Harmonics
Harmonics are integral to understanding wave phenomena, particularly in music and physical systems. They refer to the overtones of a fundamental frequency, where each harmonic is a whole-number multiple of that base frequency. In musical contexts, a vibrating string, for example, produces a fundamental frequency and various harmonics, allowing for a rich array of sounds known as complex tones. A standing wave is formed when waves reflect within a medium, creating nodes (points of minimal amplitude) and antinodes (points of maximum amplitude). The interplay of these waves defines the harmonic series, which underpins musical scales and the construction of musical tones.
The fundamental frequency corresponds to the longest wavelength produced by the medium, while higher harmonics have shorter wavelengths and higher frequencies. This relationship is crucial in numerous applications, from musical instruments to engineering, as it impacts mechanical resonance—the amplification of vibrations when an external force matches the system's natural frequency. Overall, harmonics play a significant role in both natural phenomena and human-made systems, revealing the deep connections between sound, resonance, and physical behavior.
Harmonics
FIELDS OF STUDY: Harmonics; Acoustics
ABSTRACT: All objects that are capable of vibrating have one or more natural frequencies. When they resonate at those frequencies, the vibrations form standing wave patterns characterized by a series of nodes and antinodes. Physical systems that resonate at their natural frequencies produce music, but they can also experience destructive consequences.
PRINCIPAL TERMS
- antinode: a point of maximum amplitude in a standing wave.
- fundamental frequency: the lowest frequency of a resonating medium; also called the first harmonic.
- mechanical resonance: the increase in the amplitude of motion by a medium in response to a periodic applied force or a sympathetic vibration close to the fundamental frequency.
- node: a point of minimum amplitude (typically zero) in a standing wave.
- octave: a progression of eight harmonic tones.
- overtone: a frequency that is a whole-number multiple of the fundamental frequency of a resonating medium.
- standing wave: a wave pattern that maintains a static series of nodes and antinodes within a specific wavelength.
Waves and Frequencies in Harmonics
Many natural phenomena are characterized as waves by the regularity of their behavior. Electromagnetic and sound waves obey mathematical relationships relating frequency and wavelength. Physical matter can also move as waves, especially fluid matter such as air, oil, and water.
Waves are characterized by their frequency, wavelength, and amplitude. Sound waves are produced by the vibration of an object, such as the string of a musical instrument. All vibrating objects have certain frequencies at which they naturally vibrate. The lowest of these is called the object’s fundamental frequency. If a force is applied to such an object at a rate that is equal or very close to its natural frequency, the object will undergo mechanical resonance, which increases the amplitude of its vibrations. The frequencies at which resonance occurs are called resonance frequencies and are often the same as an object’s natural frequencies. Resonance frequencies higher than the fundamental frequency are called overtones. In many cases, a note that seems to vibrate at a single frequency in fact includes overtones of that frequency as well. Such sounds are called complex tones. Most if not all naturally produced sounds are complex tones. A harmonic is an overtone that is a whole-number multiple of the fundamental frequency.
A musical string resonates when it vibrates as a standing wave. A standing wave is a pattern that forms when a wave reaches a boundary of its medium, such as the end of a string, and is reflected back in such a way that the crest (highest point) of the reflected wave meets the trough (lowest point) of the incident wave, or vice versa. (Alternately, a half-crest may meet a half-trough, a quarter-crest may meet a quarter-trough, and so on.) The two waves combine to form a resulting wave pattern whose amplitude at any given point is the sum of the amplitudes of the individual waves at that point. This effect is called interference. Thus, where a crest meets a trough (or a half-crest a half-trough, etc.), the two amplitudes add up to zero and cancel each other out. The points in a standing wave that maintain a constant amplitude of zero are called nodes. The ends of a vibrating string are also nodes, because their position is fixed. Between each pair of nodes is an antinode. These are points at which the combined amplitude of the two waves is at its maximum. For a standing wave to form, the medium must be exactly half the length of the wavelength, or some multiple of that length.
Harmonic Series
The fundamental frequency of a musical string corresponds to the wavelength that is twice the length of the string. At this frequency, the wave completes only one-half of one wave cycle before it is reflected back. A string’s fundamental frequency, therefore, is the lowest frequency at which it can form a standing wave, and its harmonics are the higher frequencies at which it can form a standing wave. These harmonic frequencies are determined by the equation
fn = νn / 2l
where fn is the frequency of the nth harmonic, ν is the velocity of the wave, and l is the length of the string. The wavelength of a harmonic can similarly be calculated using the equation
λn = 2l / n
where λn is the wavelength of the nth harmonic.
The velocity of a wave is calculated using the following equation:
v = fnλn
If the same string is used and its tension remains constant, the wave’s velocity will remain constant as well, regardless of its frequency or wavelength. Thus, as the frequency increases, the wavelength will decrease, and vice versa. This inverse relationship holds true for any type of wave.
The various harmonics of a particular fundamental frequency form a harmonic series. Tones that are not part of this series are considered inharmonic and fall into the category of noise. The overtones in a complex tone may be either harmonic or inharmonic. Cymbals and gongs are examples of instruments that produce sounds containing inharmonic tones.
In music, tones are arranged in scales according to their respective frequencies. The interval between a note at a certain frequency and another note at double that frequency is called an octave.
Sample Problem
Guitars are normally tuned to the A440 pitch standard, meaning that the A note above middle C, designated A4, vibrates at a frequency of 440 hertz (Hz), or cycles per second (1/s). Using this standard, the fundamental frequency of the low E string is 82.41 Hz. Calculate the frequency and wavelength of the fourth harmonic of the low E string if its effective length is 0.648 meters (m).
Answer:
To calculate a harmonic frequency, the wave velocity must first be determined. The fundamental frequency (f1) of the string is given as 82.41 Hz. Although no wavelength is provided, the wavelength that corresponds to a string’s fundamental frequency (λ1) is equal to twice the length (l) of that string:
λ1 = 2l
λ1 = 2(0.648 m)
λf = 1.296 m
Use the fundamental frequency and wavelength to find the wave velocity:
v = f1λ1
v = (82.41 1/s)(1.296 m)
v = 106.80336 m/s
The harmonic frequency is calculated as
The problem asks for the fourth harmonic frequency, so n = 4. Plug in this and the other appropriate values, and solve:
To calculate the wavelength of the fourth harmonic, substitute in the appropriate values for the wavelength equation:
The frequency of the fourth harmonic is 329.64 hertz, and its wavelength is 0.324 meters.
Harmonic Relationships
Every physical system has a characteristic fundamental frequency at which it will resonate. Bridges have been known to collapse because marching soldiers or high winds caused them to vibrate too close to their natural frequencies, and the resulting increase in amplitude tore the structures apart. Fluid-flow dynamics are affected by harmonics, and resonance between water waves and wind can produce large and very destructive waves. In some situations, such as the movement of fluids through a series of pipes, harmonic vibrations affect the energy economy of the system. While music is the most familiar application of harmonics, all wave and wavelike phenomena obey the fundamental physical laws of harmonic vibrations. This is especially important in electrical and electromagnetic applications such as wireless communication.
Bibliography
Beament, James. How We Hear Music: The Relationship between Music and the Hearing Mechanism. Rochester: Boydell, 2001. Print.
Hartmann, William M. Principles of Musical Acoustics. New York: Springer, 2013. Print.
Prestini, Elena. The Evolution of Applied Harmonic Analysis. Boston: Birkhauser, 2004. Print.
Smith, Walter Fox. Waves and Oscillations: A Prelude to Quantum Mechanics. New York: Oxford UP, 2010. Print.
Thompson, Daniel M. Understanding Audio. Boston: Berklee, 2005. Print.
Walker, James S., and Gary W. Don. Mathematics and Music: Composition, Perception, and Performance. Boca Raton: CRC, 2013. Print.
Wolfe, Joe. "Strings, Standing Waves and Harmonics." Music Acoustics. U of New South Wales, n.d. Web. 25 Aug. 2015.