Concept
Everywhere in daily life, there are frequencies of sound and electromagnetic waves, constantly changing and creating the features of the visible and audible world familiar to everyone. Some aspects of frequency can only be perceived indirectly, yet people are conscious of them without even thinking about it: a favorite radio station, for instance, may have a frequency of 99.7 MHz, and fans of that station knows that every time they turn the FM dial to that position, the station's signal will be there. Of course, people cannot "hear" radio and television frequencies—part of the electromagnetic spectrum—but the evidence for them is everywhere. Similarly, people are not conscious, in any direct sense, of frequencies in sound and light—yet without differences in frequency, there could be no speech or music, nor would there be any variations of color.
How It Works
Harmonic Motion and Energy
In order to understand frequency, it is first necessary to comprehend two related varieties of movement: oscillation and wave motion. Both are examples of a broader category, periodic motion: movement that is repeated at regular intervals called periods. Oscillation and wave motion are also examples of harmonic motion, or the repeated movement of a particle about a position of equilibrium, or balance.
Kinetic and Potential Energy
In harmonic motion, and in some types of periodic motion, there is a continual conversion of energy from one form to another. On the one hand is potential energy, or the energy of an object due to its position and, hence, its potential for movement. On the other hand, there is kinetic energy, the energy of movement itself.
Potential-kinetic conversions take place constantly in daily life: any time an object is at a distance from a position of stable equilibrium, and some force (for instance, gravity) is capable of moving it to that position, it possesses potential energy. Once it begins to move toward that equilibrium position, it loses potential energy and gains kinetic energy. Likewise, a wave at its crest has potential energy, and gains kinetic energy as it moves toward its trough. Similarly, an oscillating object that is as far as possible from the stable-equilibrium position has enormous potential energy, which dissipates as it begins to move toward stable equilibrium.
Vibration
Though many examples of periodic and harmonic motion can be found in daily life, the terms themselves are certainly not part of everyday experience. On the other hand, everyone knows what "vibration" means: to move back and forth in place. Oscillation, discussed in more detail below, is simply a more scientific term for vibration; and while waves are not themselves merely vibrations, they involve—and may produce—vibrations. This, in fact, is how the human ear hears: by interpreting vibrations resulting from sound waves.
Indeed, the entire world is in a state of vibration, though people seldom perceive this movement—except, perhaps, in dramatic situations such as earthquakes, when the vibrations of plates beneath Earth's surface become too forceful to ignore. All matter vibrates at the molecular level, and every object possesses what is called a natural frequency, which depends on its size, shape, and composition. This explains how a singer can shatter a glass by hitting a certain note, which does not happen because the singer's voice has reached a particularly high pitch; rather, it is a matter of attaining the natural frequency of the glass. As a result, all the energy in the sound of the singer's voice is transferred to the glass, and it shatters.
Oscillation
Oscillation is a type of harmonic motion, typically periodic, in one or more dimensions. There are two basic types of oscillation: that of a swing or pendulum and that of a spring. In each case, an object is disturbed from a position of stable equilibrium, and, as a result, it continues to move back and forth around that stable equilibrium position. If a spring is pulled from stable equilibrium, it will generally oscillate along a straight path; a swing, on the other hand, will oscillate along an arc.
In oscillation, whether the oscillator be spring-like or swing-like, there is always a cycle in which the oscillating particle moves from a certain point in a certain direction, then reverses direction and returns to the original point. Usually a cycle is viewed as the movement from a position of stable equilibrium to one of maximum displacement, or the furthest possible point from stable equilibrium. Because stable equilibrium is directly in the middle of a cycle, there are two points of maximum displacement: on a swing, this occurs when the object is at its highest point on either side of the stable equilibrium position, and on a spring, maximum displacement occurs when the spring is either stretched or compressed as far as it will go.
Wave Motion
Wave motion is a type of harmonic motion that carries energy from one place to another without actually moving any matter. While oscillation involves the movement of "an object," whether it be a pendulum, a stretched rubber band, or some other type of matter, a wave may or may not involve matter. Example of a wave made out of matter—that is, a mechanical wave—is a wave on the ocean, or a sound wave, in which energy vibrates through a medium such as air. Even in the case of the mechanical wave, however, the matter does not experience any net displacement from its original position. (Water molecules do rotate as a result of wave motion, but they end up where they began.)
There are waves that do not follow regular, repeated patterns; however, within the context of frequency, our principal concern is with periodic waves, or waves that follow one another in regular succession. Examples of periodic waves include ocean waves, sound waves, and electromagnetic waves.
Periodic waves may be further divided into transverse and longitudinal waves. A transverse wave is the shape that most people imagine when they think of waves: a regular up-and-down pattern (called "sinusoidal" in mathematical terms) in which the vibration or motion is perpendicular to the direction the wave is moving.
A longitudinal wave is one in which the movement of vibration is in the same direction as the wave itself. Though these are a little harder to picture, longitudinal waves can be visualized as a series of concentric circles emanating from a single point. Sound waves are longitudinal: thus when someone speaks, waves of sound vibrations radiate out in all directions.
Amplitude
There are certain properties of waves, such as wavelength, or the distance between waves, that are not properties of oscillation. However, both types of motion can be described in terms of amplitude, period, and frequency. The first of these is not related to frequency in any mathematical sense; nonetheless, where sound waves are concerned, both amplitude and frequency play a significant role in what people hear.
Though waves and oscillators share the properties of amplitude, period, and frequency, the definitions of these differ slightly depending on whether one is discussing wave motion or oscillation. Amplitude, generally speaking, is the value of maximum displacement from an average value or position—or, in simpler terms, amplitude is "size." For an object experiencing oscillation, it is the value of the object's maximum displacement from a position of stable equilibrium during a single period. It is thus the "size" of the oscillation.
In the case of wave motion, amplitude is also the "size" of a wave, but the precise definition varies, depending on whether the wave in question is transverse or longitudinal. In the first instance, amplitude is the distance from either the crest or the trough to the average position between them. For a sound wave, which is longitudinal, amplitude is the maximum value of the pressure change between waves.
Period and Frequency
Unlike amplitude, period is directly related to frequency. For a transverse wave, a period is the amount of time required to complete one full cycle of the wave, from trough to crest and back to trough. In a longitudinal wave, a period is the interval between waves. With an oscillator, a period is the amount of time it takes to complete one cycle. The value of a period is usually expressed in seconds.
Frequency in oscillation is the number of cycles per second, and in wave motion, it is the number of waves that pass through a given point per second. These cycles per second are called Hertz (Hz) in honor of nineteenth-century German physicist Heinrich Rudolf Hertz (1857-1894), who greatly advanced understanding of electromagnetic wave behavior during his short career.
If something has a frequency of 100 Hz, this means that 100 waves are passing through a given point during the interval of one second, or that an oscillator is completing 100 cycles in a second. Higher frequencies are expressed in terms of kilohertz (kHz; 103 or 1,000 cycles per second); megahertz (MHz; 106 or 1 million cycles per second); and gigahertz (GHz; 109 or 1 billion cycles per second.).
A clear mathematical relationship exists between period, symbolized by T, and frequency (f): each is the inverse of the other. Hence,
If an object in harmonic motion has a frequency of 50 Hz, its period is 1/50 of a second (0.02 sec). Or, if it has a period of 1/20,000 of a second (0.00005 sec), that means it has a frequency of 20,000 Hz.
Real-Life Applications
Grandfather Clocks and Metronomes
One of the best-known varieties of pendulum (plural, pendula) is a grandfather clock. Its invention was an indirect result of experiments with pendula by Galileo Galilei (1564-1642), work that influenced Dutch physicist and astronomer Christiaan Huygens (1629-1695) in the creation of the mechanical pendulum clock—or grandfather clock, as it is commonly known.
The frequency of a pendulum, a swing-like oscillator, is the number of "swings" per minute. Its frequency is proportional to the square root of the downward acceleration due to gravity (32 ft or 9.8 m/sec2) divided by the length of the pendulum. This means that by adjusting the length of the pendulum on the clock, one can change its frequency: if the pendulum length is shortened, the clock will run faster, and if it is lengthened, the clock will run more slowly.
Another variety of pendulum, this one dating to the early nineteenth century, is a metronome, an instrument that registers the tempo or speed of music. Consisting of a pendulum attached to a sliding weight, with a fixed weight attached to the bottom end of the pendulum, a metronome includes a number scale indicating the frequency—that is, the number of oscillations per minute. By moving the upper weight, one can speed up or slow down the beat.
Harmonics
As noted earlier, the volume of any sound is related to the amplitude of the sound waves. Frequency, on the other hand, determines the pitch or tone. Though there is no direct correlation between intensity and frequency, in order for a person to hear a very low-frequency sound, it must be above a certain decibel level.
The range of audibility for the human ear is from 20 Hz to 20,000 Hz. The optimal range for hearing, however, is between 3,000 and 4,000 Hz. This places the piano, whose 88 keys range from 27 Hz to 4,186 Hz, well within the range of human audibility. Many animals have a much wider range: bats, whales, and dolphins can hear sounds at a frequency up to 150,000 Hz. But humans have something that few animals can appreciate: music, a realm in which frequency changes are essential.
Each note has its own frequency: middle C, for instance, is 264 Hz. But in order to produce what people understand as music—that is, pleasing combinations of notes—it is necessary to employ principles of harmonics, which express the relationships between notes. These mathematical relations between musical notes are among the most intriguing aspects of the connection between art and science.
It is no wonder, perhaps, that the great Greek mathematician Pythagoras (c. 580-500 B.C.) believed that there was something spiritual or mystical in the connection between mathematics and music. Pythagoras had no concept of frequency, of course, but he did recognize that there were certain numerical relationships between the lengths of strings, and that the production of harmonious music depended on these ratios.
Ratios of Frequency and Pleasing Tones
Middle C—located,, appropriately enough, in the middle of a piano keyboard—is the starting point of a basic musical scale. It is called the fundamental frequency, or the first harmonic. The second harmonic, one octave above middle C, has a frequency of 528 Hz, exactly twice that of the first harmonic; and the third harmonic (two octaves above middle C) has a frequency of 792 cycles, or three times that of middle C. So it goes, up the scale.
As it turns out, the groups of notes that people consider harmonious just happen to involve specific whole-number ratios. In one of those curious interrelations of music and math that would have delighted Pythagoras, the smaller the numbers involved in the ratios, the more pleasing the tone to the human psyche.
An example of a pleasing interval within an octave is a fifth, so named because it spans five notes that are a whole step apart. The C Major scale is easiest to comprehend in this regard, because it does not require reference to the "black keys," which are a half-step above or below the "white keys." Thus, the major fifth in the C-Major scale is C, D, E, F, G. It so happens that the ratio in frequency between middle C and G (396 Hz) is 2:3.
Less melodious, but still certainly tolerable, is an interval known as a third. Three steps up from middle C is E, with a frequency of 330 Hz, yielding a ratio involving higher numbers than that of a fifth—4:5. Again, the higher the numbers involved in the ratio, the less appealing the sound is to the human ear: the combination E-F, with a ratio of 15:16, sounds positively grating.
The Electromagnetic Spectrum
Everyone who has vision is aware of sunlight, but, in fact, the portion of the electromagnetic spectrum that people perceive is only a small part of it. The frequency range of visible light is from 4.3 · 1014 Hz to 7.5 · 1014 Hz—in other words, from 430 to 750 trillion Hertz. Two things should be obvious about these numbers: that both the range and the frequencies are extremely high. Yet, the values for visible light are small compared to the higher reaches of the spectrum, and the range is also comparatively small.
Each of the colors has a frequency, and the value grows higher from red to orange, and so on through yellow, green, blue, indigo, and violet. Beyond violet is ultraviolet light, which human eyes cannot see. At an even higher frequency are x rays, which occupy a broad band extending almost to 1020 Hz—in other words, 1 followed by 20 zeroes. Higher still is the very broad range of gamma rays, reaching to frequencies as high as 1025. The latter value is equal to 10 trillion trillion.
Obviously, these ultra-ultra high-frequency waves must be very small, and they are: the higher gamma rays have a wavelength of around 10−15 meters (0.000000000000001 m). For frequencies lower than those of visible light, the wavelengths get larger, but for a wide range of the electromagnetic spectrum, the wavelengths are still much too small to be seen, even if they were visible. Such is the case with infrared light, or the relatively lower-frequency millimeter waves.
Only at the low end of the spectrum, with frequencies below about 1010 Hz—still an incredibly large number—do wavelengths become the size of everyday objects. The center of the microwave range within the spectrum, for instance, has a wavelength of about 3.28 ft (1 m). At this end of the spectrum—which includes television and radar (both examples of microwaves), short-wave radio, and long-wave radio—there are numerous segments devoted to various types of communication.
Radio and Microwave Frequencies
The divisions of these sections of the electromagnetic spectrum are arbitrary and manmade, but in the United States—where they are administered by the Federal Communications Commission (FCC)—they have the force of law. When AM (amplitude modulation) radio first came into widespread use in the early 1920s—Congress assigned AM stations the frequency range that they now occupy: 535 kHz to 1.7 MHz.
A few decades after the establishment of the FCC in 1927, new forms of electronic communication came into being, and these too were assigned frequencies—sometimes in ways that were apparently haphazard. Today, television stations 2-6 are in the 54-88 MHz range, while stations 7-13 occupy the region from 174-220 MHz. In between is the 88 to 108 MHz band, assigned to FM radio. Likewise, short-wave radio (5.9 to 26.1 MHz) and citizens' band or CB radio (26.96 to 27.41 MHz) occupy positions between AM and FM.
In fact, there are a huge variety of frequency ranges accorded to all manner of other communication technologies. Garage-door openers and alarm systems have their place at around 40 MHz. Much, much higher than these—higher, in fact, than TV broadcasts—is the band allotted to deep-space radio communications: 2,290 to 2,300 MHz. Cell phones have their own realm, of course, as do cordless phones; but so too do radio controlled cars (75 MHz) and even baby monitors (49 MHz).
Where to Learn More
Beiser, Arthur. Physics, 5th ed. Reading, MA: Addison-Wesley, 1991.
Allocation of Radio Spectrum in the United States (Web site). <http://members.aol.com/jneuhaus/fccindex/spectrum.html> (April 25, 2001).
DiSpezio, Michael and Catherine Leary. Awesome Experiments in Light and Sound. New York: Sterling Juvenile, 2001.
Electromagnetic Spectrum (Web site). <http://www.jsc.mil/images/speccht.jpg> (April 25, 2001).
"How the Radio Spectrum Works." How Stuff Works (Web site). <http://www.howstuffworks.com/radio~spectrum.html> (April 25, 2001).
Internet Resources for Sound and Light (Web site). <http://electro.sau.edu/SLResources.html> (April 25, 2001).
"NIST Time and Frequency Division." NIST: National Institute of Standards and Technology (Web site). <http://www.boulder.nist.gov/timefreq/> (April 25, 2001).
Parker, Steve. Light and Sound. Austin, TX: Raintree Steck-Vaughn, 2000.
Physics Tutorial System: Sound Waves Modules (Web site). <http://csgrad.cs.vt.edu/~chin/chin_sound.html> (April 25, 2001).
"Radio Electronics Pages" ePanorama.net (Web site). <http://www.epanorama.net/radio.html> (April 25, 2001).
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