Wave Frequency Calculator

The wave frequency calculator solves the universal wave equation v = f × λ for any of its three variables. The equation relates wave speed (v), frequency (f), and wavelength (λ) for any periodic wave travelling through a uniform medium, sound, light, water, seismic, microwaves, or waves on a string. This page explains the formula, gives the standard reference values for common media, and shows worked examples for sound and light.

Whether you need to find the wavelength of a 100 MHz FM radio station, the frequency of middle C, or the speed of sound in seawater, the math is the same. The calculator handles all three cases in one place, with preset medium speeds for the most common scenarios and full support for custom values.

How to use the wave calculator

1. Select what you want to solve for. The wave equation has three variables; choose frequency, wavelength, or wave speed. The selected variable becomes the output.
2. Enter the wave speed. Either type a custom value in m/s or click one of the preset buttons (Air, Water, Seawater, Steel, Glass, Light in vacuum, Light in water, Light in glass) to load a standard value.
3. Enter the other two known variables. When solving for wave speed, enter both frequency and wavelength. When solving for frequency or wavelength, enter the other one plus the speed.
4. Click Calculate. The result shows the solved value plus the period (T = 1/f) and angular frequency (ω = 2πf) where applicable.
5. Watch the units. The calculator uses SI units throughout: m/s for speed, Hz for frequency, m for wavelength, s for period, rad/s for angular frequency. If you're working in other units (e.g. MHz, km/h, mm), convert to SI first.

The result panel uses exponential notation for very large or very small numbers. A 1 MHz radio signal has a frequency displayed as 1,000,000 Hz; a 100 GHz radar signal is shown as 1×10¹¹ Hz; an AM radio wavelength is shown as 300 m. The formatting adjusts automatically.

The wave equation

For any periodic wave, a repeating disturbance that travels through a medium or through space, three quantities are linked by a single relationship:

v = f × λ

Where:

• v is the wave speed in metres per second (m/s)
• f is the frequency in hertz (Hz), the number of complete cycles per second
• λ (lambda) is the wavelength in metres (m), the distance between two consecutive peaks (or troughs) of the wave

This equation is universal: it works for sound, light, water, seismic, microwaves, radio, x-rays, and any other periodic wave. The constant v depends on the medium; the frequency is set by the source; the wavelength adjusts to fit.

The same equation, rearranged, gives:

• Frequency: f = v / λ
• Wavelength: λ = v / f
• Wave speed: v = f × λ

For example, sound travels through air at 343 m/s. A 440 Hz tuning fork (the A above middle C) has a wavelength of 343 / 440 ≈ 0.78 m. A 30 Hz bass note has a wavelength of 343 / 30 ≈ 11.4 m. Both are audible to humans but the bass note's wavelength is comparable to the size of a room, which is why bass is non-directional and a small speaker struggles to reproduce it.

Related quantities

Two derived quantities are often useful alongside v, f, and λ.

Period (T). The time for one complete cycle. T = 1 / f. A 50 Hz wave (mains electricity in the UK) has a period of 0.02 s = 20 ms. A 1 GHz clock signal has a period of 1 ns. The period is a more natural measure when dealing with timing or pulse generation; frequency is more natural when dealing with tuning, radio, or signal processing.

Angular frequency (ω). The rate of change of phase, measured in radians per second: ω = 2πf. A 50 Hz wave has ω = 2π × 50 = 314.16 rad/s. Angular frequency is convenient in physics because the equations of motion for oscillating systems (springs, pendulums, LC circuits) are most compact when expressed with ω. A mass on a spring with natural angular frequency ω oscillates with displacement x(t) = A cos(ωt + φ) and velocity v(t) = -Aω sin(ωt + φ). Using f instead of ω adds a 2π factor to every term.

The calculator shows both T and ω in the result panel whenever the calculation involves frequency.

Common reference values

Wave speed depends on the medium. For mechanical waves (sound, water, seismic), it depends on the medium's elasticity and density. For light, it depends on the medium's refractive index n: v = c / n. The table below lists standard values at 20°C and 1 atm (where applicable).

For sound in air, the speed increases with temperature: roughly v = 331 + 0.6 × T m/s, where T is the air temperature in °C. The calculator's "Air (20°C)" preset assumes 20°C; if you need a different temperature, enter a custom value.

The speed of light in vacuum (c) is exact by definition since 1983. All other values are measured to several significant figures.

Worked example: middle C on a piano

Middle C is 261.63 Hz. Sound in air at 20°C travels at 343 m/s. Wavelength = 343 / 261.63 ≈ 1.311 m. This wavelength is short compared to a concert grand piano (about 2.7 m long), which is why the piano body can reproduce it efficiently. A flute (about 0.6 m long) can also reproduce middle C: the air column inside the flute resonates when its length is a multiple of half a wavelength, and 0.6 m is roughly half of 1.311 m (plus end corrections).

A bass note at 30 Hz, by contrast, has a wavelength of 343 / 30 ≈ 11.4 m. The wave is much longer than the instrument. This is why subwoofers are large boxes (often a cubic foot or more), the speaker cone must be able to push enough air to create a wave that's about 11 m long. A small speaker simply cannot displace enough air to make a 30 Hz sound at meaningful volume.

Worked example: visible light

Green light has a frequency around 5.7 × 10¹⁴ Hz. In vacuum, its wavelength is:

λ = c / f = 299,792,458 / 5.7 × 10¹⁴ ≈ 5.26 × 10⁻⁷ m = 526 nm

This is in the middle of the visible spectrum, which runs from about 380 nm (violet) to 750 nm (red).

When the same light enters glass with refractive index 1.5, the speed drops to:

v = c / n = 299,792,458 / 1.5 ≈ 2 × 10⁸ m/s

The frequency does not change (it is set by the source), so the wavelength inside the glass is:

λ = v / f = 2 × 10⁸ / 5.7 × 10¹⁴ ≈ 3.51 × 10⁻⁷ m = 351 nm

The wavelength shortens by the same factor the speed reduces. The energy of each photon, E = hf, is unchanged because f is unchanged. (This is why glass slows light without absorbing it: the wave is the same, just compressed.)

Worked example: radio station

For a radio station, use v = c (electromagnetic waves in air are essentially at c, since air's refractive index is 1.0003, negligible). For an FM station at 100 MHz:

λ = c / f = 299,792,458 / 100,000,000 ≈ 3.0 m

For an AM station at 1,000 kHz (1 MHz):

λ = 299,792,458 / 1,000,000 ≈ 300 m

This is why AM radio antennas are physically long, typically 100 m or so for a quarter-wavelength monopole. FM antennas are much shorter (about 0.75 m for a quarter-wave at 100 MHz). The same logic explains why 5G cell towers use small antenna elements: their frequencies are around 28 GHz, with wavelengths around 1 cm.

Why wave speed is set by the medium

For mechanical waves (sound, water, seismic), the wave speed is determined by the medium's elasticity (resistance to deformation) and density (resistance to motion). A stiffer medium transmits force faster; a denser medium takes more force to accelerate. For sound in air, the speed depends on temperature because warmer air is "stiffer" at the molecular level (faster-moving molecules transmit vibrations faster). For sound in water, the speed is determined by the water's bulk modulus and density. For seismic waves, the speed depends on the rock's elastic moduli and density, different rock types (sedimentary vs igneous, deep vs shallow) give different speeds.

For electromagnetic waves (light, radio, x-rays, etc.), the speed in vacuum is c = 299,792,458 m/s, exact by definition. In a medium, the speed is reduced by the medium's refractive index n: v = c / n. The refractive index reflects how the electric and magnetic fields of the wave interact with the electrons in the medium; different materials slow the wave by different amounts.

In both cases, the frequency is set by the source and is unchanged when the wave passes from one medium to another. What changes is the speed and the wavelength, with λ = v / f adjusting to keep the relationship valid in the new medium.

Phase velocity vs group velocity

The wave equation v = f × λ describes the phase velocity of a wave, the speed at which a single point of constant phase (say, a wave crest) moves through the medium. For a pure single-frequency wave, this is the only velocity that exists. For a wave packet (a localised disturbance containing a range of frequencies), there is also a group velocity, the speed at which the overall envelope of the packet moves.

In non-dispersive media (vacuum for light, air for most sound frequencies), the phase and group velocities are equal, and v = f × λ completely describes the wave motion. In dispersive media (light in glass at some frequencies, deep-water ocean waves, electromagnetic waves in waveguides), the phase and group velocities differ, and a full treatment needs more than the single-equation v = f × λ.

For most everyday purposes (sound in air, light in clear media at visible frequencies, water waves in shallow tanks), the medium is essentially non-dispersive and the calculator is exact.

Common mistakes

Mixing frequency and angular frequency. Some equations (especially in physics) use ω = 2πf, not f. If your formula expects ω and you plug in f, your answer will be off by a factor of 2π. The calculator shows both, so you can check.

Using c when the wave is not light. c is the speed of light in vacuum (or very close to it for light in air). It is not the speed of sound. Sound in air travels at 343 m/s, about a million times slower. If you're calculating a sound problem and use 3 × 10⁸ m/s, the answer will be wrong by a factor of about 900,000.

Forgetting to convert units. If you have a frequency in MHz, the wavelength comes out 10⁶ too small if you forget to convert. 100 MHz = 10⁸ Hz, not 100 Hz. Convert to SI first.

Confusing wavelength with amplitude. Wavelength is the spatial period of the wave (distance between consecutive peaks). Amplitude is the height of the wave (peak displacement from zero). They are completely different quantities. The wave equation deals with wavelength, not amplitude.

Ignoring the medium. v changes with the medium. Sound in steel is 17× faster than sound in air. Light in glass is 33% slower than light in vacuum. The calculator's preset media give typical values, but if you're working in an unusual medium, enter a custom v.

Limits of the model

The single-equation v = f × λ applies to:

• Linear, non-dispersive media. The wave speed does not depend on frequency.
• Plane waves or waves in waveguides where the dispersion is negligible. In free space, far from boundaries, this is a good approximation.
• Single-frequency continuous waves. For pulses, the full Fourier analysis is needed; for chirped signals (frequency varying with time), the calculation is more complex.

It does not apply to:

• Shock waves (which are non-linear, with v depending on amplitude).
• Waves in strongly dispersive media (where v depends on f; optical fibres near their zero-dispersion wavelength, deep-water ocean swell, plasma waves).
• Quantum mechanical wave functions (where the "phase velocity" and "group velocity" of a wave packet can exceed c, though information cannot).

standard cases (sound in air, water, metal; light in vacuum, air, water, glass; water waves; radio; microwaves), the calculator is exact.

Frequently Asked Questions

What is the speed of sound in air? 343 m/s at 20°C at sea level, or about 1,235 km/h. It increases with temperature (about 0.6 m/s per °C), so it's slower on a cold morning (331 m/s at 0°C) and faster on a hot day (355 m/s at 40°C). Humidity has a small effect, sound travels slightly faster in humid air because water vapour has a lower molecular weight than nitrogen and oxygen.

How do I calculate the wavelength of a radio station? Use λ = c / f with c = 299,792,458 m/s. For an FM station at 100 MHz, λ ≈ 3.0 m. For an AM station at 1,000 kHz, λ ≈ 300 m. The AM wavelength is comparable to the size of a city block, which is why AM radio can diffract around buildings and travel further than FM.

What is the period of a 50 Hz wave? T = 1 / f = 1 / 50 = 0.02 s = 20 ms. This is the mains electricity frequency in the UK, Europe, Africa, most of Asia, and Australia. In North America the standard is 60 Hz, T = 16.67 ms. The period is also the inverse of the frequency: 60 Hz × 16.67 ms = 1.

Does the frequency change when light enters glass? No. The frequency is set by the source and is invariant across boundaries. What changes is the speed (v = c / n) and the wavelength (λ = v / f). A photon of green light at 526 nm in vacuum has a wavelength of 526 / 1.5 ≈ 351 nm in glass with n = 1.5. The energy E = hf is unchanged.

What's the difference between frequency and angular frequency? Frequency f counts full cycles per second (in Hz). Angular frequency ω counts radians per second. They differ by a factor of 2π: ω = 2πf. Use f for counting (Hz, kHz, MHz, GHz) and human-perceived phenomena; use ω for the math of oscillation (springs, pendulums, RLC circuits), where the differential equations become simpler.

Can I use this for ocean waves? Yes, the equation is universal. Ocean swells are typically described by period (T) rather than frequency; convert via f = 1/T. In deep water, the phase speed of a gravity wave is approximately v = gT / (2π) ≈ 1.56 T m/s (with T in seconds and g = 9.81 m/s²). A 10-second swell has v ≈ 15.6 m/s and λ ≈ 156 m.

What is a hertz? One hertz (Hz) is one cycle per second. Named after Heinrich Hertz, who first demonstrated radio waves in 1887. The unit is part of the SI system and is exact: 1 Hz = 1 / s. Common multiples are kHz (10³ Hz), MHz (10⁶ Hz), GHz (10⁹ Hz), and THz (10¹² Hz).

How fast does light travel in water? About 224,000,000 m/s, or about 75% of c. The reduction is governed by the refractive index of water (n ≈ 1.33). This is what causes the visual effect of objects appearing shallower than they really are when you look at them through water from above, the light bends as it leaves the water, and your brain (assuming straight-line propagation) places the object closer to the surface than it actually is.

Sources and references

• Halliday, D., Resnick, R. & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons., Standard undergraduate text covering waves, sound, and light.
• Hecht, E. (2017). Optics. Pearson., Comprehensive treatment of light, refractive index, and wave propagation.
• Kinsler, L. E., Frey, A. R., Coppens, A. B. & Sanders, J. V. (2000). Fundamentals of Acoustics. John Wiley & Sons., The standard reference for sound in various media.
• NIST (2019). CODATA Recommended Values of the Fundamental Physical Constants., Speed of light in vacuum, exact by definition.
• BIPM (2019). SI Brochure., Defines the hertz and the second.

References

• HyperPhysics, Wave Motion, Georgia State University, the v = f x lambda relationship. http://hyperphysics.phy-astr.gsu.edu/hbase/waveme.html
• NIST Special Publication 811, for the unit relationships between hertz, seconds and metres. https://www.nist.gov/pml/special-publication-811