Wave Speed Calculator v = fλ: Frequency, Wavelength, Speed

About this calculator

The wave speed calculator solves $v = f\lambda$ : given a frequency $f$ and a wavelength $\lambda$ it finds the speed $v$ at which the wave propagates; any two of the three quantities determine the third—how long the sound wave from a 440 Hz tuning fork is in air, how many metres a 100 MHz FM broadcast wavelength spans, how many centimetres Wi‑Fi's 2.4 GHz corresponds to, and how fast a 150 m ocean swell travels.

The equation applies to every periodic wave: sound, water waves, waves on a string, seismic waves, radio and visible light. It is not a law peculiar to one kind of wave but a kinematic description of “a periodic pattern on the move”: in one period the wave advances by one wavelength. The question with real physical content is what determines each of the three quantities: the speed is set by the medium, the frequency by the source, and the wavelength is the result of dividing the two. That is the main thread of this page and the key to refraction, dispersion, musical instruments and antenna dimensions.

What it does not do: it does not compute the period ( $T = 1/f$ , one hand-calculation); it does not derive the speed from medium parameters (temperature, tension, refractive index)—the speed must be entered as a known quantity; it does not distinguish phase velocity from group velocity (they differ in dispersive media; see Principles); it does not handle the Doppler effect; and it does not convert units (convert kHz, MHz, GHz and nm to Hz and m first).

Formula and variables

v = f\,\lambda = \frac{\lambda}{T}

Inverse forms:

\lambda = \frac{v}{f},\qquad f = \frac{v}{\lambda}

Symbol	Meaning	Calculator field	Unit	Range and notes
$f$	frequency: complete waves passing a fixed point per second	Frequency f	Hz $= \text{s}^{-1}$	$f > 0$ ; default 440 (the standard A4)
$\lambda$	wavelength: distance between two adjacent points in the same phase (crest to crest, for example)	Wavelength λ	m	$\lambda > 0$ ; default 0.78
$v$	wave speed: speed at which the waveform (phase) propagates through the medium	Speed v	m/s	output; an input when solving in reverse
$T$	period $= 1/f$	(not on the panel)	s	440 Hz corresponds to 2.27 ms

Dimensional check: $\text{s}^{-1} \times \text{m} = \text{m/s}$ . The hertz is an SI derived unit defined as “one per second”; using dimensional analysis as a formula self-check is covered by Newton's second law.

How to use this calculator

Open the panelthis page corresponds to “Math tools → Formulas → Wave equation” (/#/mathtools/formula?calculator=calculator.phys-wave). On open, “Frequency f” = 440 and “Wavelength λ” = 0.78 are already filled in and “Speed v” carries a “calculate” marker and shows 343.2—this time the panel opens on a meaningful result: the speed of sound in air at 20 °C.
Find the wave speedchange the frequency or the wavelength and the speed updates immediately.
Solve for wavelengthclear “Wavelength λ” and enter “Speed v” (343 for sound in air, 1500 in water, 299792458 for light in vacuum); the wavelength field switches to “calculate” and shows $v/f$ .
Solve for frequencyclear “Frequency f” and enter $v$ and $\lambda$ ; the frequency field shows $v/\lambda$ .
Large and small numbersenter GHz and THz frequencies in full (2.4 GHz as 2400000000). When the absolute result is below $10^{-8}$ or at least $10^{12}$ it is displayed in exponent form (for example 6e-7); otherwise at most 8 decimal places with trailing zeros removed.
Zero valueswith the frequency set to 0, solving for wavelength displays ∞—zero frequency is not a wave. With $v$ and $\lambda$ cleared and only $f$ filled in, both show 0, which is likewise not a result.
Copy the formulathe copy button at the right of the panel title puts the LaTeX v = f \lambda on the clipboard.

Worked examples

All four examples were recomputed by the engine's phys-wave compute routine; the display convention is at most 8 decimal places with trailing zeros removed, in units Hz, m and m/s.

Example 1: a tuning fork—the defaults are the speed of sound

Inputs: $f = 440$ , $\lambda = 0.78$ . $v = 440 \times 0.78 = 343.2$ . Interface shows: wave speed v 343.2. This is the speed of sound in dry air at 20 °C (the empirical formula $331.3\sqrt{1 + \theta/273.15}$ gives 343.2 at 20 °C). The tuning fork vibrates 440 times per second, each time pushing out a 0.78 m compression pattern, and the front of the pattern advances $440 \times 0.78 = 343.2$ m every second. The period is $T = 1/440 = 2.27\ \text{ms}$ .

Figure 1: the sound wave of Example 1. In each period T the wave advances one wavelength λ, so v = λ/T = fλ

Example 2: how long is a sound wave the ear can hear?

Solving for wavelength. Clear $\lambda$ and enter $v = 343$ , $f = 440$ : $\lambda = 343/440 = 0.77954545$ . Interface shows: wavelength λ 0.77954545—differing from the default 0.78 in the third place because 0.78 is a rounded value.

The lower hearing limit $f = 20$ : $\lambda = 343/20 = 17.15$ . Interface shows: 17.15—longer than a classroom. The upper limit $f = 20000$ : $\lambda = 0.01715$ . Interface shows: 0.01715, i.e. 1.7 cm. Three orders of magnitude of frequency correspond to three orders of magnitude of wavelength, because $v$ is fixed and $\lambda \propto 1/f$ —an inverse proportion. Bass wavelengths are much larger than a doorway and bend around obstacles (diffraction), which is why through a wall you hear mainly drums and bass; treble wavelengths of a few centimetres travel in straight lines like light and are blocked. Large bass drivers and small tweeters have the same explanation.

Solving for frequency: a tube open at both ends and 0.65 m long has a fundamental wavelength of twice the tube length, $\lambda = 1.3$ . Clear $f$ and enter $v = 343$ , $\lambda = 1.3$ : $f = 343/1.3 = 263.84615385$ . Interface shows: frequency f 263.84615385—close to middle C (261.6 Hz). Flutes, pan pipes and pipe organs all choose pitch by length in this way.

Example 3: radio, Wi‑Fi and visible light

The speed of an electromagnetic wave in vacuum is $c = 299\,792\,458\ \text{m/s}$ (a defined value). FM broadcast at 100 MHz: $f = 100000000$ , $\lambda = 2.99792458$ : $v = 299792458$ . Interface shows: wave speed v 299792458. Conversely, the wavelength at 100 MHz is exactly 3 m, so an FM half-wave dipole is about 1.5 m.

Wi‑Fi at 2.4 GHz: clear $\lambda$ and enter $v = 299792458$ , $f = 2400000000$ : $\lambda = 0.12491352$ . Interface shows: wavelength λ 0.12491352—12.5 cm, almost identical to a microwave oven (2.45 GHz, 12.2 cm), which is why microwave ovens interfere with Wi‑Fi and why router antennas are only a few centimetres long.

Visible light: $f = 500000000000000$ ( $5 \times 10^{14}$ Hz): $\lambda = 299792458 / 5 \times 10^{14} = 6 \times 10^{-7}$ . Interface shows: 6e-7, i.e. 600 nm, orange light. The frequency range of visible light is about $4 \times 10^{14}$ – $7.5 \times 10^{14}$ Hz, with wavelengths of 750–400 nm.

Example 4: ocean swell and sonar

Ocean waves: a swell with a 10 s period ( $f = 0.1$ ) and a 150 m wavelength. $f = 0.1$ , $\lambda = 150$ : $v = 15$ . Interface shows: wave speed v 15 (54 km/h). The speed of a deep-water wave is set by its wavelength ( $v = \sqrt{g\lambda / 2\pi}$ , giving 15.3 m/s for 150 m), consistent with the example. Note that this is the speed of the waveform; the water itself only circles in place. Surfers chase the waveform.

Sonar: the speed of sound in water is about 1500 m/s and the transducer works at a wavelength of 3 cm. Clear $f$ and enter $v = 1500$ , $\lambda = 0.03$ : $f = 50000$ . Interface shows: frequency f 50000—50 kHz, a common fish-finder band. The wavelength sets the smallest target that can be resolved, so seeing small fish requires a higher frequency and a shorter wavelength, at the cost of stronger absorption by the water and a shorter range.

The inputs of Example 1 can be entered directly into the panel to reproduce it; change the frequency to 880 (an octave up) and see that the wavelength must halve while the speed is unchanged (clear the wavelength, enter 343.2, then change the frequency); change the wavelength to 3.41 to see the speed of 440 Hz in water.

Principle and derivation

One wavelength advanced per period

Take a periodic wave travelling to the right and watch one crest. After one period $T$ the source completes a full vibration and the next crest takes the place of the first—the whole pattern has shifted to the right by exactly one wavelength $\lambda$ . The speed of the pattern is therefore $v = \lambda / T$ . The frequency is the reciprocal of the period, $f = 1/T$ ; substituting gives $v = f\lambda$ .

The derivation uses no assumption about the medium or the type of wave, only periodicity and a translating pattern. That is why it holds for sound, light, water and string waves alike—and also why, being purely kinematic, it cannot tell you what $v$ is.

What determines what

The medium sets the speed. For a string wave $v = \sqrt{F/\mu}$ (tension over linear density); for sound in air $v = \sqrt{\gamma R T / M}$ , depending only on temperature and gas composition, not on pitch or loudness; for electromagnetic waves in vacuum $v$ is always $c$ , and in a medium of refractive index $n$ it is $c/n$ . Nothing the source does can change any of these.
The source sets the frequency. The 440 Hz of a tuning fork is its own natural frequency; the 100 MHz of a broadcast station is the oscillator frequency of the transmitter. When a wave enters a new medium the vibrations on the two sides of the boundary must stay in step, so the frequency cannot jump.
The wavelength is a result. $\lambda = v/f$ : the medium supplies $v$ , the source supplies $f$ , and the wavelength follows. The same tuning fork produces sound of three different wavelengths in air, water and steel.

These three rules explain why antenna dimensions change with band (Example 3), why instruments choose pitch by length (Example 2), and why the frequency is unchanged on refraction (the note).

Units: the hertz and the second

The hertz is an SI derived unit, $1\ \text{Hz} = 1\ \text{s}^{-1}$ , adopted by the 11th General Conference on Weights and Measures in 1960 to replace “cycles per second”. The second itself has been defined since 1967 by the caesium-133 ground-state hyperfine transition at exactly 9 192 631 770 Hz—frequency is today's most precisely measured physical quantity—and the speed of light $c$ was in turn defined as exactly 299 792 458 m/s in 1983, with the metre derived from $c$ and the second. That $f\lambda$ in Example 3 equals $c$ exactly is not a coincidence but the definition of the metre.

Dispersion: when the wave speed depends on frequency

The statement that “the medium sets the speed” implicitly assumes it is independent of frequency. This is almost exactly true for sound in air and light in vacuum, but not in dispersive media: violet light is slower than red in glass (hence prism spectra), and long waves are faster than short ones in deep water (the $v = \sqrt{g\lambda/2\pi}$ of Example 4, which is why long swells from a distant storm arrive first). Under dispersion $v = f\lambda$ still holds point by point, but the $v$ computed is the phase velocity—the speed of a single-frequency pattern; a wave packet made of many frequencies travels as a whole at the group velocity, and the two can differ greatly (for deep-water waves the group velocity is only half the phase velocity). This calculator computes phase velocity only.

A little history

In 1636 Mersenne summarised the relations between a string's frequency, length, tension and linear density from experiments with vibrating strings—the earliest quantitative law of waves. In 1687 Newton computed the speed of sound in air as about 979 feet per second (298 m/s) under an “isothermal” assumption, about 15% below measurement; in 1816 Laplace pointed out that the compression of sound is adiabatic, and only after including the $\sqrt{\gamma}$ factor did theory match experiment—the first precise confirmation that the speed is set by the properties of the medium. In 1887 Hertz measured the wavelength of radio waves with standing waves and multiplied by the known oscillator frequency to obtain the speed of light, confirming Maxwell's prediction that light is an electromagnetic wave: the most famous application of $v = f\lambda$ .

Assumptions

Periodic single-frequency wave: the pattern translates without changing shape; pulses and noise have no unique wavelength.
Non-dispersive, or phase velocity is the quantity of interest: $v$ is independent of $f$ , or the result is accepted as a phase velocity.
Uniform medium: the wave speed is the same all along the path; across media, calculate each segment separately.
SI units: Hz, m, m/s; no conversion is performed.
Interface conventions: in inverse modes the engine divides directly, $v/(f)$ or $v/(\lambda)$ ; an unfilled $v$ participates as 0, and $f = 0$ or $\lambda = 0$ displays ∞.

Scope and limitations

Can calculate: any one of $f$ , $\lambda$ , $v$ from the other two, for any kind of periodic wave.
Approximate only: in dispersive media the result is the phase velocity; the speed of sound varies with temperature (about +0.6 m/s per °C), so 343 is accurate only at 20 °C.
Cannot calculate: the speed from medium parameters (each kind of wave needs its own formula); group velocity; frequency after a Doppler shift ( $v = f\lambda$ holds in the medium's frame; a moving observer sees a changed $f$ , but $\lambda$ and $v$ in the medium are unchanged); the “speed” of a standing wave (a standing wave does not propagate, although its $\lambda_n = 2L/n$ and $f_n$ still satisfy $v = f\lambda$ ).
Does not: convert units; compute the period $T = 1/f$ .
Numerics: very large or very small values are shown in exponent form; substituting an 8-decimal inverse result back leaves a rounding tail in the last digit.

Common pitfalls

Assuming a higher frequency means a faster wave: in one medium all frequencies of sound travel at the same speed (otherwise a distant band would be heard treble-first). Higher frequency only means a shorter wavelength.
Changing the frequency when crossing a medium: what changes on entering a new medium is $v$ and $\lambda$ ; $f$ stays put (the note).
Entering kHz or MHz as plain numbers: 2.4 GHz entered as 2.4 gives a wavelength of $1.25 \times 10^8$ m. Convert to Hz first.
Entering nm as m: 600 nm is $6 \times 10^{-7}$ m.
Taking the wave speed for the speed of the medium: the water in Example 4 only circles in place; the air does not flow as a whole when sound passes.
Confusing wavelength with amplitude: $\lambda$ is the distance between adjacent points in the same phase along the direction of travel, and has nothing to do with how “tall” the wave is (loudness, brightness).
Treating one speed as the only speed in a dispersive medium: deep-water waves and light in a prism give a phase velocity; energy travels at the group velocity.

Typical use cases

Teaching

Deriving $v = f\lambda$ and checking dimensions; the medium sets the speed and the source the frequency; refraction and dispersion; the speed of sound and temperature.

Acoustics and music

Instrument tube and string lengths versus pitch; driver sizes versus frequency bands; room dimensions and low-frequency standing waves (resonance when $\lambda/2$ matches a room dimension).

Wireless and communications

Frequency band versus antenna size (half-wave dipole $\lambda/2$ ); why Wi‑Fi, Bluetooth and microwave ovens share the neighbourhood of 2.4 GHz; satellite and radar wavelengths.

Oceanography and geophysics

Swell period, wavelength and arrival-time estimates; resolution and frequency choice in sonar and ultrasonic testing; locating earthquake sources from the different speeds of seismic wave types.

FAQ

Why is the wave speed 343.2 when the panel opens?

The defaults 440 Hz and 0.78 m multiply to exactly the speed of sound in air at 20 °C—the default scenario is meaningful in itself, unlike other formula panels that open on 0.

How do I compute the period?

$T = 1/f$ . 440 Hz corresponds to 0.00227 s (2.27 ms); the panel does not display the period.

Why is the speed of sound 343? Does it change?

343 m/s is the value for dry air at 20 °C. The speed of sound rises with temperature (about 331 at 0 °C and 349 at 30 °C), hardly changes with pressure, and depends very little on humidity. In water it is about 1480–1500, and in steel about 5000–6000.

Does the frequency change when light enters water? Does the colour?

The frequency is unchanged and the speed and wavelength shrink by a factor of 1/1.333. Colour is set by frequency, so it does not change.

Does a high-frequency wave travel faster?

Not in a single non-dispersive medium. All sound frequencies travel at the same speed in air, and all colours of light at the same speed in vacuum. Only in dispersive media (glass, deep water) does the speed depend on frequency, and then the result is a phase velocity.

Can the panel compute the speed of sound from temperature?

No—the wave speed must be entered as a known quantity. The empirical formula $v \approx 331.3\sqrt{1 + \theta/273.15}$ ( $\theta$ in degrees Celsius) can be evaluated by hand first.

References and further reading

The phys-wave definition in the CalcX engine source src/data/formulasScience.ts (compute(known, solveFor) solves for v / f / λ, with defaults f = 440, λ = 0.78); every figure on this page was recomputed by that engine.
Wikipedia, Wave （访问日期：2026-09-09）—definitions of wavelength, frequency, phase velocity and group velocity.
Wikipedia, Speed of sound （访问日期：2026-09-09）—the temperature dependence of sound speed, the Newton–Laplace formula and tables of speeds in various media.
Wikipedia, Dispersion (water waves) （访问日期：2026-09-09）—the deep-water $v = \sqrt{g\lambda/2\pi}$ and the derivation that the group velocity is half the phase velocity.
NIST, CODATA 2018: speed of light in vacuum （访问日期：2026-09-09）— $c = 299\,792\,458\ \text{m/s}$ (exact value).
BIPM, The International System of Units (SI), 9th edition （访问日期：2026-09-09）—the definition of the hertz, the caesium frequency definition of the second and the light-speed definition of the metre.
Isaac Newton, Principia (1687), Book II, Section VIII, translated by Motte （访问日期：2026-09-09）—the first theoretical calculation of the speed of sound.

Privacy

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Sources & review

Reviewed by: CalcX 编辑组
Updated: 2026-09-09

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