# Sound Frequency, Wavelength and Octave

## An introduction to the nature of sound - frequency, wavelength and octaves

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Sound energy is transmitted through air (or other particles) as a traveling pressure wave. In air the displacement wave amplitude may range from *10 ^{-7} mm* to a few mm per second.

### Frequency

The frequency - cycles per second - of a sound is expressed in hertz - *Hz*. The frequency can be expressed as

f = 1 / T(1)

where

f= frequency (s^{-1}, 1/s, Hz)

T= time for completing one cycle (s)

#### Example - Frequency

The time for completing one cycle for a *500 Hz* tone can be calculated using *(1)* as

T= 1 / (500 Hz)

= 0.002 s

The frequency range for human hearing is *20* to *20.000 Hz*. By age the upper limit for many is reduced to *12-13.000 Hz*.

#### Example - Alternating Current

An alternating current completes *5 cycles* in *100 ms*.

The time for completing one cycle can be calculated

*T = (100 ms/cycle) / (5 cycles)*

* = 20 ms*

The frequency of the alternating current can be calculated as

*f = 1 / ((20 ms) (10 ^{-3} s/ms))*

* = 50 Hz*

### Wavelength

The wavelength of sound is the distance between the analogous points of two successive waves.

λ = c / f(2)

where

λ =wavelength (m)

c= speed of sound (m/s)

f= frequency (s^{-1}_{,}Hz)

#### Example - the Wavelength of a Tone

In air at normal atmosphere at *0 ^{ o}C* the speed of sound is

*331.2 m/s*. The wavelength of a

*500 Hz*tone can be calculated as

λ =(331.2 m/s)/(500 Hz)

= 0.662 m

### Octave

An octave is the interval between two points where the frequency at the second point is twice the frequency of the first.

Octave Center Frequency (Hz) | 31.25 | 62.5 | 125 | 250 | 500 | 1000 | 2000 | 4000 | 8000 | 16000 |

Wavelength in air (70^{o}F, 21^{o}C) (ft) | 17.92 | 9.03 | 4.52 | 2.26 | 1.129 | 0.56 | 0.28 | 0.14 | ||

Wavelength in air (70^{o}F, 21^{o}C) (m) | 5.46 | 2.75 | 1.38 | 0.69 | 0.34 | 0.17 | 0.085 | 0.043 |

Although the octave is the eighth note of a series in any key - an octave is separated by 5 whole notes and 2 half notes.

- only six whole notes separates an octave

#### Example - Guitar

A guitar's open E string ascends by whole notes as

- E F# G# A# C D E
- 1 2 3 4 5 6

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