# What is the wave speed?

Wave speed calculator: v = f × λ from frequency and wavelength, λ ÷ T from the period, distance ÷ time, or √(F ÷ μ) on a stretched string, with the period, frequency and string fundamental.

- Page: https://www.acalculator.org/physics/wave-speed-calculator
- JSON spec: https://www.acalculator.org/physics/wave-speed-calculator.json
- Version: 84115fd71b90

## Default answer

Example with the default inputs (From Frequency and wavelength, Frequency (f) 2 Hz, Wavelength (λ) 8.20209973753281 ft): The wave speed is 5 m/s.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| by | From | What you know about the wave. |
| f | Frequency (f) | How many waves pass a point each second. |
| wl | Wavelength (λ) | The distance from one crest to the next. |
| p | Period (T) | The time for one full wave to pass a point. |
| d | Distance | How far a crest travels. |
| t | Time | How long the crest takes to travel that distance. |
| F | Tension (F_T) | The pulling force in the string. |
| mu | Linear density (μ) | The mass of the string per unit length, in the unit chosen below. |
| muu | Linear density unit | The unit of the linear density you type. |
| len | String length (L) | Optional: the length between the two fixed ends, for the fundamental frequency. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| speed | Wave speed | How fast the wave travels, in metres per second. |
| kmh | Wave speed (km/h) | The same speed in kilometres per hour. |
| fps | Wave speed (ft/s) | The same speed in feet per second. |
| mph | Wave speed (mph) | The same speed in miles per hour. |
| frequency | Frequency (Hz) | f: as typed, 1 ÷ T from the period, or the fundamental v ÷ 2L of a string of length L. |
| period | Period (s) | T = 1 ÷ f, the time for one full wave. |
| wavelength | Wavelength (m) | λ: as typed, or 2L for the fundamental of a string of length L. |
| omega | Angular frequency (rad/s) | ω = 2π × f. |
| k | Wave number (rad/m) | k = 2π ÷ λ. |

## Method

v = f × λ = λ ÷ T; v = distance ÷ time; on a string v = √(F_T ÷ μ), with f₁ = v ÷ 2L and λ₁ = 2L for a string of length L; T = 1 ÷ f, ω = 2πf, k = 2π ÷ λ.

## Assumptions

- A wave of one frequency travelling at a steady speed; the string is uniform and perfectly flexible.
- For a string, the fundamental assumes both ends are fixed (nodes at each end).

## Worked examples

1. by = dt, d = 30, t = 6 gives speed = 5, kmh = 18. Source: OpenStax, University Physics Volume 1, §16.1 Traveling Waves (Equation 16.1, v = λ ÷ T = λf; Examples 16.1 and 16.2), https://openstax.org/books/university-physics-volume-1/pages/16-1-traveling-waves (retrieved 2026-10-18), Example 16.1: v = 30.00 m ÷ 6.00 s = 5.00 m/s.
2. by = fl, f = 2, wl = 2.5 gives speed = 5, p = 0.5, f = 2, wl = 2.5, omega = 12.566371. Source: OpenStax, University Physics Volume 1, §16.1 Traveling Waves (Equation 16.1, v = λ ÷ T = λf; Examples 16.1 and 16.2), https://openstax.org/books/university-physics-volume-1/pages/16-1-traveling-waves (retrieved 2026-10-18), Example 16.1: T = 1 ÷ 2.00 Hz = 0.50 s and λ = vT = 2.50 m, so v = fλ = 5.00 m/s.
3. by = period, wl = 0.08, p = 4 gives speed = 0.02, f = 0.25, k = 78.539816. Source: OpenStax, University Physics Volume 1, §16.1 Traveling Waves (Equation 16.1, v = λ ÷ T = λf; Examples 16.1 and 16.2), https://openstax.org/books/university-physics-volume-1/pages/16-1-traveling-waves (retrieved 2026-10-18), Example 16.2: λ = 8.00 cm, T = 4.00 s, v = 2.00 cm/s, f = 0.25 Hz.
4. by = string, F = 56.4, mu = 0.000309, muu = kg-m gives speed = 427.228594. Source: OpenStax, University Physics Volume 1, §16.3 Wave Speed on a Stretched String (Equation 16.8, v = √(F_T ÷ μ); Example 16.5), https://openstax.org/books/university-physics-volume-1/pages/16-3-wave-speed-on-a-stretched-string (retrieved 2026-10-18), Example 16.5: v = √(56.40 N ÷ 3.09 × 10⁻⁴ kg/m) = 427.23 m/s.
5. by = string, F = 19.6, mu = 0.006, muu = kg-m, len = 2 gives speed = 57.154761, f = 14.28869, wl = 4. Source: OpenStax, University Physics Volume 1, §16.6 Standing Waves and Resonance (Equations 16.15 and 16.16, λ₁ = 2L, f₁ = v ÷ 2L; Example 16.7), https://openstax.org/books/university-physics-volume-1/pages/16-6-standing-waves-and-resonance (retrieved 2026-10-18), Example 16.7: v = √(19.6 ÷ 0.006) = 57.15 m/s, λ₁ = 4.00 m, f₁ = 14.29 Hz.

## FAQ

### How do you calculate wave speed?

Multiply the frequency by the wavelength: v = f × λ. A wave of 2 Hz with a wavelength of 2.5 m travels at 2 × 2.5 = 5 m/s. If you know the period T instead of the frequency, divide: v = λ ÷ T.

### What is the formula for the speed of a wave on a string?

v = √(F_T ÷ μ), with F_T the tension in newtons and μ the mass per metre of the string in kg/m. A guitar string under 56.4 N with μ = 0.000309 kg/m carries waves at √(56.4 ÷ 0.000309) = 427.2 m/s.

### How do you find wave speed from distance and time?

Divide the distance a crest travels by the time it takes. A crest that moves 30 m in 6 s travels at 30 ÷ 6 = 5 m/s.

### Does a higher frequency make a wave faster?

No. The speed is set by the medium (the string tension and mass, or the air). At a higher frequency the waves are shorter, so f × λ stays the same.

### What is the difference between frequency and period?

The frequency f is how many waves pass a point each second, in hertz. The period T is the time for one wave, in seconds. Each is 1 divided by the other: T = 1 ÷ f.

### How do you get the note a string plays from its wave speed?

A string fixed at both ends plays its lowest note when its length holds half a wavelength: λ₁ = 2L and f₁ = v ÷ 2L. A 2 m string with v = 57.15 m/s plays f₁ = 57.15 ÷ 4 = 14.29 Hz.

## Sources

- OpenStax, University Physics Volume 1, §16.1 Traveling Waves: Equation 16.1, v = λ ÷ T = λf, f = 1 ÷ T; Examples 16.1 and 16.2, retrieved 2026-10-18. https://openstax.org/books/university-physics-volume-1/pages/16-1-traveling-waves
- OpenStax, University Physics Volume 1, §16.3 Wave Speed on a Stretched String: Equation 16.7, μ = m ÷ l; Equation 16.8, v = √(F_T ÷ μ); Example 16.5, retrieved 2026-10-18. https://openstax.org/books/university-physics-volume-1/pages/16-3-wave-speed-on-a-stretched-string
- OpenStax, University Physics Volume 1, §16.6 Standing Waves and Resonance: Equations 16.15 and 16.16, λₙ = 2L ÷ n, fₙ = nv ÷ 2L; Example 16.7, retrieved 2026-10-18. https://openstax.org/books/university-physics-volume-1/pages/16-6-standing-waves-and-resonance
- NIST Special Publication 811 (2008), Appendix B.8: 1 ft = 0.3048 m, 1 in = 0.0254 m, 1 mi = 1,609.344 m, 1 lb = 0.45359237 kg, 1 lbf = 4.4482216152605 N, 1 kgf = 9.80665 N, retrieved 2026-10-18. https://www.nist.gov/pml/special-publication-811