What is the wavelength formula?

Wavelength (λ) = Wave Speed (v) / Frequency (f). For sound in air at 20°C, speed is about 343 m/s. For light in vacuum, speed is 299,792,458 m/s. The formula applies to all waves: sound, light, radio, water, seismic, even matter waves.

What is the wavelength of audible sound?

At 20 Hz (lowest audible), wavelength is about 17.15 m. At 20,000 Hz (highest audible), wavelength is about 1.7 cm. Middle A (440 Hz) has a wavelength of about 0.78 m. This wide range is why bass is room-dependent (long λ) but treble isn't (short λ).

Why is visible light called "visible"?

Human eyes have cone cells sensitive to electromagnetic radiation roughly in the 380-700 nm wavelength range — what we call "visible light". Other wavelengths exist but aren't directly visible to humans (some animals see UV or IR). Different wavelengths in this range correspond to different colors.

How does wavelength change in different media?

In media with refractive index n, light slows down: v = c/n. Frequency stays the same (color), but wavelength shortens: λ_medium = λ_vacuum / n. Green light (550 nm vacuum) becomes 414 nm in water (n=1.33). Color perception unchanged because rods/cones respond to frequency.

How does wavelength relate to antenna size?

Antennas are typically λ/4 or λ/2 long. For 2.4 GHz WiFi (λ = 12.5 cm): quarter-wave = 3.1 cm. For 100 MHz FM (λ = 3 m): quarter-wave = 75 cm. Higher frequency = shorter wavelength = smaller antenna. Why cellular phones (GHz frequencies) can have small antennas while old shortwave radios needed large ones.

What's the wavelength of WiFi?

2.4 GHz WiFi: λ = 3×10⁸ / 2.4×10⁹ = 0.125 m = 12.5 cm. 5 GHz WiFi: λ ≈ 6 cm. 60 GHz WiFi (WiGig): λ ≈ 5 mm. Shorter wavelengths offer more bandwidth but worse penetration of walls and obstacles.

How is photon energy related to wavelength?

E = hc/λ. Shorter λ = higher energy. Red light (700 nm): ~1.77 eV. Violet (400 nm): ~3.1 eV. UV (100 nm): ~12 eV. X-rays (1 nm): ~1.24 keV. Gamma rays (1 pm): ~1.24 MeV. Higher energy photons cause more biological damage — why X-rays are dangerous but radio waves aren't.

Do all waves have wavelength?

Yes — anything periodic in space has a wavelength. Sound waves (compressions/rarefactions in air). Light/radio (oscillating electromagnetic field). Water waves (oscillating height). Seismic waves (oscillating displacement in rock). Even matter has de Broglie wavelength: λ = h/p (Planck constant / momentum) — basis of electron microscopy.

Wavelength Calculator

Find the wavelength of a wave given its speed and frequency using the formula wavelength = speed / frequency. Supports sound waves, light waves, and radio waves with preset speeds.

Last updated: May 30, 2026

About this calculator

Wavelength is the spatial period of a periodic wave — the distance between identical points in successive cycles. For sound waves in air, wavelength ranges from ~17 m (low bass) to ~1.7 cm (high treble). For visible light, it spans ~400-700 nanometers — colors are perceived as different wavelengths. For radio waves, wavelengths can stretch from millimeters (5G mmWave) to thousands of kilometers (submarine communications).

The fundamental relationship is c = f × λ, where c is wave speed, f is frequency, and λ is wavelength. This applies to all waves: sound, light, water waves, seismic waves, even quantum mechanical matter waves. Knowing any two quantities, you can find the third. The calculator does this conversion automatically.

Wavelength is critical for many practical applications. Antenna size scales with wavelength (typically λ/2 or λ/4 dipoles). Microwave ovens use 2.45 GHz (λ ≈ 12.2 cm) for efficient water absorption. Light wavelength determines color, vision, photography, and laser characteristics. Sound wavelength affects loudspeaker design, room acoustics, and noise control.

The wave equation c = fλ has a beautiful consequence: in vacuum, light's speed is constant (c = 299,792,458 m/s), so wavelength and frequency are strictly inversely related. In media (water, glass), light slows down, so wavelength shortens for the same frequency. Color (frequency) stays the same; visual wavelength in the medium differs from vacuum.

Common applications: antenna design, audio engineering, optics, spectroscopy, radar systems, microwave technology, and any wave-related engineering problem.

Formula

**Wavelength formula:** λ = v / f Where: - λ = wavelength (m) - v = wave speed (m/s) - f = frequency (Hz) **Rearranged:** - Frequency: f = v / λ - Speed: v = f × λ **Worked example: middle A in air** Concert pitch A = 440 Hz. Speed of sound at 20°C = 343 m/s. λ = 343 / 440 ≈ 0.78 m A4 sound wave is 78 cm long in air. Lower notes are longer; higher are shorter. **Worked example: green light** Frequency = 5.45 × 10¹⁴ Hz (green light). λ = 3 × 10⁸ / 5.45 × 10¹⁴ = 5.5 × 10⁻⁷ m = 550 nm Visible spectrum: ~380-750 nm in vacuum. **Wave speeds in common media:** | Wave type | Speed | |---|---| | Sound in air (20°C) | 343 m/s | | Sound in water | 1,480 m/s | | Sound in steel | 5,960 m/s | | Light in vacuum | 3 × 10⁸ m/s | | Light in water | 2.25 × 10⁸ m/s (75%) | | Light in glass | 2.0 × 10⁸ m/s (67%) | | Light in diamond | 1.24 × 10⁸ m/s (41%) | | Radio waves (vacuum) | 3 × 10⁸ m/s | | Seismic P-waves (rock) | 5-13 km/s | | Water waves (deep ocean) | varies with λ | **Electromagnetic spectrum:** | Type | Frequency | Wavelength | |---|---|---| | Radio | 3 kHz - 300 GHz | 100 km - 1 mm | | Microwave | 300 MHz - 300 GHz | 1 m - 1 mm | | Infrared | 300 GHz - 430 THz | 1 mm - 700 nm | | Visible | 430-750 THz | 700-400 nm | | Ultraviolet | 750 THz - 30 PHz | 400-10 nm | | X-rays | 30 PHz - 30 EHz | 10 nm - 10 pm | | Gamma rays | > 30 EHz | < 10 pm | **Visible spectrum (in vacuum):** | Color | Wavelength (nm) | Frequency (THz) | |---|---|---| | Violet | 380-450 | 668-789 | | Blue | 450-485 | 618-668 | | Cyan | 485-500 | 600-618 | | Green | 500-565 | 530-600 | | Yellow | 565-590 | 508-530 | | Orange | 590-625 | 480-508 | | Red | 625-700 | 428-480 | **Audible sound (in air, 343 m/s):** | Frequency | Wavelength | |---|---| | 20 Hz (very low) | 17.15 m | | 100 Hz | 3.43 m | | 200 Hz | 1.72 m | | 440 Hz (A4) | 0.78 m | | 1 kHz | 0.343 m | | 5 kHz | 6.86 cm | | 10 kHz | 3.43 cm | | 20 kHz (very high) | 1.72 cm | **Radio bands (common):** | Band | Frequency | Wavelength | Use | |---|---|---|---| | VLF | 3-30 kHz | 100-10 km | Submarine comms | | LF | 30-300 kHz | 10-1 km | AM long wave | | MF (AM radio) | 0.3-3 MHz | 1 km-100 m | AM broadcasting | | HF (shortwave) | 3-30 MHz | 100-10 m | Amateur radio | | VHF | 30-300 MHz | 10-1 m | FM, TV | | UHF | 0.3-3 GHz | 1 m-10 cm | Cellular, TV, WiFi | | SHF | 3-30 GHz | 10-1 cm | Satellite, radar | | EHF (mmWave) | 30-300 GHz | 10-1 mm | 5G, radar | **Antenna sizing:** Typical antenna length: λ/4 or λ/2. | Frequency | λ | Quarter-wave | |---|---|---| | 100 MHz (FM) | 3 m | 75 cm | | 900 MHz (cellular) | 33 cm | 8.3 cm | | 2.4 GHz (WiFi) | 12.5 cm | 3.1 cm | | 5 GHz (WiFi) | 6 cm | 1.5 cm | | 28 GHz (5G mmWave) | 10.7 mm | 2.7 mm | This is why higher frequencies → smaller antennas. Why cell phones can have small antennas hidden inside, but old radio antennas were prominent. **Light in media (refractive index):** Wavelength in medium: λ_medium = λ_vacuum / n Where n = refractive index. In water (n = 1.33): 550 nm green light has λ = 414 nm. Frequency unchanged. In glass (n = 1.5): 550 nm becomes 367 nm. Color perception unchanged because rods/cones respond to frequency, not wavelength. **Photon energy:** E = h × f = h × c / λ Where h = 6.626 × 10⁻³⁴ J·s (Planck's constant). | Photon | λ | Energy | |---|---|---| | Red (700 nm) | 700 nm | 1.77 eV | | Green (500 nm) | 500 nm | 2.48 eV | | Violet (400 nm) | 400 nm | 3.10 eV | | UV-A (350 nm) | 350 nm | 3.54 eV | | X-ray (1 nm) | 1 nm | 1.24 keV | | Gamma (1 pm) | 1 pm | 1.24 MeV | Higher frequency = higher energy = more damaging biologically. **Doppler effect on wavelength:** For approaching source: λ_observed = λ_source × (c − v_source) / c (compressed). For receding: λ stretched. Used in astronomy (redshift) to measure cosmic distances. **Standing waves:** When wave reflects back, standing wave forms with nodes and antinodes spaced λ/2 apart. Determines: - Resonant frequencies of strings, pipes. - Standing waves in rooms (acoustic). - Antenna resonance. - Microwave cavity modes. **de Broglie wavelength (quantum matter):** λ = h / p For matter particles, where p = momentum. Electron at 100 V acceleration: λ ≈ 0.12 nm — comparable to atomic spacings, basis of electron microscopy. **Sound wavelength and room acoustics:** Low frequencies (long λ) interact strongly with room dimensions: - 50 Hz (λ ≈ 6.9 m): rooms have strong modes. - 100 Hz (λ ≈ 3.4 m): typical room standing waves. - 1 kHz (λ ≈ 34 cm): less affected by room. This is why bass response is room-dependent; high frequencies less so.

How to use this calculator

Choose wave type: sound (343 m/s), light (3×10⁸ m/s), or custom.
Enter frequency in Hz.
For custom: enter wave speed manually.
Calculator returns wavelength.
Common units: meters for radio, mm/cm for sound/microwave, nm for light.
Light in media: divide vacuum λ by refractive index.

Worked examples

WiFi 2.4 GHz antenna size

**Scenario:** Design an antenna for 2.4 GHz WiFi. Quarter-wave length? **Calculation:** λ = 3×10⁸ / 2.4×10⁹ = 0.125 m = 12.5 cm. Quarter-wave: 3.125 cm. **Result:** ~3.1 cm quarter-wave antenna. Most laptops, phones, and routers have hidden antennas this size or shorter (using folded designs). For higher-gain antennas (Yagi, parabolic), more complex geometry but still scaled to wavelength.

Green light wavelength

**Scenario:** Photon at 5.45 × 10¹⁴ Hz (green light). Wavelength in vacuum and water? **Calculation:** In vacuum: λ = 3×10⁸ / 5.45×10¹⁴ = 5.5×10⁻⁷ m = 550 nm. In water (n=1.33): 550/1.33 = 414 nm. **Result:** 550 nm in vacuum, 414 nm in water (shorter). Color appearance unchanged because eye responds to frequency, not wavelength. Used in spectroscopy and optical design.

Subwoofer wavelength

**Scenario:** Subwoofer producing 50 Hz tone in 20°C air. Wavelength? **Calculation:** λ = 343/50 = 6.86 m. **Result:** ~6.9 m wavelength — longer than most rooms! This means bass interacts strongly with room dimensions, creating standing wave nodes (dead spots) and antinodes (boomy spots). Why bass response varies dramatically with listener position. Room treatment (bass traps) typically targets these standing waves.

When to use this calculator

**Use wavelength calculations for:**

- **Antenna design**: sizing for specific frequencies. - **Radio engineering**: matching impedances, wave propagation. - **Optics**: lens, filter, and grating design. - **Audio engineering**: loudspeaker design, room acoustics. - **Spectroscopy**: identifying materials by absorbed/emitted λ. - **Telecommunications**: channel spacing, modulation. - **Microwave technology**: oven design, radar systems. - **Physics problems**: wave mechanics, quantum mechanics.

**Wavelength size vs interaction:**

When λ is comparable to obstacle size: strong interaction (diffraction, scattering). When λ << obstacle: shadows, simple geometric optics. When λ >> obstacle: barely affected (waves pass around).

**Examples:** - Visible light (~500 nm) and large objects: shadows, optics. - Visible light and dust (1 μm): scattering (sky blue). - Long radio waves (km) and buildings: passes through easily. - WiFi (12 cm) and walls (10-30 cm): partial blocking.

**Antenna sizing (more detail):**

| Type | Size relative to λ | |---|---| | Hertz dipole | λ/2 | | Marconi antenna | λ/4 above ground | | Yagi-Uda | several elements spaced ~λ/4 | | Parabolic dish | aperture much greater than λ | | Phased array | many elements spaced ~λ/2 | | Metasurface | sub-wavelength elements |

Antennas smaller than λ/4 are inefficient (low radiation resistance). Antennas comparable to λ work best.

**Microwave oven physics:**

2.45 GHz → λ = 12.2 cm. Wavelength enters food, oscillating electric field rotates water molecules → friction → heat.

Why food has hot/cold spots: standing wave pattern at λ/4 ≈ 3 cm intervals. Turntable spreads heating.

**Light through media:**

When light enters glass, frequency stays the same (color), wavelength shortens (because v decreases). This is why: - Glass appears clear (light passes through, slowed). - Light bends at interfaces (Snell's law). - Lenses focus by retarding waves in glass.

**Standing waves in rooms (sound):**

Room mode frequencies: f = (c/2) × √((n_x/L_x)² + (n_y/L_y)² + (n_z/L_z)²)

For typical room 4×5×3 m: lowest modes at 34 Hz, 43 Hz, 57 Hz, ... — clusters in bass region.

Result: bass response is room-dependent. Bass traps reduce these modes.

**Spectroscopy:**

Each element absorbs/emits characteristic wavelengths. Hydrogen Balmer series: - 656 nm (red, H-alpha). - 486 nm (blue-green, H-beta). - 434 nm (blue, H-gamma).

Used in chemistry, astronomy, materials science.

**Photon energy and biological effects:**

| Wavelength | Energy | Effect | |---|---|---| | Radio | < 10⁻⁵ eV | warming only | | Microwave | 10⁻⁵ - 10⁻³ eV | molecular rotation | | IR | 10⁻³ - 1.7 eV | molecular vibration | | Visible | 1.7-3 eV | electronic transitions | | UV-A | 3-3.5 eV | tanning, DNA damage | | UV-B | 3.5-4 eV | sunburn | | UV-C | 4-12 eV | germicidal, ionization | | X-ray | 1.2 keV+ | ionization, tissue damage | | Gamma | 1.2 MeV+ | nuclear, severe damage |

**Common applications:**

- **Radio broadcasting**: 535-1605 kHz AM, 88-108 MHz FM. - **TV broadcasting**: 54-806 MHz. - **Cellular**: 700 MHz - 39 GHz. - **WiFi**: 2.4, 5, 60 GHz. - **Radar**: 1-100 GHz. - **5G mmWave**: 24-100 GHz. - **Optical fiber**: ~1550 nm (low-loss window). - **Lasers**: He-Ne 632 nm, Nd:YAG 1064 nm.

**Software:**

- **Antenna design**: 4nec2, MMANA-GAL (free). - **Optics**: Zemax, OpticStudio. - **Acoustics**: Room EQ Wizard (REW). - **Spectroscopy**: NIST databases.

**Pitfalls:**

- **Wrong wave speed**: light vs sound vs in different media. - **Frequency vs wavelength**: inverse — don't confuse. - **Vacuum vs medium**: λ shrinks in glass/water; f stays same. - **Doppler shift**: moving sources change observed λ. - **Mixing units**: Hz vs kHz vs MHz vs GHz. - **Sound speed varies with temperature**: 343 m/s is at 20°C. - **Standing waves**: nodes/antinodes spaced λ/2.

Common mistakes to avoid

Mixing wave speeds (light vs sound vs medium-specific).
Confusing frequency and wavelength (they're inverse).
Forgetting that wavelength shortens in dense media (light in glass).
Mixing unit prefixes (Hz vs kHz vs MHz vs GHz).
Using vacuum wavelength when medium's value is needed.
Ignoring Doppler shift for moving sources/observers.
Calculating antenna size with wrong wavelength.
Confusing wavelength with period (T = 1/f, time; λ = v/f, space).

Frequently Asked Questions

Sources & further reading

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