WAVES OF ONE KIND OR ANOTHER are found at work everywhere in the universe, ranging from gamma rays of minute wavelength emitted by nuclear particles to the immense undulations in clouds of dust scattered thinly between the stars. Because waves of all kinds have in common the function of carrying energy, it is not surprising that all waves behave much alike. They move in straight lines and at constant velocities through uniform mediums and to some extent change direction and velocity at junctions where the physical properties of the mediums change. The part of a sound wave in air that strikes a hard object such as a brick wall, for example, bounces back to the source as an echo, an effect identical in principle with the image cast by a mirror and with the seismic disturbances that ricochet from layers of rock in the earth's interior. All are examples of wave reflection. Disjunctions in wave mediums account for the power transformers that hang from poles on city streets. Offhand they may seem to bear little resemblance to the bluish coating on camera lenses or to the megaphones used by cheerleaders. But the three devices have a function in common: helping waves to travel across junctions between mediums of differing characteristics without reflecting energy back to the source. All three are transformers.

Such similarities suggest the underlying simplicity and order that characterize nature. By learning how waves of one kind behave the experimenter learns what behavior to expect of others, and problems solved by the study of waves in one medium can be applied, with appropriate modification, to those in other mediums. Some acoustic properties of an auditorium, for instance, can be investigated by observing the action of waves in a shallow pan of water. When the pan is fitted with a glass bottom, illuminated by a point source of light and equipped with a motor-driven agitator to generate waves, it becomes a ripple tank, a fascinating apparatus that is widely used for investigating wave behavior of all kinds.

The pan of a simple ripple tank that can be made in the home consists of a picture frame about two inches thick and two feet square, closed at the bottom by a sheet of glass calked to hold water, as shown in Figure 1. The tank is supported about two feet above the floor by four sheetmetal legs. A source of light to cast shadows of ripples through the glass onto a screen below is provided by a 100-watt clear lamp with a straight filament. Because the lamp is suspended above the tank with the filament axis vertical, the end of the filament approximates a point source and casts sharp shadows. The lamp, partly enclosed by a fireproof cardboard housing, is suspended about two feet above the tank on a framework of dowels. The wave generator hangs on rubber bands from a second framework made of a pair of metal brackets notched at the upper end to receive a wooden crossbar. The distance between the wave generator and the water can be adjusted either by changing the angle of the metal brackets or by lifting the crossbar from the supporting notches and winding the rubber bands up or down as required. The agitator of the wave generator is a rectangular wooden rod. A wooden clothespin at its center grasps a 1 1/2-volt toy motor driven by a No. 6 dry-cell battery. Several glass or plastic beads are attached to the agitator by stiff wires, bent at right angles, that fit snugly into any of a series of holes spaced about two inches apart. Details of the wave generator are shown in Figure 2. Attached to the shaft of the motor is an eccentric weight, a 10-24 machine screw about an inch long. The shaft runs through a transverse hole drilled near the head of the screw, which is locked to the motor by a nut run tight against the shaft; another nut is run partly up the screw. The speed of the motor is adjusted by a simple rheostat: a helical spring of thin steel wire (approximately No. 26 gauge) and a small alligator clip. One end of the spring is attached to a battery terminal, and the alligator clip is made fast to one lead of the motor. The desired motor speed is selected by clipping the motor lead to the spring at various points determined experimentally. (A 15-ohm rheostat of the kind used in radio sets can be substituted for the spring-and-clip arrangement.)

The inner edges of the tank are lined with four lengths of aluminum fly screening three inches wide bent into a right angle along their length and covered with a single layer of cotton gauze bandage, either spiraled around the screening as shown in Figure 3 or draped as a strip over the top. The combination of gauze and screening absorbs the energy of ripples launched by the generator and so prevents reflection at the edges of the tank that would otherwise interfere with wave patterns of interest.

The assembled apparatus is placed in operation by leveling the tank and filling it with water to a depth of about 3/4 inch, turning on the lamp, clipping the motor lead to the steel spring and adjusting the height of the wave generator until the tip of one glass bead makes contact with the water. The rotation of the eccentric weight makes the rectangular bar oscillate and the bead bob up and down in the water. The height, or amplitude, of the resulting ripples can be adjusted by altering the position of the free nut on the machine screw. The wavelength, which is the distance between the crests of adjacent waves, can be altered by changing the speed of the motor. The amount of contrast between light and shadow in the wave patterns projected on the screen can be altered by rotating the lamp. The wave generator should be equipped with at least one pair of beads so that ripples can be launched from two point sources. Waves with straight fronts (analogues of plane waves that travel in mediums of three dimensions) are launched by turning the bead supports up and lowering the rectangular bar into the water.

As an introductory experiment, set up the generator to launch plane waves spaced about two inches from crest to crest. If the apparatus functions properly, the train of ripples will flow smoothly across the tank from the generator and disappear into the absorbing screen at the front edge. Adjust the lamp for maximum contrast. Then place a series of parafffin blocks (of the kind sold in grocery stores for sealing jelly), butted end to end, diagonally across the tank at an angle of about 45 degrees. Observe how the paraffin barrier reflects waves to one side, as in the upper illustration Figure 5. In particular, note that the angle made between the path of the incident waves and a line perpendicular to the barrier (_i) equals the angle made by the path of the reflected rays and the same perpendicular (_r). Set the barrier at other angles larger and smaller than 45 degrees with respect to the wave generator and also vary the wavelength and amplitude of the waves. It will be found that the angle of incidence equals the angle of reflection whatever the position of the barrier, a law of reflection that describes waves of all kinds.

Next replace the paraffin barrier with a slab of plate glass about six inches wide and a foot long and supported so that its top surface is about 1/2 inch above the tank floor. Adjust the water level until it is between 1/16 and 1/8 inch above the glass and launch a series of plane waves. Observe how the waves from the generator slow down when they cross the edge of the glass and encounter shallow water, as shown in Figure 6. As a result of the change in speed the waves travel in a new direction above the glass, just as a rank of soldiers might do if they marched off a dry pavement obliquely into a muddy field. In this experiment waves have been diverted from their initial direction by refraction, an effect observed in waves of all kinds when they cross obliquely from one medium to another which they travel at a different velocity. Water waves are unique in that they travel at different speeds when the thickness, or depth, of the medium changes. To a very good approximation the ratio of wave velocity in shallow and deep water is proportional to the ratio of the of the water. This ratio is in effect the "index of refraction" of the two "mediums." In the case of electromagnetic waves (such as light) or mechanical waves (such as sound) the velocity of wave propagation varies with the density of the mediums. Light waves travel at 186,200 miles per second in a vacuum but somewhat slower in air and much slower in dense materials, such as flint glass. A dense medium shaped in the form of a lens will focus or disperse light rays, radio waves and even mechanical vibrations.

Note in the illustration that the edge of the glass facing the generator reflects waves of low amplitude toward the side. Some energy is reflected by waves of all kinds when they encounter an abrupt difference in the refractive index of mediums. The net reflection at the disjunction between the deep and shallow water can be minimized by beveling the edge of the glass (or any other smooth, solid material substituted for glass) as shown in Figure 4. Reflection still occurs. But reflections cast by the front part of the wave oppose those set up by following parts of the wave and cancellation takes place. The beveled edge thus acts as a transformer that in effect matches the impedance of the two mediums. Similarly, the optical density of the bluish coating on camera lenses is made intermediate between that of air and glass to minimize the loss of light by partial reflection, much as the tapered megaphone minimizes the reflection back into a cheerleader's throat from the acoustic disjunction between his lips and open air.

Wave energy can also be focused, dispersed and otherwise distributed as desired by barriers of appropriate shape, as exemplified by the parabolic reflectors used in telescopes, searchlights, radars and even orchestra

shells. The effect can be demonstrated in two dimensions by the ripple tank. Make a barrier of paraffin blocks or rubber hose in the shape of a parabola and direct plane waves toward it. At every point along the barrier the angle made by the incident waves and the perpendicular to the parabola is such that the reflected wave travels to a common point: the focus of the parabola. Conversely, a circular wave that originates at the focus reflects as a plane wave from the parabolic barrier, as shown in the upper illustration on the left. In this experiment the wave was generated by a drop of water.

Waves from two or more sources travel through a uniform medium independently of one another, although the wave energy at any point in the medium is at each instant the algebraic sum of all waves impinging on that point Where the crests of waves from two sources coincide, the amplitudes add, and where the crest of one wave coincides with the trough of another of equal amplitude, they cancel. The effect is called wave interference. Two reflecting surfaces can be spaced in such a way that light waves of all colors except one cancel Such optical interference accounts for the rainbow hues of soap bubbles, which change with variations in the thickness of the soap film.

Interference effects can be demonstrated in the ripple tank by adjusting a pair of beads so that they make contact with the water about two inches apart. A typical interference pattern made by two beads vibrating in step with each other is shown in the lower illustration in Figure 8. Observe that maximum amplitude occurs along paths where the wave crests coincide and that nodes appear along paths where crests coincide with troughs. The angles at which maxima and nodes occur can be calculated easily. The trigonometric sine of the angles for maxima, for example, is equal to n/d, where n is the order of the maximum (the central maximum, extending as a perpendicular to the line joining the source, is the "zeroth" order, and the curving maxima extending radially on each side are numbered "first," "second," "third" and so on consecutively), A is the wavelength in inches and d is the distance between sources in inches. Similarly, minima lie along angular paths given by the equation sin= (m - H) /d, where m is the order of the minima and the other terms are as previously defined. In the illustration the ratio /d was approximately .39. Wave interference finds application in the determination of standards of length by means of such instruments as the interferometer, in receiving antennas that favor television signals from a desired direction and in many other fields.

Barriers need not be solid to reflect waves. A two-dimensional lattice of uniformly spaced pegs arranged as in the illustration in Figure 9 will reflect waves in the ripple tank that bear a required geometrical relation to the lattice. When a train of plane waves impinges obliquely against the lattice, circular waves are scattered by each peg and interfere to produce a coherent train of plane waves. The maximum amplitude of this train makes an angle with respect to the rows making up the lattice such that sin max = n/2d, where sin max designates the direction of maximum wave amplitude, n the order, A the wavelength and d the spacing between adjacent rows of pegs (the lattice spacing). This equation, known as Bragg's law in honor of its British discoverers, the father-and-son team of Sir William Bragg and Sir Lawrence Bragg, has been widely applied in computing the lattice structure of crystal solids from photographs of wave maxima made by the reflection of X-ray waves from crystals. X-ray waves are reflected from the three-dimensional lattice of atoms constituting crystals in the same way and for the same reason that water waves reflect from the two-dimensional array of brass pegs in the ripple tank. The phenomenon is seen in the series of three illustrations in Figures 10 - 12. In the top illustration, which shows strong reflection, the ratio of /d is 1.36 and n = 1. After this picture was made the ripple generator was speeded up to launch waves of shorter length. The middle illustration was then made at the /d ratio of .96. Substantially no reflection was recorded. The wavelength was then made still shorter for a /d ratio of .74, and relatively strong reflection was recorded in the second order, as seen in the bottom illustration.

Another of the many aspects of wave behavior that can be investigated with the ripple tank is the Doppler effect, first studied intensively by the Austrian physicist Christian Johann Doppler. He recognized the similarity in wave behavior that explains the apparent increase in pitch of an onrushing train whistle and the slight shift toward the blue end of the spectrum in the color of a star speeding toward the earth.

Both effects are observed because it is possible for moving wave sources to overtake and in some cases to outrun their own wave disturbances. To demonstrate the Doppler effect in the ripple tank, substitute for the agitator bar a small tube that directs evenly timed puffs of air from a solenoid-actuated bellows against the surface of the water while simultaneously moving across the tank at a controlled and uniform speed. (A few lengths of track from a toy train can be mounted along the edge of the tank and a puffer can be improvised on a toy car.)

When the puffer moves across the tank at a speed slower than that of the waves, crests in front of the puffer crowd closely together, whereas those behind spread apart, as shown in the upper illustration in Figure 13. If the puffer is imagined to be the whistle of an approaching train, an observer stationed at point A in the photograph would hear more sound waves per second than an observer at B and consequently would hear a higher pitch. Similarly, an earthbound observer at A would describe the color of an approaching star (represented by the puffer) as containing more blue than would an observer at B simultaneously looking at the receding star. The Doppler effect is observed in waves of all kinds, including radio signals. By means of relatively simple apparatus the effect can be applied to determine the direction and velocity of an artificial satellite from its radio signals [see "The Amateur Scientist, SCIENTIFIC AMERICAN, January, 1958].

In the first Doppler-effect illustration the puffer was moving about a third as fast as the ripples. When the speed of the puffer across the tank exceeds that of the ripples, shock waves appear, as shown in the lower illustration on the right. During this experiment the source was moving across the stationary medium at a speed about 1.6 times the wave speed, or at Mach 1.6. The source is clearly outdistancing its own waves, just as a speedboat outdistances its bow waves. If the medium were air and the puffer a supersonic aircraft, an observe at point A would be outside the "Mac cone" and would hear nothing. A observer at point B, however, would hear a violent shock wave and one at would hear a continuous rumble.

These experiments merely suggest the many wave phenomena that can b demonstrated by the ripple tank. Th apparatus is not capable of duplicating all forms of wave behavior by analog because water waves arise from the elliptical motion of water molecules; sound waves, on the other hand, are propagated by molecules of gas oscillating in the direction of the advancing wave front, and electromagnetic waves are propagated by transverse changes in the strength of electric and magnetic fields, somewhat as a kink travels on a stretched clothesline when one end is snapped sideways. Nevertheless, anyone who builds and operates a ripple tank will find it appropriate for enough fascinating experiments to occupy many rainy afternoons.

Bibliography

THE SCIENTIFIC AMERICAN BOOK OF PROJECTS FOR THE AMATEUR SCIENTIST. C. L. Stong. Simon and Schuster, 1960.

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