# Fundamental and Harmonics

The lowest resonant frequency of a vibrating object is called its fundamental frequency. Most vibrating objects have more than one resonant frequency and those used in musical instruments typically vibrate at harmonics of the fundamental. A harmonic is defined as an integer (whole number) multiple of the fundamental frequency. Vibrating strings, open cylindrical air columns, and conical air columns will vibrate at all harmonics of the fundamental. Cylinders with one end closed will vibrate with only odd harmonics of the fundamental. Vibrating membranes typically produce vibrations at harmonics, but also have some resonant frequencies which are not harmonics. It is for this class of vibrators that the term overtone becomes useful - they are said to have some non-harmonic overtones.

The nth harmonic = n x the fundamental frequency.

 Harmonics in cents
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# Sinusoidal Waves

A single- frequency traveling wave will take the form of a sine wave.

 Sine waves can be represented mathematically and it can be shown that any wave can be constructed from an appropriate combination of sine waves (Fourier synthesis) This transverse wave is typical of that caused by a small pebble dropped into a still pool.

The position of an object vibrating in simple harmonic motion will trace out a sine wave as a function of time. (Or if a mass on a spring is carried at constant speed across a room, it will trace out a sine wave.)

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Wave concepts

Resonance concepts

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