Harmonic
Introduction
Harmonics are integral components of physics, mathematics, and music, representing a phenomenon that occurs when waves interact to create a complex pattern. The term 'harmonic' is derived from the Greek word 'harmonikos', meaning 'fitting together' or 'joining'. In the context of physics, harmonics are the frequencies that are integral multiples of the fundamental frequency. In mathematics, they are used in the study of Fourier series and other areas, while in music, they are responsible for the richness and complexity of sound.
Physics
In physics, a harmonic is a wave whose frequency is an integral (whole-number) multiple of the frequency of some reference wave (the fundamental). The fundamental frequency is the lowest frequency in a resonating system. For instance, in a vibrating string, the fundamental frequency is produced by the vibration of the whole string. Higher frequencies, called overtones, are created by the vibration of parts of the string. The first overtone is the second harmonic, the second overtone is the third harmonic, and so forth.
Harmonics are significant in various areas of physics, including acoustics, signal processing, and optics. In acoustics, harmonics contribute to the timbre of a sound. In signal processing, harmonics are often unwanted distortions. In optics, harmonics are used in lasers for frequency doubling or frequency tripling.
Acoustics
In acoustics, a harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency. For a wave with a frequency of f, the harmonics have frequencies 2f, 3f, 4f, etc. The harmonics have the property that they are all periodic at the fundamental frequency, therefore the sum of harmonics is also periodic at that frequency.
Harmonics in music are notes that are produced in a special way. They are integer multiples of the fundamental frequency, and they give the note its characteristic timbre. This is why a note played on a piano sounds different from the same note played on a violin.
Signal Processing
In signal processing, the harmonics of a signal are the frequencies in the signal that are whole multiples of a specified fundamental frequency. The fundamental frequency is the lowest frequency, and it is also the inverse of the period of the signal. Harmonics in signal processing are generally considered to be unwanted distortion or noise.
Optics
In optics, a harmonic is a signal at a frequency that is a whole number multiple of a fundamental frequency. The second harmonic is twice the fundamental frequency, the third harmonic is three times the fundamental frequency, and so on. Harmonics in optics are used in lasers for frequency doubling or frequency tripling.
Mathematics
In mathematics, particularly in Fourier analysis, a harmonic is a component of a periodic function. If the function is defined as a sum of sine and cosine functions with different frequencies, each of these components is a harmonic. The fundamental frequency is the lowest frequency, and the harmonics are the frequencies that are integer multiples of the fundamental frequency.
Harmonics are used in the study of Fourier series, wave equations, and other areas of mathematics. They are also used in signal processing and in the analysis of physical systems that exhibit periodic behavior.
Music
In music, a harmonic is a tone that is a whole number multiple of the fundamental frequency. The fundamental frequency is the lowest frequency, and it is the frequency of the note that is perceived as the pitch of the sound. The harmonics are the overtones of this fundamental frequency, and they contribute to the timbre of the sound.
The study of harmonics in music is part of the field of music theory. Musicians use harmonics to enrich the sound of their music and to create special effects. For example, guitarists can produce harmonics by lightly touching the string at certain points and then plucking the string.