What is the relationship between frequency and period?

Frequency (f) = 1 / Period (T). If a wave completes one cycle every 0.01 seconds, its frequency is 100 Hz. Angular frequency: ω = 2πf, with units rad/s. Angular frequency is preferred in physics and AC circuit analysis.

What are common frequency units?

1 kHz = 1,000 Hz, 1 MHz = 1,000,000 Hz, 1 GHz = 1,000,000,000 Hz, 1 THz = 10¹² Hz. US power grid is 60 Hz (50 Hz in Europe), human hearing 20 Hz - 20 kHz, FM radio 88-108 MHz, WiFi 2.4/5 GHz, visible light ~500 THz.

How is frequency related to wavelength?

For any wave: speed = frequency × wavelength (c = fλ). For light: c = 3 × 10⁸ m/s, so red light (700 nm) ≈ 430 THz. For sound in air: c = 343 m/s, so 1 kHz tone has wavelength of 0.343 m. Higher frequency = shorter wavelength at constant speed.

What is the Nyquist frequency?

The minimum sampling rate needed to capture a signal: f_sample ≥ 2 × f_max. Below this, aliasing distorts the recording. CD audio samples at 44.1 kHz, capturing up to 22.05 kHz — just above human hearing. Phone audio samples at 8 kHz, limiting to 4 kHz.

What is angular frequency?

ω = 2πf, measured in rad/s. It's used heavily in physics and circuit analysis because it relates to circular motion (each cycle = 2π radians). For a 60 Hz signal: ω = 377 rad/s. Use ω in impedance formulas: Z_L = jωL, Z_C = 1/(jωC).

What is concert pitch?

Concert A (A4) is standardized at 440 Hz. Other notes follow equal temperament: each semitone is 2^(1/12) times the previous. C4 (middle C) ≈ 262 Hz; C5 ≈ 523 Hz. Historic concert pitch varied (380-465 Hz); 440 Hz adopted by ISO in 1955.

Why is the AC power frequency 60 Hz in the US?

Historical choice. Tesla and Westinghouse selected 60 Hz for early US AC systems. Europe later standardized on 50 Hz partly for compatibility with telephone equipment. 60 Hz reduces motor size slightly; 50 Hz is sometimes preferred for transformer efficiency. Both work fine.

What frequencies do humans hear?

Healthy young adults: 20 Hz to 20 kHz. Most people lose high-frequency sensitivity with age — typical 60-year-old hears up to ~12 kHz. Below 20 Hz is infrasound (felt, not heard); above 20 kHz is ultrasound. Dogs hear up to 60 kHz; bats up to 200 kHz.

Frequency Calculator

Convert between frequency and period. Enter frequency in Hz, kHz, MHz, or GHz, or enter the period in seconds to find the frequency. Also calculates angular frequency (omega).

Frequency measures how many cycles of a periodic phenomenon occur per unit of time. A heartbeat, a pendulum swing, a sound wave's compression-rarefaction cycle, an AC power signal, a radio carrier wave, or a laser's electromagnetic oscillation — all are characterized by frequency. The unit, named for Heinrich Hertz who first demonstrated radio waves in 1887, is 1 Hz = 1 cycle per second.

The basic relationship f = 1/T is one of physics' simplest formulas: frequency and period are reciprocals. A 60 Hz power signal has a period of 1/60 ≈ 16.67 milliseconds. A 1 GHz processor clock has a period of 1 nanosecond. A 2 Hz heart rhythm corresponds to a period of 500 ms (120 bpm).

Frequency ranges in nature span an astonishing 40+ orders of magnitude. The slow swing of a long pendulum: ~0.5 Hz. AC power: 50-60 Hz. Audible sound: 20-20,000 Hz. AM radio: ~1 MHz. FM radio: ~100 MHz. WiFi: 2.4 / 5 GHz. Visible light: ~500 THz. X-rays: 10¹⁸ Hz. Gamma rays: 10²⁰ Hz and above. The decibel-style logarithmic axis is essential for visualizing this range.

Common applications: audio engineering (instrument tuning, equalization), radio communications (carrier frequencies, channel spacing), electrical engineering (AC analysis, oscillator design), medical (EEG, ECG, ultrasound), and any field involving waves or oscillations.

Inputs

Solve For

Frequency (Hz)

Period (seconds)

Results

Frequency

59.99 Hz

Period

0.016670 s

Angular Freq

376.92 rad/s

Frequency Results

Parameter	Value
Frequency	59.988002 Hz
Frequency (kHz)	0.059988 kHz
Frequency (MHz)	0.000060 MHz
Frequency (GHz)	0.000000060 GHz
Period	0.0166700000 seconds
Period (ms)	16.670000 ms
Period (µs)	16670.0000 µs
Angular Frequency (ω)	376.9157 rad/s
Formula	f = 1/T, ω = 2πf

Last updated: May 29, 2026

Formula

**Frequency and period:** f = 1 / T T = 1 / f Where: - f = frequency (Hz) - T = period (s) **Angular frequency:** ω = 2π × f = 2π / T Units: rad/s. Useful when treating waves as rotating phasors (e.g., AC circuit analysis with j×ω notation). **Worked example: 60 Hz mains** T = 1/60 ≈ 0.01667 s = 16.67 ms Each AC cycle takes ~16.67 ms. Half-cycle (0 → peak → 0): 8.33 ms. ω = 2π × 60 ≈ 377 rad/s **Wave frequency-wavelength relationship:** c = f × λ Where: - c = wave speed (m/s; light = 3 × 10⁸ m/s; sound in air ≈ 343 m/s) - f = frequency (Hz) - λ = wavelength (m) For light: - Visible red (700 nm): f = 3e8 / 700e-9 ≈ 4.3 × 10¹⁴ Hz = 430 THz. - Visible violet (400 nm): f ≈ 750 THz. For sound in air: - 1 kHz: λ = 343/1000 = 0.343 m (~1 foot). - 100 Hz: λ = 3.43 m. - 10 kHz: λ = 3.43 cm. **Common frequencies in nature/technology:** | Frequency | Source/Use | |---|---| | 0.001 Hz | Tides, slow ocean cycles | | 1 Hz | Heartbeat (resting) | | 50/60 Hz | AC power | | 100-200 Hz | Bass instruments | | 440 Hz | Concert pitch A | | 1 kHz | Phone tones | | 10 kHz | Sibilants in speech | | 20 Hz – 20 kHz | Human hearing range | | 25 Hz – 150 kHz | Bat echolocation | | 100 kHz | AM radio (long) | | 540-1700 kHz | AM radio band | | 88-108 MHz | FM radio band | | 470-700 MHz | UHF TV | | 900/1800 MHz | GSM cellular | | 2.4 GHz | Microwave oven, WiFi | | 5 GHz | 5 GHz WiFi | | 24 GHz | Police radar | | 60-100 GHz | 5G mmWave | | ~5 × 10¹⁴ Hz | Visible light | | 10¹⁵ – 10¹⁷ Hz | UV | | 10¹⁷ – 10²⁰ Hz | X-rays | | > 10²⁰ Hz | Gamma rays | **Common SI multipliers:** | Symbol | Multiplier | |---|---| | mHz | 10⁻³ | | Hz | 1 | | kHz | 10³ | | MHz | 10⁶ | | GHz | 10⁹ | | THz | 10¹² | | PHz | 10¹⁵ | | EHz | 10¹⁸ | **Music intervals (frequency ratios):** - **Octave**: 2:1 (frequency doubles). A4 (440 Hz) → A5 (880 Hz). - **Perfect fifth**: 3:2 ratio. - **Major third**: 5:4. - **Semitone**: 2^(1/12) ≈ 1.0595 ratio. **Note frequencies (equal temperament, A4 = 440 Hz):** | Note | Frequency (Hz) | |---|---| | C4 (middle C) | 261.63 | | D4 | 293.66 | | E4 | 329.63 | | F4 | 349.23 | | G4 | 392.00 | | A4 | 440.00 | | B4 | 493.88 | | C5 | 523.25 | **Bandwidth and resolution:** For digital sampling (Nyquist theorem): - Sample rate ≥ 2× highest frequency captured. - CD audio: 44.1 kHz sample rate → up to 22.05 kHz captured (just above human hearing). - Phone: 8 kHz sample rate → 4 kHz cutoff. **Doppler frequency shift:** For sound (low speeds): Δf/f ≈ v/c_sound For light: Δf/f ≈ v/c (low speeds; relativistic at high) **Resonance:** For a simple pendulum: f = (1/2π) × √(g/L) For mass-spring: f = (1/2π) × √(k/m) For LC circuit: f = 1 / (2π × √(LC)) **Worked example: Q-factor in resonance** Q = f₀ / Δf Where Δf = bandwidth at half-power. High-Q resonators are very selective (narrow peak). Crystal oscillators: Q ~ 10,000-100,000. Atomic clocks (Cs-133): Q ~ 10⁹.

How to use this calculator

Choose to solve for frequency or period.
Enter the known value in Hz or seconds.
Calculator returns the reciprocal.
Use SI prefixes: 1 kHz = 1000 Hz, 1 GHz = 10⁹ Hz.
For waves: f = c/λ (speed = frequency × wavelength).
For angular form: ω = 2πf rad/s.

Worked examples

AC power period

**Scenario:** US household power is 60 Hz. Period? **Calculation:** T = 1/60 ≈ 0.01667 s = 16.67 ms. ω = 2π × 60 ≈ 377 rad/s. **Result:** Each AC cycle lasts ~16.67 ms. Most circuit-level designs use ω = 377 rad/s for impedance calculations (Z_L = jωL, Z_C = 1/(jωC)). In Europe (50 Hz), period is 20 ms and ω ≈ 314 rad/s.

WiFi channel center frequency

**Scenario:** Convert 2.4 GHz WiFi to period and wavelength. **Calculation:** T = 1 / 2.4e9 ≈ 0.417 ns. λ = c/f = 3e8 / 2.4e9 ≈ 0.125 m (12.5 cm). **Result:** Wavelength of 12.5 cm means optimal antenna size ~6.25 cm (quarter wavelength). Standard 2.4 GHz omnidirectional antennas are typically 6-15 cm long. The microwave oven, also at 2.45 GHz, has wavelength of the same scale.

Tuning to concert A

**Scenario:** Concert pitch A4 = 440 Hz. Period and wavelength in air? **Calculation:** T = 1/440 ≈ 2.27 ms. λ = 343/440 ≈ 0.78 m (78 cm). **Result:** Sound wave of 78 cm wavelength. Each cycle takes 2.27 ms. Open-pipe instrument tuned to A4 has length λ/2 = 39 cm — typical violin or guitar string scale. The 12-tone equal temperament puts adjacent semitones at frequency ratio 2^(1/12) ≈ 1.0595.

When to use this calculator

**Use frequency calculations for:**

- **Audio engineering**: instrument tuning, EQ, crossover design. - **Radio engineering**: channel allocation, antenna sizing. - **Electrical**: AC analysis, oscillator design. - **Medical**: ECG (1-100 Hz), EEG (0.5-100 Hz), ultrasound (1-20 MHz). - **Physics education**: wave problems, simple harmonic motion. - **Optics**: photon energy E = hf, color/wavelength relations. - **Vibration analysis**: machinery diagnostics, modal analysis.

**Frequency ranges by application:**

| Field | Range | |---|---| | Seismology | 0.001-100 Hz | | Heart rate | 0.5-3 Hz | | Brain waves (EEG) | 0.5-100 Hz | | AC power | 50-60 Hz | | Audio | 20-20 kHz | | Ultrasound (medical) | 1-20 MHz | | AM radio | 0.5-1.7 MHz | | FM radio | 88-108 MHz | | TV broadcast | 470-700 MHz | | Cellular | 0.7-39 GHz | | WiFi | 2.4 / 5 / 60 GHz | | Radar | 1-100 GHz | | Visible light | 430-750 THz | | X-rays | 10-10,000 PHz | | Gamma rays | > 10⁴ PHz |

**Hearing perception:**

Healthy young human: 20 Hz to 20 kHz. By age 60: typically 20 Hz to 12 kHz (high-frequency loss is normal aging). Phone bandwidth: 300-3400 Hz (sufficient for speech intelligibility). CD: 20 Hz – 22 kHz. Hi-Res Audio: 20 Hz – 40+ kHz.

**Photon energy:**

E = h × f = h × c / λ

Where h = 6.626 × 10⁻³⁴ J·s (Planck's constant).

Visible photon (500 nm): E ≈ 4 × 10⁻¹⁹ J ≈ 2.5 eV. X-ray (1 nm): E ≈ 1.2 keV. Gamma (1 pm): E ≈ 1.2 MeV.

**Resonance design:**

Build oscillators by matching: - Pendulum length L for desired f: L = g / (4π²f²). - LC circuit: choose L and C so 1/(2π√(LC)) = target f. - Mechanical: spring constant k vs mass m.

**Software:**

- **Audio**: REW, MATLAB Signal Processing Toolbox, Python scipy.signal. - **RF**: ADS, AWR Microwave Office, Keysight Genesys. - **Vibration**: Brüel & Kjær PULSE, LMS Test.Lab. - **Frequency spectrum analyzers**: hardware (Keysight, Rohde & Schwarz) or USB SDRs (RTL-SDR).

**Pitfalls:**

- **Confusing units**: kHz vs MHz vs GHz. - **Hz vs rad/s**: factor of 2π difference. - **Period vs frequency**: reciprocals, easy to mix up. - **Sampling errors**: violating Nyquist causes aliasing. - **Wavelength medium**: c differs by medium (light slows in glass). - **Beat frequency**: combining slightly different frequencies creates beats (|f₁ - f₂|). - **Harmonics**: integer multiples of fundamental; distinct from frequency itself.

Common mistakes to avoid

Confusing frequency with angular frequency (factor 2π).
Mixing SI prefixes (Hz vs kHz vs MHz vs GHz).
Forgetting that frequency and period are reciprocals.
Using wrong wave speed (light vs sound vs medium-specific).
Sampling below Nyquist rate causing aliasing.
Confusing fundamental with harmonics in musical/acoustic contexts.
Using f when ω is required (radians/s in circuit analysis).
Forgetting Doppler shift in moving sources/observers.

Frequently Asked Questions

Sources & further reading

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