Radio waves and mathematics

Summary: Radio waves have numerous applications and are described, analyzed, encoded, and “jammed” using mathematics.

Radio is a means of sending information by transmitting signals using radio waves, which are a type of electromagnetic radiation with frequencies in the spectrum of approximately 3 kilohertz (kHz) or 1000 cycles per second, to 300 gigahertz (GHz), or 1 billion cycles per second. These units are named for German experimental physicist Heinrich Hertz. Radio waves are used not only to carry radio and television signals but are also used in many other common technologies including wireless computer networks, wildlife tracking systems, cordless and cellular phones, baby monitors, and garage door openers. One interesting way that mathematics connects to radio is through mathematically based radio shows, like Math Medley, which was hosted by Patricia Kenschaft. Mathematicians have also spoken on programs like National Public Radio’s Science Friday.

Radio waves are sinusoidal, meaning that they are characterized by a smooth, repetitive oscillation whose function at time t can be described algebraically as

y(t) = (A) sin (ωt + φ)

where A is the wave’s amplitude (peak deviation), ω is the wave’s angular frequency (described in radians per second), and φ is the wave’s phase (where the wave cycle is at time t=0).

Brief History and Unique Properties

In 1864, the British physicist James Clerk Maxwell predicted the existence of radio waves as part of his theory of electromagnetism. Hertz confirmed Maxwell’s theory between 1886 and 1888 and is generally credited with being the first person to send and receive radio waves. Several individuals played an important role in developing a practical system of radio transmission including the Serbian-American engineer Nikola Tesla, who demonstrated wireless radio communication in 1893; the British physicist Oliver Lodge, who demonstrated the transmission of Morse Code using radio waves in 1894; and the Italian physicist Guglielmo Marconi, who in 1896 was granted the first patent for a radio. Radio communications between ships and coastal stations were in use by 1897, and the first radio time signal (used to synchronize clocks) was transmitted from a U.S. Naval Observatory clock in 1904.

Radio waves may be broadcast over long distances because of the Heaviside Layer (also called the “Kennelly–Heaviside layer”), a conducting layer in the ionosophere predicted independently in 1902 by the British mathematician and physicist Oliver Heaviside and the British physicist Arthur Edwin Kennelly. The existence of the Heaviside Layer was established in 1924 by the British physicist Edward Appleton, who also determined that the height of this reflective layer was about 100 kilometers (62 miles) above the Earth’s surface. The Heaviside Layer allows radio signals to follow the curvature of the Earth (rather than disappearing into space) because they are reflected by the Heaviside layer and thus “bounce back” to Earth.

Applications

Radio astronomy, which led to the discovery of objects such as pulsars and quasars, dates from the 1931 discovery by American physicist Karl Guthe Jansky of radio waves emitted from the Milky Way galaxy. American astronomer Grote Reber created the first radio frequency sky map in 1941, and in the 1950s, the British astronomers Martin Ryle and Antony Hewish produced two notable catalogues of celestial radio sources.

Historically, most radio broadcasts used one of two techniques for sending their signals: amplitude modulation (AM) or frequency modulation (FM). AM is the older technology (the first AM broadcast took place in 1906) and it was the dominant radio technology for most of the twentieth century. AM encodes information by modifying the amplitude of the transmitted signal. The technology for FM broadcasting, which encodes information by varying the frequency of the transmitted signal, was developed in the 1930s and became common by the late 1970s. The information in these analog signals is inherently part of the signal itself—the information influences the wave’s shape, and thus information loss can occur with any disruption of the signal. One example is the audible static that occurs when a radio receiver begins to travel beyond the range of a radio transmitter. In the twenty-first century, digital modulation has been increasingly used to minimize this problem. Digital modulation transfers digitized information using a broad spectrum of radio frequencies—far more than the AM or FM systems. Further, each signal is sent many times, reducing the chance of interference and signal loss because separate bits from many streams may be pieced together. Further, since the radio waveforms are not altered by the information, multiple signals may be carried at the same time in the form of one composite signal that is decoded by the receiver, a technique called “multiplexing.” Satellite radio systems take advantage of multiplexing and the wider angle of coverage to offer many hundreds of specialized channels across broad geographic areas. Television is also transitioning from analog to digital signals.

Radio transmissions are used for communication during wartime, but because a radio signal may be picked up by anyone with a receiver, various coding methods have been developed. One famous example is the code talkers used by the American Army during World War I and World War II. This program capitalized on the fact that Native-American languages such as Navajo and Choctaw were almost unknown outside those tribes and also developed a simple code for terms like “tank” and “submarine,” which allowed them to code and encode messages rapidly and with little risk of comprehension by the enemy. Also in World War II, the German Army used mechanical circuits to encrypt information. Although supposedly unbreakable because of the large number of combinations possible, the British mathematician William Tutte was able to deduce the pattern of the encoding machines after British intelligence intercepted two long coded messages, each of which was transmitted twice (the second time with corrected punctuation).

Interference

Radio waves can be blocked by weather formations, geographic features, and many other natural phenomena. Further, if several stations are broadcasting on a similar frequency, they may interfere with each other. Use of an antenna tuned to a particular frequency (so it will pick up the signal at the frequency more strongly than signals at other frequencies) and aimed at the source of the signal can improve reception. Radio signals can be deliberately jammed by broadcasting noise on the same frequency as the signal. For example, the Soviet Union regularly jammed broadcasts by Radio Free Europe and Voice of America.

To minimize unintentional interference, different parts of the radio spectrum are reserved for different uses and broadcast stations are assigned specific frequencies for their use. In the United States, AM radio uses frequencies from 535 to 1700 kHz, and FM radio uses frequencies between 88 megahertz (mHz) and 108 mHz. A radio station that identifies itself as “90.7 FM” is broadcasting at the frequency of 90.7 mHz, or 90,700,000 cycles per second (technically, 90.7 mHz is the station’s mean frequency). Other parts of the spectrum are reserved for other uses. For instance, 30–30.56 mHz is allocated for military air-to-ground and air-to-air communications systems for tactical and training operations and for land mobile radio communication in support of wildlife telemetry and natural resource management.

Bibliography

Regal, Brian. Radio: The Life Story of a Technology. Westport, CT: Greenwood Press, 2005.

Richards, John. Radio Wave Propagation: An Introduction for the Non-Specialist. New York: Springer, 2008.

Weightman, Gavin. Signor Marconi’s Magic Box: The Most Remarkable Invention of the 19th Century and the Amateur Inventor Whose Genius Sparked a Revolution. Cambridge, MA: Da Capo Press, 2003.