Frequency and Wavelength Calculator

The fundamental wave equation λ = v/f (or f = v/λ) relates wavelength λ, frequency f, and wave speed v. For electromagnetic waves in vacuum: λ = c/f where c = 299,792,458 m/s.

Common frequency units and their conversions:

• 1 kHz = 10³ Hz
• 1 MHz = 10⁶ Hz
• 1 GHz = 10⁹ Hz
• 1 THz = 10¹² Hz

Common wavelength units:

• 1 mm = 10⁻³ m
• 1 μm = 10⁻⁶ m
• 1 nm = 10⁻⁹ m
• 1 Å = 10⁻¹⁰ m

A 2.4 GHz WiFi signal has wavelength λ = 3×10⁸ / 2.4×10⁹ = 0.125 m = 12.5 cm. This wavelength determines antenna size (quarter-wave antenna = 3.1 cm for 2.4 GHz) and penetration through materials.

The EM spectrum spans over 15 orders of magnitude in frequency:

• Radio waves (3 kHz - 300 GHz, wavelengths km to mm) carry broadcasts, WiFi, and cellular signals.
• Microwaves (300 MHz - 300 GHz) heat food and enable radar.
• Infrared (300 GHz - 400 THz) carries heat radiation and fiber optic data.
• Visible light (400-790 THz, 380-750 nm) — red at 700nm, violet at 380nm.
• Ultraviolet (790 THz - 30 PHz) causes sunburn and drives fluorescence.
• X-rays (30 PHz - 30 EHz) penetrate soft tissue for medical imaging.
• Gamma rays (above 30 EHz) arise from nuclear reactions and cosmic events.

Each region has distinct generation mechanisms, interactions with matter, and practical applications determined by the wavelength-to-object-size relationship.

Sound waves travel much slower than light — approximately 343 m/s in air at 20°C, 1,480 m/s in water, and 5,960 m/s in steel. Human hearing spans 20 Hz to 20 kHz, corresponding to wavelengths from 17 meters (20 Hz) to 17 millimeters (20 kHz) in air.

Musical note A4 (440 Hz) has a wavelength of 343/440 = 0.78 meters. Low bass frequencies (40-80 Hz) have wavelengths of 4-8 meters, which is why bass sounds pass through walls easily and are difficult to absorb acoustically.

Sound speed increases with temperature: v ≈ 331.3 + 0.606 × T(°C) m/s. At body temperature (37°C), sound travels at 353.8 m/s, relevant for medical ultrasound imaging which uses frequencies of 1-20 MHz (wavelengths 0.08-1.5 mm) to resolve anatomical structures.

To calculate wavelength from frequency, divide the wave speed by the frequency: λ = v / f. For electromagnetic waves in vacuum, use v = c = 299,792,458 m/s (NIST-defined exact value), so λ = c / f.

Worked example: a 100 MHz FM radio signal has λ = 3×10⁸ / 1×10⁸ = 3 meters.

To reverse the calculation and find frequency from wavelength, use f = v / λ. Both forms come from rearranging the wave equation v = f × λ, described by HyperPhysics (Georgia State University). The result is expressed in meters when speed is in m/s and frequency is in hertz. Always match units before dividing to avoid scale errors.

The SI unit of frequency is the hertz (Hz), defined by BIPM and NIST as one cycle per second, or 1 Hz = 1 s⁻¹. Wavelength is a length, measured in meters (m), the SI base unit. Wave speed combines these as meters per second (m/s), since v = f × λ gives Hz × m = s⁻¹ × m = m/s, which is dimensionally consistent.

Common metric prefixes scale these units:

• 1 kHz = 10³ Hz
• 1 MHz = 10⁶ Hz
• 1 GHz = 10⁹ Hz
• 1 THz = 10¹² Hz

For wavelength, 1 nm = 10⁻⁹ m and 1 ångström (Å) = 10⁻¹⁰ m. The hertz honors physicist Heinrich Hertz, per Encyclopaedia Britannica.

The period T is the time for one complete wave cycle, and it is the reciprocal of frequency: T = 1 / f, measured in seconds. A higher frequency means a shorter period.

Worked example: a 100 MHz signal has a period T = 1 / (1×10⁸) = 1×10⁻⁸ s = 10 nanoseconds.

Conversely, frequency equals one over period: f = 1 / T. Angular frequency, used in oscillation and AC circuit math, is ω = 2πf in radians per second; for 100 MHz, ω ≈ 6.283×10⁸ rad/s.

Khan Academy and HyperPhysics both present period and frequency as inverse quantities, a relationship fundamental to timing, signal processing, and resonance analysis.

The energy of a single photon is directly proportional to its frequency through the Planck relation: E = h × f, where h is Planck's constant, 6.62607015×10⁻³⁴ J·s (an exact value fixed by the 2019 SI redefinition, per BIPM and NIST). You can also write it in terms of wavelength as E = hc / λ.

Worked example: green light at 5.45×10¹⁴ Hz (about 550 nm) carries E = 6.626×10⁻³⁴ × 5.45×10¹⁴ ≈ 3.61×10⁻¹⁹ joules per photon.

Because energy rises with frequency, gamma rays are far more energetic than radio waves. This relationship underpins photoelectric effects, spectroscopy, and solar cell physics as explained by HyperPhysics.

When a wave enters a denser medium, its frequency stays constant but its speed and wavelength decrease. For light, the wavelength in a medium is λ_medium = λ_vacuum / n, where n is the refractive index.

Worked example: 600 nm red light entering glass with n = 1.5 shortens to 600 / 1.5 = 400 nm, while its frequency is unchanged because the light source, not the medium, sets frequency.

The wave speed in the medium is v = c / n, so in n = 1.5 glass light travels at about 2.0×10⁸ m/s. This is why light refracts and lenses focus, a principle detailed by Encyclopaedia Britannica and HyperPhysics. Always specify the medium when reporting wavelength.

Frequency-wavelength conversions drive practical engineering across many fields:

• In radio and antenna design, a quarter-wave antenna length equals λ/4; for a 5 GHz WiFi band, λ = 3×10⁸ / 5×10⁹ = 0.06 m, so the quarter-wave element is about 1.5 cm, per IEEE antenna practice.
• In fiber-optic communications, the 1550 nm telecom band is chosen for minimal glass attenuation.
• Medical ultrasound uses 1-20 MHz sound waves (millimeter-scale wavelengths) to resolve fine tissue detail.
• Astronomers classify sources by spectral region, and audio engineers size acoustic treatment to bass wavelengths of several meters.

Each application depends on the same v = f × λ relationship applied at the correct wave speed for the medium.

While frequency f counts cycles per second, wavenumber describes cycles per unit distance. The angular wavenumber is k = 2π / λ in radians per meter, and the spectroscopic wavenumber is ν̃ = 1 / λ, often given in cm⁻¹.

Worked example: 500 nm light has ν̃ = 1 / (500×10⁻⁷ cm) = 20,000 cm⁻¹.

Wavenumber is popular in chemistry and infrared spectroscopy because it is proportional to photon energy, so higher wavenumber means higher energy. The relationships k = 2π / λ = 2πf / v tie spatial and temporal descriptions together. NIST and IUPAC reference cm⁻¹ as the standard spectroscopic unit, making wavenumber a convenient stand-in for both wavelength and energy.

Several errors trip up frequency and wavelength calculations:

• The most frequent error is mixing units before dividing — for example dividing meters per second by megahertz without converting MHz to Hz, which throws the answer off by a factor of a million.
• Another mistake is using the speed of light for sound problems; sound travels at roughly 343 m/s in air, not 3×10⁸ m/s.
• Many learners also forget that frequency stays fixed when a wave crosses into a new medium, incorrectly recalculating it from the shortened wavelength.
• Confusing period (seconds) with frequency (hertz) is common too; they are reciprocals, not equals.
• Finally, avoid rounding c prematurely — HyperPhysics and NIST recommend carrying full precision until the final step to prevent compounding errors.