Frequency Calculator

How to use this calculator

Three modes let you calculate frequency from different starting information. Select the mode that matches what you know.

From Period (f = 1/T): Enter the time for one complete cycle in seconds. A period of 0.01 s gives a frequency of 100 Hz.

From Wavelength and Wave Speed (f = v/λ): Enter the wave speed in metres per second and the wavelength in metres. Use the wave speed preset dropdown to quickly fill in the speed of sound in air (343 m/s), speed of sound in water (1480 m/s), or the speed of light (299,792,458 m/s).

From Angular Velocity (f = ω/2π): Enter angular velocity in radians per second. This mode is useful for rotating machinery, alternating current circuits, and vibration analysis.

Results include frequency in Hz, kHz, and MHz; period in seconds; angular frequency in rad/s; and wavelength in air calculated at 343 m/s.

What is frequency

Frequency is the number of complete cycles of a periodic event that occur per unit of time. A cycle is one full repetition: in a sound wave, it means the air pressure rises to a peak, falls through equilibrium to a trough, and returns to equilibrium. In AC electricity, it means the voltage rises from zero to positive peak, falls through zero to negative peak, and returns to zero.

The standard unit of frequency is the hertz, symbol Hz, defined as one cycle per second. The unit is named after Heinrich Hertz, the German physicist who in 1887 became the first person to transmit and detect radio waves experimentally, providing direct evidence for James Clerk Maxwell’s 1865 electromagnetic theory.

Larger units are used for high frequencies: 1 kilohertz (kHz) = 1,000 Hz, used for audio and radio; 1 megahertz (MHz) = 1,000,000 Hz, used for AM radio and older computer processors; 1 gigahertz (GHz) = 1,000,000,000 Hz, used for modern processors and microwave communication; 1 terahertz (THz) = 10¹² Hz, used for infrared and millimetre-wave research.

Before the hertz was adopted as the SI unit in 1960, frequency was commonly expressed as “cycles per second” (cps), and that term still appears in older technical literature.

The frequency-period relationship

Frequency and period are reciprocals of each other. If you know one, you can always find the other:

Period T is measured in seconds. A 50 Hz power supply completes one cycle every 1/50 = 0.02 seconds (20 milliseconds). A 2.4 GHz Wi-Fi signal completes one cycle every 1/(2.4 × 10⁹) ≈ 4.2 × 10⁻¹⁰ seconds (about 0.4 nanoseconds).

Doubling the frequency halves the period. This inverse relationship appears everywhere: musical pitch (each octave doubles frequency while halving the period of each note), motor speed (faster rotation means shorter cycle time), and radio carrier waves (higher frequency bands have shorter cycles and shorter wavelengths).

Angular frequency

Angular frequency ω (omega) expresses how many radians of phase a wave or oscillation accumulates per second. Since one complete cycle equals 2π radians:

Angular frequency appears naturally in the mathematics of oscillations and waves. The position of a simple pendulum over time is x(t) = A×cos(ωt), and the voltage of a sinusoidal AC supply is V(t) = V_peak×sin(ωt). Using ω rather than f eliminates the repeated factor of 2π from these equations.

For a standard 50 Hz power supply: ω = 2π × 50 ≈ 314.16 rad/s. For a 60 Hz supply: ω ≈ 376.99 rad/s. Electric motor nameplate data often specifies angular velocity rather than rotations per minute for this reason.

Frequency and wavelength

In any wave, frequency relates to wavelength through wave speed:

Where v is wave speed (m/s) and λ is wavelength (m). This means frequency and wavelength are inversely proportional for a given medium: doubling the frequency halves the wavelength.

For sound in air at 20°C (v = 343 m/s): a 343 Hz tone has a 1-metre wavelength. Bats using 50 kHz echolocation produce sound with a 6.86 mm wavelength, short enough to reflect from small insects.

For electromagnetic waves in a vacuum (v = c = 299,792,458 m/s): AM radio at 1 MHz has a 300-metre wavelength. FM radio at 100 MHz: 3 metres. Wi-Fi at 2.4 GHz: 12.5 cm. Visible light at 500 THz: 600 nanometres.

The key point is that wave speed does not change when you change frequency (within the same medium). Instead, wavelength adjusts to compensate.

The electromagnetic spectrum

The electromagnetic spectrum covers an enormous range of frequencies, all travelling at the speed of light in a vacuum. Each band has distinct properties and applications.

Radio waves (3 Hz to 300 GHz): The lowest frequencies. Used for broadcasting (AM, FM), television, mobile phones, radar, satellite communication, and Wi-Fi. Wavelengths range from thousands of kilometres down to millimetres.

Microwaves (300 MHz to 300 GHz): A subset of radio waves used for radar, microwave ovens (2.45 GHz), satellite communication, and 5G cellular in millimetre-wave bands.

Infrared (300 GHz to 400 THz): Thermal radiation. All warm objects emit infrared. Night-vision cameras, remote controls, and fibre-optic data links at 200 THz use infrared.

Visible light (400 THz to 790 THz): The narrow band the human eye detects. Violet is around 750 THz (400 nm wavelength); red is around 430 THz (700 nm).

Ultraviolet (790 THz to 30 PHz): Beyond visible. UV-A and UV-B cause tanning and sunburn. UV-C (100-280 nm) kills bacteria and is used for sterilisation.

X-rays (30 PHz to 30 EHz): High enough energy to penetrate soft tissue but absorbed by dense materials like bone and metal. Used in medical imaging and security scanners.

Gamma rays (above 10 EHz): The highest frequency electromagnetic radiation. Produced by nuclear reactions and certain radioactive decays. Used in cancer radiotherapy.

Human hearing and audible frequency

The human auditory range spans roughly 20 Hz to 20,000 Hz (20 kHz), though this varies considerably by age, individual, and hearing health. At birth, the upper limit may extend to 20 kHz; by age 60, many people cannot hear above 8-10 kHz.

The ear is not equally sensitive across this range. Maximum sensitivity is around 1-4 kHz, which is no coincidence: this range corresponds to the primary frequencies of the human voice, and evolution has tuned our hearing to communication.

Sound below 20 Hz is called infrasound. Elephants communicate using infrasound at 14-35 Hz that travels kilometres through the ground and air. Some large marine animals and earthquakes also produce infrasound. Exposure to intense infrasound can cause nausea and disorientation.

Sound above 20 kHz is ultrasound. Dogs hear up to about 65 kHz; cats up to 79 kHz; bats echolocate at 20-200 kHz. Medical ultrasound imaging uses frequencies of 2-18 MHz, far above any biological hearing.

Musical notes and frequency

Western music divides the octave into 12 equal semitones using equal temperament tuning. Since each octave doubles the frequency, each semitone multiplies frequency by the twelfth root of 2 (approximately 1.0595).

The international reference pitch is A4 = 440 Hz, standardised in 1939. From there, other notes follow:

A full piano spans 27.5 Hz (lowest A0) to 4186 Hz (highest C8). Organ pipes can extend down to 16 Hz, at the edge of human hearing, where the notes are felt as vibrations as much as heard as pitch.

The relationship between frequency and pitch is logarithmic: equal ratios of frequency correspond to equal musical intervals. A doubling (any octave) sounds like the same interval regardless of starting pitch because perception of pitch is itself logarithmic.

AC power frequency

Alternating current power systems operate at either 50 Hz or 60 Hz depending on country. The split dates to the late 19th century when different electrical standards were adopted independently in Europe and the United States.

50 Hz countries: Most of Europe, UK, Australia, Asia, Africa, and most of South America. One complete AC cycle takes 20 milliseconds.

60 Hz countries: United States, Canada, Mexico, most of Central America, and parts of Japan and South Korea. One complete AC cycle takes 16.67 milliseconds.

The frequency choice involves trade-offs. Transformers and motors are slightly more efficient at 60 Hz. Power transmission losses are slightly lower at 50 Hz. Neither difference is large enough to justify converting one system to the other.

The frequency of AC power has historically affected devices that use it as a timing reference: older clocks, film projectors run at 24 frames per second on a 48 Hz cycle, and television frame rates (25 fps in 50 Hz countries; 30 fps in 60 Hz countries) were chosen to avoid beat-frequency flicker against mains lighting.

Radio frequency bands

Radio frequencies cover an enormous practical range, each segment allocated to specific uses by international agreement through the International Telecommunication Union.

Very Low Frequency (VLF, 3-30 kHz): Can penetrate seawater to limited depth. Used for communication with submerged submarines.

AM broadcast (535-1605 kHz): Medium-wave AM radio. Propagates over long distances at night via ionospheric reflection.

Shortwave (3-30 MHz): Can travel globally via multiple ionospheric reflections. Used for international broadcasting and aviation communication.

FM broadcast (87.5-108 MHz): Line-of-sight propagation. High audio quality compared to AM due to frequency modulation.

Wi-Fi 2.4 GHz (2400-2484 MHz): Better range and wall penetration than 5 GHz but more crowded spectrum.

Wi-Fi 5 GHz (5180-5825 MHz): Higher data rates, less interference, but shorter range.

Cellular networks: 600 MHz to 3.5 GHz for sub-6GHz 5G; 24-86 GHz for millimetre-wave 5G with very high data rates but limited range.

Resonance and natural frequency

Every physical system capable of oscillation has one or more natural frequencies at which it vibrates most readily when disturbed. At resonance, even a small periodic driving force can build up large oscillations if the driving frequency matches the natural frequency.

The natural frequency of a simple spring-mass system is:

Where k is the spring constant (N/m) and m is the mass (kg). Stiffer springs (higher k) and lighter masses (lower m) both raise the natural frequency.

Resonance has practical importance in engineering: bridge designers must ensure that the fundamental frequency of a bridge does not match the frequency of wind-induced vortex shedding or marching soldiers (the original Broughton Suspension Bridge collapsed in 1831 when soldiers marched across it in step). Buildings in earthquake zones are designed so their fundamental frequency does not match common earthquake frequencies. Radio tuners work by adjusting an LC circuit until its resonant frequency matches the carrier frequency of the desired station.

The Doppler effect and frequency shift

The Doppler effect is the change in observed frequency when a wave source and observer are in relative motion. When source and observer approach each other, the observer hears or measures a higher frequency than is being emitted. When they move apart, a lower frequency is observed.

For sound, the relationship is:

Where v is wave speed, v_observer is observer speed (positive toward source), and v_source is source speed (positive toward observer).

The Doppler effect has applications across science and technology: police radar guns measure vehicle speed from the frequency shift of reflected radio waves; medical Doppler ultrasound measures blood flow velocity; astronomers measure the recession velocity of distant galaxies from the redshift of their light (a Doppler shift toward lower frequencies as galaxies move away); and bats use Doppler-shifted echoes to detect the wing-beat frequency of flying prey.