Frequency Formulas
Types of Frequency
Ordinary (cyclic) frequency (f) counts complete cycles per second in Hertz. Angular frequency (ω = 2πf) counts radians per second — it eliminates the 2π factor from trigonometric wave equations.
Wave number (k = 2π/λ) is the spatial analogue of angular frequency — radians per metre instead of per second. Together, ω and k fully describe a sinusoidal wave: y = A sin(kx − ωt).
Frequency Spectrum Reference
| Band | Frequency Range | Example Uses |
|---|---|---|
| Infrasound | < 20 Hz | Earthquakes, elephants |
| Audible sound | 20 Hz – 20 kHz | Speech, music |
| Ultrasound | 20 kHz – 1 GHz | Medical imaging, cleaning |
| AM Radio | 530 kHz – 1.7 MHz | Broadcast radio |
| FM Radio | 88 – 108 MHz | Broadcast radio |
| Wi-Fi | 2.4 / 5 / 6 GHz | Wireless networking |
| 5G mmWave | 24 – 39 GHz | High-speed cellular |
| Infrared | 300 GHz – 430 THz | Heat, remote controls |
| Visible light | 430 – 790 THz | Human vision |
| X-rays | 30 PHz – 30 EHz | Medical imaging |
Musical Note Frequencies (A4 = 440 Hz)
| Note | Frequency (Hz) | Note | Frequency (Hz) |
|---|---|---|---|
| C4 (Middle C) | 261.63 | A4 | 440.00 |
| D4 | 293.66 | B4 | 493.88 |
| E4 | 329.63 | C5 | 523.25 |
| F4 | 349.23 | A5 | 880.00 |
| G4 | 392.00 | A3 | 220.00 |
Equal temperament tuning. Each semitone is a factor of 2^(1/12) ≈ 1.05946.
Worked Examples
Period of 50 Hz mains power
f = 1/T → T = 1/50 = 0.02 s = 20 ms. One cycle of 50 Hz AC takes 20 milliseconds.
Frequency of red light (700 nm)
f = v/λ = 299,792,458 / 700 × 10⁻⁹ ≈ 4.28 × 10¹⁴ Hz = 428 THz.
Beat frequency: tuning forks
Two tuning forks at 440 Hz and 443 Hz produce f_beat = |443 − 440| = 3 Hz — three beats per second.
Engineering Applications
| Field | Frequency Use | Typical Range |
|---|---|---|
| Power systems | AC mains frequency | 50 or 60 Hz |
| Audio engineering | Sample rate, filtering | 20 Hz – 48 kHz |
| RF engineering | Carrier frequencies | kHz – GHz |
| Medical imaging | Ultrasound probe | 1 – 15 MHz |
| Spectroscopy | EM absorption lines | THz – PHz |
Accuracy and Limitations
This calculator uses ideal frequency relationships. Real-world factors like Doppler shift, relativistic effects, dispersion, and temperature dependence are not included. Musical note frequencies use A4 = 440 Hz equal temperament — other tuning systems exist.
Frequently Asked Questions
What is frequency?›
Frequency is the number of complete cycles (oscillations) that occur per second. It is measured in Hertz (Hz), where 1 Hz = 1 cycle per second. Higher frequency means more oscillations per second.
What is the difference between frequency and period?›
Frequency (f) and period (T) are reciprocals: f = 1/T. Frequency tells you how many cycles per second; period tells you how long one cycle takes. For example, 60 Hz AC power has a period of 1/60 ≈ 0.0167 seconds.
What is angular frequency?›
Angular frequency (ω) = 2πf measures oscillation rate in radians per second rather than cycles per second. It appears in physics equations like x = A sin(ωt) and is convenient because one full cycle = 2π radians.
What is a beat frequency?›
When two waves of slightly different frequencies f₁ and f₂ interfere, the result oscillates at the average frequency with an amplitude that pulses at the beat frequency f_beat = |f₁ − f₂|. Musicians use this to tune instruments — zero beats means the frequencies match.
What is the wave number?›
The angular wave number k = 2π/λ is the spatial equivalent of angular frequency. It measures the number of radians per metre. Together, ω and k describe the wave equation: y = A sin(kx − ωt).
What frequencies can humans hear?›
The typical human hearing range is 20 Hz to 20,000 Hz (20 kHz). Frequencies below 20 Hz are infrasound; above 20 kHz are ultrasound. Hearing sensitivity peaks around 2,000–5,000 Hz.
What is the frequency of visible light?›
Visible light ranges from about 430 THz (red, 700 nm) to 790 THz (violet, 380 nm). These frequencies are far too high to see individual oscillations — we perceive them as colour instead.
What frequency is 5G?›
5G cellular networks operate in three bands: sub-6 GHz (typically 3.5 GHz), mid-band (around 6–24 GHz), and mmWave (24–39 GHz). The mmWave band offers very high speeds but shorter range.
