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 cliched 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 X (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 /, 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
Transverse waves produced by a water droplet penetrating the surface of a body of liquid.
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 = fk.
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 = 11 f. Furthermore, period is equal to the ratio of wavelength to wave speed; in other words, T = k/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 wave- length
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.
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 given distance. 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 k (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. 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 overpass? 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.
