The wavelength to frequency and frequency to wavelength calculations are important in physics and engineering. Here is the equation relating wavelength and frequency, example calculations, and a table of common values.

### Relationship Between Wavelength and Frequency

A simple equation relates wavelength to frequency:

v = λf

- v = wave velocity (how fast the wave propagates in a medium)
- λ = wavelength (distance over which a wave shape repeats)
- f = wave frequency (number of waves per unit of time)

For light and other electromagnetic radiation in a vacuum, the wave velocity is the speed of light (c):

c = λf

But, the wave speed is different for other kinds of waves and for light passing through a medium.

- Light in air or vacuum: 299,792,458 meters per second
- Light in water: 224,901,000 m/s
- Sound in air: 343.2 m/s
- Sound in water (20 °C): 1,481 m/s

Wavelength and frequency are inversely proportional. As wavelength increases, frequency decreases. As frequency increases, wavelength decreases.

### How to Calculate Wavelength From Frequency

Rearrange the equation and calculate wavelength from frequency:

λ = v/f

For example, find the wavelength of the musical note A4, which has a frequency of 440 Hz.

The only tricky part in the calculation is keeping the units straight. Usually, you work with meters and Hertz and then convert to other units (e.g., nanometers, THz, GHz). In this problem, the wave velocity is the speed of sound in air (343.2 m/s). The frequency is 440 Hz. One hertz unit equal one cycle (wave) per second, so a frequency of 440 Hz is 440 s^{-1}.

λ = v/f

λ = (343.2 m/s)/(440 s^{-1})

λ = 0.78 m or 78 cm

As another example, find the frequency of the green light of the aurora borealis, which has a frequency of 5.38 x 10^{14} Hz.

Here, the equation is:

λ = c/f

λ = (3 x 10^{8} m/sec)/(5.38 x 10^{14} s^{-1})

λ = 5.576 x 10^{-7} m = 557.6 nm

### How to Calculate Frequency From Wavelength

Rearrange the equation and calculate frequency from wavelength:

f = v/λ

For example, find the wavelength of orange light with a frequency of 4.8×10^{14} Hz.

f = v/λ (but v is c for light)

f = c/λ

f = (3.00 × 10^{8 }m/s)/(4.8×10^{14} s^{-1})

f = 6.2 x 10^{-7} m = 620 nm

### Wavelength to Frequency Chart

This chart shows the wavelength to frequency relationship for electromagnetic radiation:

Electromagnetic Radiation | Wavelength | Frequency |

Gamma radiation | 1 pm | 300 EHz |

X-ray | 1 nm | 300 PHz |

Ultraviolet | 100 nm | 3 PHz |

Visible light | 400-700 nm | 430-750 THz |

Infrared | 100 μm | 3 THz |

EHF (Extremely high frequency) | 1 mm | 300 GHz |

SHF (Super high frequency) | 1 cm | 30 GHz |

UHF (Ultra high frequency) | 1 dm | 3 GHz |

VHF (Very high frequency) | 10 m | 30 MHz |

ELF (Extremely low frequency) | 100,000 km | 3 Hz |

### References

- Avison, John (1999).
*The World of Physics*. Nelson Thornes. ISBN 978-0-17-438733-6. - Cassidy, David C.; Holton, Gerald James; Rutherford, Floyd James (2002).
*Understanding Physics*. Birkhäuser. ISBN 0-387-98756-8. - Hecht, Eugene (1987).
*Optics*(2nd ed.). Addison Wesley. ISBN 0-201-11609-X.