pin vise and thereby changes the effective length of the pendulum.

The length is initially adjusted so that the bob just grazes a magnet assembly located exactly below the pendulum at h the center of its swing. The magnet consists of a concentric yoke, also machined of mild steel and annealed. The yoke has the form of a circular pan 2 1/2 inches deep with an inner diameter of eight inches. A cylindrical pole two inches in diameter extends up from the center of the pan 1/8 inch higher than the edge of the pan [see illustration above]. The rectangular toroidal cavity thus formed between the center pole and the edge is fitted with a coil of closely wound, plastic-coated No. 14 copper magnet wire-the maximum number of turns the space will accommodate. The coil is served by a wrapping of plastic insulating tape to make a snug fit with the yoke. Leads from the coil are brought out through two small holes in the bottom of the assembly. The yoke is surrounded by a circular dial divided by hour graduations appropriate to the geographical latitude of the location-36 divisions in the case of Bingham's installation.

The Charron ring, made of nickel, serves both to suppress the pendulum's tendency to vibrate in elliptical or oval paths instead of the desired plane path and to actuate the electric driving circuit. The nickel ring, perhaps appropriately, should have the dimensions of a U.S. five-cent piece, with a 3/16-inch hole exactly in the center. (Do not make the part from a coin; it is illegal to deface U.S. currency.) The Charron ring is clamped, by three-adjustment screws spaced 120 degrees apart; to a circular plate three inches in diameter that is in turn fastened to a length of angle iron screwed to ceiling joists. A bracket made of two stud bolts and a second length of strap iron supports the three-jawed chuck.

The height of the chuck must be adjusted so that the tips of the jaws are exactly 2 and 7/16 inches above the lower face of the circular plate that clamps the Charron ring. The various assemblies are then aligned in such a way that the suspension wire hangs exactly in the middle of the chuck, before the jaws are clamped, and exactly in the center of the Charron ring. The center of the magnetic yoke must be placed exactly below the center of the bob when the pendulum is hanging free and motionless.

The electric circuit [see illustration above] can be energized by any miniature transformer capable of delivering eight volts at 10 watts, or more. One side of the secondary winding of the transformer is connected to the collector terminal of a 2N155 transistor (or its equivalent) and to the Charron ring. The three-jawed chuck connects to the base lead of the transistor and the coil assembly is placed in series with the emitter lead, a 50-ohm rheostat and the remaining lead of the secondary transformer winding. (Alligator clips should be used for making connections to the transistor unless the amateur is familiar with the technique of soldering to these devices. Transistors can be permanently damaged by excessive heat.)

When the installation is complete, adjust the length of the pendulum so that the bob clears the center pole of the yoke by about an eighth of an inch. Clamp the wire in this position by closing the jaws of the chuck. Then start the pendulum with a gentle push, just enough to bring the wire into contact with the Charron ring. This energizes the magnet, which attracts the bob. As the bob approaches the magnet the wire separates from the Charron ring and the magnetic field collapses. The bob now swings free to the opposite side, where the wire again makes contact with the Charron ring energizing the magnet to initiate the next half of the cycle.

At first the pendulum may follow an elliptical path or a figure eight, but within an hour it will settle into plane vibration. The plane will rotate in a clockwise direction. When the vibration has stabilized, observe its azimuth in relation to the hour circle and also make a note of the time as indicated by a reliable clock. Let the unit operate for 24 hours. If the pendulum runs slow, shorten the wire; if it is fast, lengthen the wire. The amplitude of the swing should be maintained at about 12 to 14 inches, just enough for the wire to make solid contact with the Charron ring. The amplitude can be increased or decreased by adjusting the rheostat in the emitter circuit of the transistor.

Bingham reports that the rate of his pendulum appears to deviate from clock time when the swing is parallel to the earth's magnetic field and to be synchronous with the clock when the swing is at a right angle to the magnetic field. The plane of the swing rotates faster than the clock as it proceeds from west to north and slower in the east-to-south segment. The maximum deviation never exceeds 4.5 per cent of true clock time, however-well under the 15 per cent error that is considered normal for Foucault pendulums.

Bibliography

FUNDAMENTALS OF OPTICS. Francis A. Jenkins and Harvey E. White. McGraw-Hill Book Company, 1967.

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