Harmonic

Harmonic is an adjective and noun used across disciplines to describe phenomena related to harmony, proportional relationships, or periodic motion. It denotes regularity and patterns that repeat or balance in a predictable way.

In music, a harmonic refers to a component of a sound whose frequency is an integer multiple of a fundamental frequency. The collection of these overtones forms the harmonic series, shaping timbre and pitch relationships. Musical harmony also concerns how notes and chords sound together. Instruments naturally produce harmonics when strings, air columns, or resonant bodies vibrate.

In mathematics, harmonic has several precise meanings. A harmonic number H_n is the sum of reciprocals 1 + 1/2 + ... + 1/n. A harmonic function u satisfies Laplace's equation ∇^2 u = 0 in a domain, and harmonic conjugates pair with them in two dimensions to form analytic functions. The harmonic mean is a type of average defined as n divided by the sum of reciprocals, used for averaging rates.

In physics and engineering, simple harmonic motion describes motion with a restoring force proportional to displacement, producing sinusoidal oscillations such as x(t) = A cos(ω t + φ). The simple harmonic oscillator model appears in mechanical systems, springs, and electrical circuits.

In signal processing, harmonics are integer multiples of a fundamental frequency; Fourier analysis expresses complex signals as sums of harmonics, revealing frequency content and periodic structure.