WAVE MOTION

CONCEPT

Wave motion is activity that carries energy from one place to another without actually moving any matter. Studies of wave motion are most commonly associated with sound or radio transmissions, and, indeed, these are among the most common forms of wave activity experienced in daily life. Then, of course, there are waves on the ocean or the waves produced by an object falling into a pool of still water—two very visual examples of a phenomenon that takes place everywhere in the world around us.

HOW IT WORKS

Related Forms of Motion

In wave motion, energy—the ability to perform work, or to exert force over distance—is transmitted from one place to another without actually moving any matter along the wave. In some types of waves, such as those on the ocean, it might seem as though matter itself has been displaced; that is, it appears that the water has actually moved from its original position. In fact, this is not the case: molecules of water in an ocean wave move up and down, but they do not actually travel with the wave itself. Only the energy is moved.

A wave is an example of a larger class of regular, repeated, and/or back-and-forth types of motion. As with wave motion, these varieties of movement may or may not involve matter, but, in any case, the key component is not matter, but energy. Broadest among these is periodic motion, or motion that is repeated at regular intervals called periods. A period might be the amount of time that it takes an object orbiting another (as, for instance, a satellite going around Earth) to complete one cycle of orbit. With wave motion, a period is the amount of time required to complete one full cycle of the wave, from trough to crest and back to trough.

HARMONIC MOTION.

Harmonic motion is the repeated movement of a particle about a position of equilibrium, or balance. In harmonic motion—or, more specifically, simple harmonic motion—the object moves back and forth under the influence of a force directed toward the position of equilibrium, or the place where the object stops if it ceases to be in motion. A familiar example of harmonic motion, to anyone who has seen an old movie with a clichéd depiction of a hypnotist, is the back-and-forth movement of the hypnotist's watch, as he tries to control the mind of his patient.

One variety of harmonic motion is vibration, which wave motion resembles in some respects. Both wave motion and vibration are periodic, involving the regular repetition of a certain form of movement. In both, there is a continual conversion and reconversion between potential energy (the energy of an object due to its position, as for instance with a sled at the top of a hill) and kinetic energy (the energy of an object due to its motion, as with the sled when sliding down the hill.) The principal difference between vibration and wave motion is that, in the first instance, the energy remains in place, whereas waves actually transport energy from one place to another.

OSCILLATION.

Oscillation is a type of harmonic motion, typically periodic, in one or more dimensions. Suppose a spring is fixed in place to a ceiling, such that it hangs downward. At this point, the spring is in a position of equilibrium. Now, consider what happens if the spring is grasped at a certain point and lifted, then let go. It will, of course, fall downward with the force of gravity until it comes to a stop—but it will not stop at the earlier position of equilibrium. Instead, it will continue downward to a point of maximum tension, where it possesses maximum potential energy as well. Then, it will spring upward again, and as it moves, its kinetic energy increases, while potential energy decreases. At the high point of this period of oscillation, the spring will not be as high as it was before it was originally released, but it will be higher than the position of equilibrium.

Once it falls, the spring will again go lower than the position of equilibrium, but not as low as before—and so on. This is an example of oscillation. Now, imagine what happens if another spring is placed beside the first one, and they are connected by a rubber band. If just the first spring is disturbed, as before, the second spring will still move, because the energy created by the movement of the first spring will be transmitted to the second one via the rubber band. The same will happen if a row of springs, all side-by-side, are attached by multiple rubber bands, and the first spring is once again disturbed: the energy will pass through the rubber bands, from spring to spring, causing the entire row to oscillate. This is similar to what happens in the motion of a wave.

Types and Properties of Waves

There are some types of waves that do not follow regular, repeated patterns; these are discussed below, in the illustration concerning a string, in which a pulse is created and reflected. Of principal concern here, however, is the periodic wave, a series of wave motions, following one after the other in regular succession. Examples of periodic waves include waves on the ocean, sound waves, and electromagnetic waves. The last of these include visible light and radio, among others.

Electromagnetic waves involve only energy; on the other hand, a mechanical wave involves matter as well. Ocean waves are mechanical waves; so, too, are sound waves, as well as the waves produced by pulling a string. It is important to note, again, that the matter itself is not moved from place to place, though it may move in place without leaving its position. For example, water molecules in the crest of an ocean wave rotate in the same direction as the wave, while those in the trough of the wave rotate in a direction opposite to that of the wave, yet there is no net motion of the water: only energy is transmitted along the wave.

FIVE PROPERTIES OF WAVES.

There are three notable interrelated characteristics of periodic waves. One of these is wave speed, symbolized by v and typically calculated in meters per second. Another is wavelength, represented as λ (the Greek letter lambda), which is the distance between a crest and the adjacent crest, or a trough and the adjacent trough. The third is frequency, abbreviated as f, which is the number of waves passing through a given point during the interval of 1 second.

Frequency is measured in terms of cycles per second, or Hertz (Hz), named in honor of nineteenth-century German physicist Heinrich Rudolf Hertz (1857-1894). If a wave has a frequency of 100 Hz, this means that 100 waves are passing through a given point during the interval of 1 second. Higher frequencies are expressed in terms of kilohertz (kHz; 10³ or 1,000 cycles per second) or megahertz (MHz; 10⁶ or 1 million cycles per second.)

Frequency is clearly related to wave speed, and there is also a relationship—though it is not so immediately grasped—between wavelength and speed. Over the interval of 1 second, a given number of waves pass a certain point (frequency), and each wave occupies a certain distance (wavelength). Multiplied by one another, these two properties equal the speed of the wave. This can be stated as a formula: v = f λ.

Earlier, the term "period" was defined in terms of wave motion as the amount of time required to complete one full cycle of the wave. Period, symbolized by T, can be expressed in terms of frequency, and, thus, can also be related to the other two properties identified above. It is the inverse of frequency, meaning that T = 1/f. Furthermore, period is equal to the ratio of wavelength to wave speed; in other words, T = λ/v.

A fifth property of waves—one not mathematically related to wavelength, wave speed, frequency, or period, is amplitude. Amplitude can be defined as the maximum displacement of oscillating particles from their normal position. For an ocean wave, amplitude is the distance from either the crest or the trough to the level that the ocean would maintain if it were perfectly still.

WAVE SHAPES.

When most people think of waves, naturally, one of the first images that comes to mind is that of waves on the ocean. These are an example of a transverse wave, or one in which the vibration or motion is perpendicular to the direction the wave is moving. (Actually, ocean waves are simply perceived as transverse waves; in fact, as discussed below, their behavior is rather more complicated.) In a longitudinal wave, on the other hand, the movement of vibration is in the same direction as the wave itself.

Transverse waves are easier to visualize, particularly with regard to the aspects of wave motion—for example, frequency and amplitude—discussed above. Yet, longitudinal waves can be understood in terms of a common example. Sound waves, for instance, are longitudinal: thus, when a stereo is turned up to a high volume, the speakers vibrate in the same direction as the sound itself.

A longitudinal wave may be understood as a series of fluctuations in density. If one were to take a coiled spring (such as the toy known as the "Slinky") and release one end while holding the other, the motion of the springs would produce longitudinal waves. As these waves pass through the spring, they cause some portions of it to be compressed and others extended. The distance between each point of compression is the wavelength.

Now, to return to the qualified statement made above: that ocean waves are an example of transverse waves. We perceive them as transverse waves, but, in fact, they are also longitudinal. In fact, all types of waves on the surface of a liquid are a combination of longitudinal and transverse, and are known as surface waves. Thus, if one drops a stone into a body of still water, waves radiate outward (longitudinal), but these waves also have a component that is perpendicular to the surface of the water, meaning that they are also transverse.

REAL-LIFE APPLICATIONS

Pulses on a String

There is another variety of wave, though it is defined in terms of behavior rather than the direction of disturbance. (In terms of direction, it is simply a variety of transverse wave.) This is a standing wave, produced by causing vibrations on a string or other piece of material whose ends are fixed in place. Standing waves are really a series of pulses that travel down the string and are reflected back to the point of the original disturbance.

Suppose you hold a string in one hand, with the other end attached to a wall. If you give the string a shake, this causes a pulse—an isolated, non-periodic disturbance—to move down it. A pulse is a single wave, and the behavior of this lone wave helps us to understand what happens within the larger framework of wave motion. As with wave motion in general, the movement of the pulse involves both kinetic and potential energy. The tension of the string itself creates potential energy; then, as the movement of the pulse causes the string to oscillate upward and downward, this generates a certain amount of kinetic energy.

TENSION AND REFLECTION.

The speed of the pulse is a function of the string and its properties, not of the way that the pulse was originally delivered. The tighter the string, and the less its mass per unit of length, the faster the pulse travels down it. The greater the mass per unit of length, however, the greater the inertia resisting the movement of the pulse. Furthermore, the more loosely you hold the string, the less it will respond to the movement of the pulse.

In accordance with the third law of motion, there should be an equal and opposite reaction once the pulse comes into contact with the wall. Assuming that you are holding the string tightly, this reaction will be manifested in the form of an inverted wave, or one that is upside-down in relation to the original pulse. In this case, the tension on the end attached to the support is equal and opposite to the tension exerted by your hand. As a result, the pulse comes back in the same shape as before, but inverted.

If, on the other hand, you hold the other end of the string loosely; instead, once it reaches the wall, its kinetic energy will be converted into potential energy, which will cause the end of the string closest to the wall to move downward. This will result in sending back a pulse that is reversed in horizontal direction, but the same in vertical direction.

In both cases, the energy in the string is reflected backward to its source—that is, to the place from which the pulse was originally produced by the action of your hand. If, however, you hold the string so that its level of tension is exactly between perfect rigidity and perfect looseness, then the pulse will not be reflected. In other words, there will be no reflected wave.

TRANSMISSION AND REFLECTION.

If two strings are joined end-to-end, and a pulse is produced at one end, the pulse would, of course, be transmitted to the second string. If, however, the second string has a greater mass per unit of length than the first one, the result would be two pulses: a transmitted pulse moving in the "right" direction, and a reflected, inverted pulse, moving toward the original source of energy. If, on the other hand, the first string has a greater mass per unit of length than the second one, the reflected pulse would be erect (right side up), not inverted.

For simplicity's sake, this illustration has been presented in terms of a string attached to a wall, but, in fact, transmission and reflection occur in a number of varieties of wave motion—not just those involving pulses or standing waves. A striking example occurs when light hits an ordinary window. The majority of the light, of course, is transmitted through the window pane, but a portion is reflected. Thus, as one looks through the window, one also sees one's reflection.

Similarly, sound waves are reflected depending on the medium with which they are in contact. A canyon wall, for instance, will reflect a great deal of sound, and, thus, it is easy to produce an echo in such a situation. On the other hand, there are many instances in which the desire is to "absorb" sound by transmitting it to some other form of material. Thus, for example, the lobby of an upscale hotel will include a number of plants, as well as tapestries and various wall hangings. In addition to adding beauty, these provide a medium into which the sound of voices and other noises can be transmitted and, thus, absorbed.

Sound Waves

PRODUCTION.

The experience of sound involves production, or the generation of sound waves; transmission, or the movement of those waves from their source; and reception, the principal example of which is hearing. Sound itself is discussed in detail elsewhere. Of primary concern here is the transmission, and to a lesser extent, the production of sound waves.

In terms of production, sound waves are, as noted, longitudinal waves: changes in pressure, or alternations between condensation and rarefaction. Vibration is integral to the generation of sound. When the diaphragm of a loudspeaker pushes outward, it forces nearby air molecules closer together, creating a high-pressure region all around the loudspeaker. The loudspeaker's diaphragm is pushed backward in response, thus freeing up a volume of space for the air molecules. These, then, rush toward the diaphragm, creating a low-pressure region behind the high-pressure one. As a result, the loudspeaker sends out alternating waves of high pressure (condensation) and low pressure (rarefaction).

FREQUENCY AND WAVELENGTH.

As sound waves pass through a medium such as air, they create fluctuations between condensation and rarefaction. These result in pressure changes that cause the listener's eardrum to vibrate with the same frequency as the sound wave, a vibration that the ear's inner mechanisms translate and pass on to the brain. The range of audibility for the human ear is from 20 Hz to 20 kHz. The lowest note of the eighty-eight keys on a piano is 27 Hz and the highest 4.186 kHz. This places the middle and upper register of the piano well within the optimal range for audibility, which is between 3 and 4 kHz.

Sound travels at a speed of about 1,088 ft (331 m) per second through air at sea level, and the range of sound audible to human ears includes wavelengths as large as 11 ft (3.3 m) and as small as 1.3 in (3.3 cm). Unlike light waves, which are very small, the wavelengths of audible sound are comparable to the sizes of ordinary objects. This creates an interesting contrast between the behaviors of sound and light when confronted with an obstacle to their transmission.

It is fairly easy to block out light by simply holding up a hand in front of one's eyes. When this happens, the Sun casts a shadow on the other side of one's hand. The same action does not work with one's ears and the source of a sound, however, because the wavelengths of sound are large enough to go right past a relatively small object such as a hand. However, if one were to put up a tall, wide cement wall between oneself and the source of a sound—as is often done in areas where an interstate highway passes right by a residential community—the object would be sufficiently large to block out much of the sound.

Radio Waves

Radio waves, like visible light waves, are part of the electromagnetic spectrum. They are characterized by relatively long wavelengths and low frequencies—low, that is, in contrast to the much higher frequencies of both visible and invisible light waves. The frequency range of radio is between 10 KHz and about 2,000 MHz—in other words, from 10,000 Hz to as much as 2 billion Hz—an impressively wide range.

AM radio broadcasts are found between 0.6 and 1.6 MHz, and FM broadcasts between 88 and 108 MHz. Thus, FM is at a much, much higher frequency than AM, with the lowest frequency on the FM dial 55 times as great as the highest on the AM dial. There are other ranges of frequency assigned by the FCC (Federal Communications Commission) to other varieties of radio transmission: for instance, citizens' band (CB) radios are in a region between AM and FM, ranging from 26.985 MHz to 27.405 MHz.

Frequency does not indicate power. The power of a radio station is a function of the wattage available to its transmitter: hence, radio stations often promote themselves with announcements such as "operating with 100,000 watts of power…." Thus, an AM station, though it has a much lower frequency than an FM station, may possess more power, depending on the wattage of the transmitter. Indeed, as we shall see, it is precisely because of its high frequency that an FM station lacks the broadcast range of an AM station.

AMPLITUDE AND FREQUENCY MODULATIONS.

What is the difference between AM and FM? Or to put it another way, why is it that an AM station may be heard halfway across the country, yet its sound on a car radio fades out when the car goes under an over-pass? The difference relates to how the various radio signals are modulated.

A radio signal is simply a carrier: it may carry Morse code, or it may carry complex sounds, but in order to transmit voices and music, its signal must be modulated. This can be done, for instance, by varying the instantaneous amplitude of the radio wave, which is a function of the radio station's power. These variations in amplitude are called amplitude modulation, or AM, and this was the first type of commercial radio to appear. Developed in the period before World War I, AM made its debut as a popular phenomenon shortly after the war.

Ironically, FM (frequency modulation) was developed not long after AM, but it did not become commercially viable until well after World War II. As its name suggests, frequency modulation involves variation in the signal's frequency. The amplitude stays the same, and this—combined with the high frequency—produces a nice, even sound for FM radio.

But the high frequency also means that FM signals do not travel as far. If a person is listening to an FM station while moving away from the station's signal, eventually the station will be below the horizon relative to the car, and the car radio will no longer be able to receive the signal. In contrast to the direct, or line-of-sight, transmissions of FM stations, AM signals (with their longer wavelengths) are reflected off of layers in Earth's ionosphere. As a result, a nighttime signal from a "clear channel station" may be heard across much of the continental United States.

WHERE TO LEARN MORE

Ardley, Neil. Sound Waves to Music. New York: Gloucester Press, 1990.

Berger, Melvin and Gilda Berger. What Makes an Ocean Wave?: Questions and Answers About Oceans and Ocean Life. New York: Scholastic, 2001.

Catherall, Ed. Exploring Sound. Austin, TX: Steck-Vaughn Library, 1990.

Glover, David. Sound and Light. New York: Kingfisher Books, 1993.

"Longitudinal and Transverse Wave Motion" (Web site). <http://www.kettering.edu/~drussell/Demos/waves/wavemotion.html> (April 22, 2001).

"Multimedia Activities: Wave Motion." ExploreScience.com (Web site). <http://www.explorescience.com/activities/activity_list.cfm?categoryID=3> (April 22, 2001).

Ruchlis, Hyman. Bathtub Physics. Edited by Donald Barr; illustrated by Ray Skibinski. New York: Harcourt, Brace, and World, 1967.

"Wave Motion" (Web site). <http://www.media.uwe.ac.uk/masoud/projects/water/wave.html> (April 22, 2001).

"Wave Motion and Sound." The Physics Web (Web site). <http://www.hcrhs.hunterdon.k12.nj.us/disk2/Physics/wave.html> (April 22, 2001).

"Wave Motion Menu." Carson City-Crystal High School Physics and Chemistry Departments (Web site). <http://members.aol.com/cepeirce/b2.html> (April 22, 2001).

KEY TERMS

AMPLITUDE:

The maximum displacement of particles in oscillation from their normal position. For an ocean wave, amplitude is the distance from either the crest or the trough to the level that the ocean would maintain if it were perfectly still.

ENERGY:

The ability to perform work, which is the exertion of force over a givendistance. Work is the product of force and distance, where force and distance are exerted in the same direction.

FREQUENCY:

The number of waves passing through a given point during the interval of one second. The higher the frequency, the shorter the wavelength. Frequency can also be mathematically related to wave speed and period.

HARMONIC MOTION:

The repeated movement of a particle about a position of equilibrium, or balance.

HERTZ:

A unit for measuring frequency, equal to one cycle per second. If a sound wave has a frequency of 20,000 Hz, this means that 20,000 waves are passing through a given point during the interval of one second. Higher frequencies are expressed in terms of kilohertz (kHz; 10³ or 1,000 cycles per second) or megahertz (MHz; 10⁶ or 1 million cycles per second).

KINETIC ENERGY:

The energy that an object possesses due to its motion, as with a sled when sliding down a hill. This is contrasted with potential energy.

LONGITUDINAL WAVE:

A wave in which the movement of vibration is in the same direction as the wave itself. This is contrasted to a transverse wave.

MATTER:

Physical substance that has mass; occupies space; is composed of atoms; and is ultimately convertible to energy.

MECHANICAL WAVE:

A type of wave that involves matter. Ocean waves are mechanical waves; so, too, are the waves produced by pulling a string. The matter itself may move in place, but, as with all types of wave motion, there is no net movement of matter—only of energy.

OSCILLATION:

A type of harmonic motion, typically periodic, in one or more dimensions.

PERIOD:

For wave motion, a period is the amount of time required to complete one full cycle of the wave, from trough to crest and back to trough. Period can be mathematically related to frequency, wavelength, and wave speed.

PERIODIC MOTION:

Motion that is repeated at regular intervals. These intervals are known as periods.

PERIODIC WAVE:

A wave in which a uniform series of crests and troughs follow one after the other in regular succession. By contrast, the wave produced by applying a pulse to a stretched string does not follow regular, repeated patterns.

POTENTIAL ENERGY:

The energy that an object possesses due to its position, as for instance with a sled at the top of a hill. This is contrasted with kinetic energy.

PULSE:

An isolated, non-periodic disturbance that takes place in wave motion of a type other than that of a periodic wave.

STANDING WAVE:

A type of transverse wave produced by causing vibrations on a string or other piece of material whose ends are fixed in place.

SURFACE WAVE:

A wave that exhibits the behavior of both a transverse wave and a longitudinal wave.

TRANSVERSE WAVE:

A wave in which the vibration or motion is perpendicular to the direction in which the wave is moving. This is contrasted to a longitudinal wave.

WAVELENGTH:

The distance between a crest and the adjacent crest, or the trough and an adjacent trough, of a wave. Wavelength, abbreviated λ (the Greek letter lambda) is mathematically related to wave speed, period, and frequency.

WAVE MOTION:

Activity that carries energy from one place to another without actually moving any matter.

Wave motion

A wave is nothing more than a disturbance that moves from place to place in some medium, carrying energy with it. Since the behavior of waves is so closely related to the concept of oscillations , that is a good place to start.

There are many examples of simple oscillations, but a very good one is that of an object attached to the end of a spring. Assume that the other end is held fixed, perhaps by a clamp. Suppose the spring hangs vertically and slowly lowers the object until it becomes stationary. The spring is now stretched enough for its upward pull to balance the weight of the object, which at that location is in equilibrium. Now disturb the object by lifting it a short distance above that point and letting go. The object then begins to oscillate vertically as it first falls, until the spring stops it and pulls it back upward to the original position, then it falls again, etc. The energy in the oscillating motion came from the original disturbance, which in this case moved the object to a position above the equilibrium point. At that instance, the object was at its maximum displacement whose size is the amplitude. The larger the amplitude, the greater the energy in the motion.

If we hang a duplicate object and spring side-by-side with the first one, but without the two being in contact in any way. If we disturb the first object, it oscillates just as before and the second object remains stationary at its equilibrium position. However, the situation becomes different if we connect the two objects with a rubber band and then disturb only the first object. It begins to oscillate as before, but soon the second object starts to oscillate also. The rubber band allows energy to transfer to the second object, which will move with the same frequency as the first oscillation. We can make the experiment more complicated if we use a large number of springs hanging in a row, with each object connected to the one before and after it with rubber bands. If only the first object is disturbed, the oscillation will pass its energy through all the springs. After the energy has been transferred to the next few springs, the first spring will become still with its object back at its equilibrium point. This demonstrates exactly how a disturbance can move as a wave through a medium. In this example, the medium is composed of the objects on springs, which act as coupled oscillators.

The way that a disturbance is transferred from one part of a medium to another in the spring model is very similar to the way that waves move in water , air, or a guitar string. We can think of the water, perhaps in a bathtub, as being composed of a great number of H2O molecules lying very close side-by-side. If the water has been undisturbed for a long time , its surface will be calm and the molecules will be relatively stationary (of course, they are not completely stationary, but their motion is microscopic). Now suppose we disturb the water by tapping it with a finger. This produces a disturbance as many molecules are forced downward, the opposite of what happened in our spring example. We have all seen a wave move on the surface of the water away from the position of our tap, and this is simply the disturbance being transferred from one molecule to another. Just as before, the larger the original displacement, the greater the energy in the disturbance and the bigger the energy that the wave carries. You might ask how the transfer of energy takes place. Actually, the molecules of water all exert some force on their neighbors which are quite close. This holds them together and provides the same effect as the rubber bands.

The wave we just made is an example of a traveling wave since the disturbance moves from place to place in the medium. It can also be classified as a transverse wave because the direction of the disturbance (vertical in this case) is perpendicular to the direction that the wave travels (horizontally). There are also longitudinal waves,
like those which carry the energy of sound in air, in which the direction of the disturbance is the same as that of the wave motion. This is easy to visualize if you have ever seen a speaker move when the volume is turned up very loud. The sudden movements of the speaker compress the nearby air and that disturbance moves in the same direction toward you and your ears.

A repeated pattern of individual waves, a wave train, often occurs. A wave train can be produced in water by tapping the surface with a specific rhythm, or frequency. A complementary characteristic of a wave train is the wavelength. Suppose a friend taps the water with a specific frequency and you take a picture of the resulting waves. In effect, this lets you "freeze" the wave train in time and examine it. You will notice that there is a constant distance between the individual waves; this is the wavelength. The frequency tells how often the wave train repeats itself in time and the inverse of the wavelength tells how often the wave train repeats itself in space . Multiplying the numerical values of the frequency and the wavelength gives the speed at which the waves move.

Waves have many interesting properties. They can reflect from surfaces and refract, or change their direction, when they pass from one medium into another. If these properties seem familiar, that is because we are accustomed to light behaving in exactly this way. Obviously light reflects, and an example of refraction is the bending of light when it passes from water into air. For this reason, when you look into water, objects appear to be at different locations than their real positions. Light is therefore considered in these instances to behave as though it was a wave produced by moving disturbances of electric and magnetic fields.

Waves can also combine, or interfere. For example, two waves can reach a particular point at just the right time for both to disturb the medium in the same way (such as if two water waves both try to lift the surface at the same time). This is constructive interference . Likewise, destructive interference happens when the disturbances of different waves cancel. Interference can also lead to standing waves which appear to be stationary—the medium is still disturbed, but the disturbances are oscillating in place. This can only occur within confined regions, like a bathtub or a guitar string (fixed at both ends). For just the right wavelengths, traveling wave trains and their reflections off the boundaries can interfere to produce a wave that appears stationary.

See also Acoustics; Fluid dynamics.

Resources

books

Clark, J.O.E. Matter and Energy: Physics in Action. New York: Oxford University Press, 1994.

Ehrlich, R. Turning the World Inside Out, and 174 Other Simple Physics Demonstrations. Princeton, NJ: Princeton University Press, 1990.

Epstein, L.C. Thinking Physics: Practical Lessons in Critical Thinking, Second Edition. San Francisco: Insight Press, 1994.

Gough, W., et al. Vibrations and Waves. 2nd ed. Englewood Cliffs, NJ: Prentice Hall, 1995.

James J. Carroll

KEY TERMS

Frequency

—The number of cycles, or repetitions, of an oscillating motion which occur per second.

Medium

—A material in which a disturbance (oscillation) at one location can transfer its energy to nearby locations, for example from one molecule of water to another.

Wave train

—A repeated pattern of individual waves which are produced with a specific rhythm, or frequency.

Wave Motion

Resources

A wave is a disturbance that moves from place to place in a medium, carrying energy with it. The behavior of waves is closely related to the concept of oscillations, which will be discussed first.

A good example of an oscillation is that of an object attached to the end of a spring hanging from a fixed clamp. When the object is at rest, the spring is stretched so that its upward pull balances the weight of the object. This is the equilibrium position. If the object is lifted a short distance above the equilibrium position and released, it will fall until the force of the spring stops it and pulls it back upward through the equilibrium position and nearly to the point where it was lifted. Then it will fall again and spring upward again, etc. The energy in the oscillating motion came from the original disturbance, which in this case moved the object to a position above the equilibrium point. At that position, the object was at its maximum displacement. The distance between the equilibrium point and the maximum displacement is the amplitude. The larger the amplitude, the greater the energy in the motion.

Consider a duplicate and identical object-spring system hung side-by-side with the first one, but without the two being in contact in any way. If the first object is disturbed, it oscillates just as before and the second object remains stationary at its equilibrium position. However, the situation becomes different when the two objects are connected with a rubber band and then only the first object is disturbed. It begins to oscillate as before, but soon the second object starts to oscillate also. The rubber band allows energy to transfer to the second object, which will eventually move with the same frequency as the first oscillation. We can make the experiment more complicated if we use a large number of springs hanging in a row, with each object

connected to the one before and after it with rubber bands. If only the first object is disturbed, the oscillation will pass its energy through all the springs. After the energy has been transferred to the next few springs, the first spring will become still with its object back at its equilibrium point. This shows exactly how a disturbance can move as a wave through a medium. In this example, the medium is composed of the objects on springs, which act as coupled oscillators.

The way that a disturbance is transferred from one part of a medium to another in the spring model is very similar to the way that waves move in water, air, or a guitar string. Water is composed of a great number of H₂O molecules lying very close to each other. If the water has been undisturbed for a long time, its surface will be calm and the molecules will be relatively stationary (of course, they are not completely stationary, but their motion is microscopic). Now suppose the water is disturbed by an object impinging on it. This produces a disturbance as many molecules are forced downward. We have all seen a wave move on the surface of the water away from the position of our tap, and this is simply the disturbance being transferred from one molecule to another. The molecules of water all exert an electrical force on their neighbors, which are quite close. This holds them together and provides the same effect as the rubber bands in the example above. Just as before, the larger the original displacement, the greater the energy in the disturbance and the bigger the energy that the wave carries.

The water wave discussed above is an example of a traveling wave since the disturbance moves from place to place in the medium. It can also be classified as a transverse wave because the direction of the disturbance (vertical in this case) is perpendicular to the

KEY TERMS

Frequency —The number of cycles, or repetitions, of an oscillating motion which occur per second.

Medium —A material in which a disturbance (oscillation) at one location can transfer its energy to nearby locations, for example from one molecule of water to another.

Wave train —A repeated pattern of individual waves which are produced with a specific rhythm, or frequency.

direction that the wave travels (horizontally). There are also longitudinal waves, like those that carry the energy of sound in air, in which the direction of the disturbance is the same as that of the wave motion. This is easy to visualize if you have ever seen a speaker move when the volume is turned up very loud. The sudden movements of the speaker compress the nearby air and that disturbance moves in the same direction toward you and your ears.

A repeated pattern of individual waves, a wave train, often occurs in water bodies. A wave train is produced in water by tapping the surface with a specific rhythm, or frequency. A complementary characteristic of a wave train is the wavelength. Suppose the surface of the water is tapped with a specific frequency and a picture of the resulting waves is taken. In effect, this “freezes” the wave train in time. Examination of the picture will show that there is a constant distance between the individual waves; this is the wavelength. The frequency tells how often the wave train repeats itself in time and the inverse of the wavelength tells how often the wave train repeats itself in space. Multiplying the values of the frequency and the wavelength results in the speed at which the wave moves.

Waves have many interesting properties. They can reflect from surfaces and refract, or change their direction, when they pass from one medium into another. These properties are commonly observed in the behavior of light, which behaves as a wave in many everyday circumstances. Light reflects off of shiny surfaces such as mirrors and glass. An example of refraction is the bending of light when it passes from water into air. For this reason, when looking at objects underwater, they appear to be at different locations than their real positions.

Waves can also combine, or interfere. For example, if two waves reach a particular point so that they both disturb the medium in the same way the amplitude of the wave will increase additively. This is constructive interference. Likewise, destructive interference happens when the disturbances of different waves cancel. Interference can also lead to standing waves which appear to be stationary—the medium is still disturbed, but the disturbances are oscillating in place. This occurs within confined regions, like a bathtub or a guitar string (fixed at both ends). For just the right wavelengths, traveling wave trains and their reflections off the boundaries can interfere to produce a wave that appears stationary.

See also Acoustics; Fluid dynamics.

Resources

BOOKS

Halliday, David, Robert Resnick, and Jearl Walker. Fundamentals of Physics. Danvers, MA: John Wiley & Sons, 2005.

Pain, H. J. The Physics of Vibrations and Waves. West Sussex, England: John Wiley & Sons Ltd., 2005.

James J. Carroll

Wave motion

Wave motion is a disturbance that moves from place to place in some medium, carrying energy with it. Probably the most familiar example of wave motion is the action of water waves. A boat at rest on the ocean moves up and down as water waves pass beneath it. The waves appear to be moving toward the shore. But the water particles that make up the wave are actually moving in a vertical direction. The boat itself does not move toward the shore or, if it does, it's at a much slower rate than that of the water waves themselves.

The energy carried by a water wave is obvious to anyone who has watched a wave hit the shore. Even small waves have enough energy to move bits of sand. Much larger waves can, of course, tear apart the shore and wash away homes.

Types of wave motion

Two types of waves exist: transverse and longitudinal. A transverse wave is one that causes the particles of the surrounding medium to vibrate in a direction at right angles to the direction of the wave. A water wave is an example of a transverse wave. As water particles move up and down, the water wave itself appears to move to the right or left.

Words to Know

Amplitude: The maximum displacement (difference between an original position and a later position) of the material that is vibrating. Amplitude can be thought of visually as the highest and lowest points of a wave.

Condensation: A region of space with a higher-than-normal density.

Crest: The highest point reached by a wave.

Frequency: The number of wave crests (or wave troughs) that pass a given point per unit of time (usually per second).

Longitudinal wave: A wave that causes the particles of the surrounding medium to vibrate in the same direction as that in which the wave is moving.

Rarefaction: A region of space with a lower-than-normal density.

Transverse wave: A wave that causes the particles of the surrounding medium to vibrate in a direction at right angles to the direction of the wave motion.

Trough: The lowest point reached by a wave.

Wavelength: The distance between any two adjacent wave crests (wave crests that are next to each other) or any two adjacent wave troughs in a wave.

A longitudinal wave is one that causes the particles of the surrounding medium to vibrate in the same direction as that in which the wave is moving. A sound wave is an example of a longitudinal wave. A sound wave is produced when the pressure in a medium is suddenly increased or decreased. That pressure change causes pulses of rarefactions and condensations to spread out away from the source of the sound. A rarefaction is a region of space with a lower-than-normal density; a condensation is a region with a higher-than-normal density. The sound wave travels from one place to another as particles vibrate back and forth in the medium in the same direction as the sound wave.

Characteristics of a wave

Any wave can be fully characterized by describing three properties: wavelength, frequency, and amplitude. Like any wave, a water wave appears to move up and down in a regular pattern. The highest point reached by the wave is known as the wave crest; the lowest point reached is the wave trough (pronounced trawf).

The distance between any two adjacent (next to each other) wave crests or any two adjacent wave troughs is known as the wavelength of the wave. The wavelength is generally abbreviated with the Greek letter lambda, λ. The number of wave crests (or wave troughs) that pass a given point per unit of time (usually per second) is known as the frequency of the wave. Frequency is generally represented by the letter f. The highest point reached by a wave above its average height is known as the amplitude of the wave. The speed at which a wave moves is the product of its wavelength and its frequency, or, v = λ f.

Two kinds of waves most commonly encountered in science are sound waves and electromagnetic waves. Electromagnetic radiation includes a wide variety of kinds of energy, including visible light, ultraviolet light, infrared radiation, X rays, gamma rays, radar, microwaves, and radio waves. As different as these forms of energy appear to be, they are all alike in the way in which they are transmitted. They travel as transverse waves with the same velocity, about 3 × 10¹⁰ centimeters (1.2 × 10¹⁰ inches) per second, but with different wavelengths and frequencies.

Properties of waves

Waves have many interesting properties. They can reflect from surfaces and refract, or change their direction, when they pass from one medium into another. An example of reflection is the light we observe that bounces off an object, allowing us to see that object. An example of refraction is the apparent dislocation of objects when they are placed underwater.

Waves also can interfere, or combine, with each other. For example, two waves can reach a particular point at just the right time for both to disturb the medium in the same way. This effect is known as constructive interference. Similarly, destructive interference occurs when the disturbances of different waves cancel each other out. Interference can also lead to standing waves—waves that appear to be stationary. The medium is still disturbed, but the disturbances are oscillating in place. Standing waves can occur only within confined regions, such as in water in a bathtub or on a guitar string that is fixed at both ends.

[See also Acoustics; Fluid dynamics; Interference; Light ]

Wave motions

With regard to Earth science , wave motion describes the physical transmission of force or energy potential through a medium of transmission. The transmission disturbs the medium by displacing the medium. For example, water waves propagate through displacement (not linear movement) of water molecules; sound waves propagate via displacement of air molecules. Light also propagates via wave—but not in the same manner as water and sound. Light is transmitted via electromagnetic waves, the alternating of disturbances in electrical and magnetic fields.

A single equation is all that is needed to understand wave motion. The first attempt to mathematically describe wave motion was made by Jean Le Rond d'Alembert in 1747. His equation sought to explain the motion of vibrating strings. While d'Alembert's equation was correct, it was overly simplistic. In 1749, the wave equation was improved upon by Leonhard Euler; he began to apply d'Alembert's theories to all wave forms, not just strings. For more than seventy years the equations of Euler and d'Alembert were debated among the European scientific community, most of whom disagreed upon the universality of their mathematics.

In 1822, Jean-Baptiste-Joseph Fourier proved that an equation governing all waves could be derived using an infinite series of sines and cosines. The final equation was provided by John William Strutt (Lord Rayleigh) in 1877, and it is his law of wave motion that is used today. All waves have certain properties in common: they all transmit a change in energy state, whether it be mechanical, electromagnetic, or other; they all require some point of origin and energy source; and almost all move through some sort of medium (with the exception of electromagnetic waves, which travel most efficiently through a vacuum).

There are three physical characteristics that all wave forms have in common—wavelength, frequency, and velocity—and it is this common bond that allows the wave equation to apply to all wave types. In order to understand these physical characteristics, consider one of the most familiar wave forms, the water wave. As a wave passes through water, it forms high and low areas called, respectively, crests and troughs. The wavelength of the water wave is the minimum distance between two identical points, for example, the distance between two consecutive crests or two consecutive troughs. Imagine the water wave striking a barrier, such as a sea wall: the wave will splash against the wall, followed shortly by another, and so on. The amount of time between each splash (the rate at which the wave repeats itself) is the frequency of the wave. Generally, wavelength and frequency are inversely proportional: the higher the frequency, the shorter the wavelength. The final physical characteristic, velocity, is dependent upon the type of wave generated. A mechanical wave, such as our water wave, will move relatively slowly; a sound wave will move much faster (about 1,129 feet or 344 meters per second) while a light wave moves faster still (186,000 miles or 299,200 km per second in a vacuum). It is important to note that while a wave will move through a medium, it does not carry the medium with it. This is hard to picture in our water example, since it appears as if the water does move with the wave. A cork placed in the water moves up and down with the passing of the wave but returns essentially to the same location.

See also Electricity and magnetism; Quantum electrodynamics (QED); Quantum theory and mechanics; Relativity theory; Solar energy