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/sec 2 ) 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.
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.
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.
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 · 10 14 Hz to 7.5 · 10 14 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 10 20 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 10 25 . 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 10 10 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.
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).
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