Home › Calculators › Wave Speed Calculator
Wave Speed Calculator
Solve the wave equation v = f × λ for any one of the three, get wave speed from a distance and travel time, or work out how fast a wave runs along a stretched string from its tension and linear density.
Last updated:
Method
kg/m
g
m
m
Fixed-to-fixed length; gives the fundamental and harmonics
Hz
Also show the wavelength at this frequency
Enter your values to see the result.
Advertisement
Advertisement
How to use the Wave Speed Calculator
- Pick a method: wave equation, distance and time, or string.
- Choose what to solve for and enter the other two values with units.
- For a string, enter the tension and either μ or the string's mass and length.
- Add a vibrating length to see the fundamental and harmonics.
- Copy the results or download the table as CSV.
Wave speed formulas
- Wave equation: v = f × λ, so f = v / λ and λ = v / f
- From a measurement: v = d / t
- Transverse wave on a string: v = √(T / μ), with μ = string mass ÷ length
- Standing waves on a string fixed at both ends: fₙ = n × v / (2L), λₙ = 2L / n
- Period and wave number: T = 1 / f, ω = 2πf, k = 2π / λ
Example: you see lightning and hear thunder 2.92 seconds later from a strike 1 km away. v = 1,000 / 2.92 ≈ 342 m/s, which matches the speed of sound in air at about 20 °C. This is the basis of the "count seconds, divide by 3 for kilometres" rule.
Speed of waves in common media
| Wave and medium | Approximate speed |
|---|---|
| Sound in air, 0 °C | 331 m/s |
| Sound in air, 20 °C | 343 m/s |
| Sound in fresh water, 20 °C | 1,482 m/s |
| Sound in steel (longitudinal) | about 5,900 m/s |
| Seismic P-waves in the crust | about 5,000–7,000 m/s |
| Deep-water ocean swell, 10 s period | about 15.6 m/s |
| Light and radio in vacuum | 299,792,458 m/s |
Tips
- Speed depends on the medium, not the source. When a wave moves into a new medium its frequency stays the same while its speed and wavelength change.
- Sound in air speeds up by roughly 0.6 m/s for each °C of warming.
- Doubling the tension on a string raises its wave speed and pitch by a factor of √2, about 41%.
Frequently asked questions
What is the formula for wave speed?
Wave speed equals frequency times wavelength: v = f × λ. It can also be found directly as distance divided by travel time, v = d / t.
How do I find wavelength from speed and frequency?
Divide the speed by the frequency: λ = v / f. A 440 Hz tone in air at 343 m/s has a wavelength of 343 / 440 ≈ 0.78 m.
How fast does a wave travel on a string?
v = √(T / μ), where T is the tension in newtons and μ is the mass per metre in kg/m. A string at 70 N with μ = 6.3 g/m carries waves at about 105 m/s.
Does frequency change the speed of a wave?
Usually not. In a given medium the speed is set by the medium's properties; raising the frequency shortens the wavelength instead. Exceptions are dispersive waves such as deep-water ocean waves.
What is the difference between period and frequency?
Frequency is the number of cycles per second (hertz); the period is the time for one cycle. They are reciprocals: T = 1 / f.
How do I find the frequency of a guitar string?
The fundamental is f₁ = v / (2L) with L the vibrating length. Combining this with v = √(T/μ) gives f₁ = (1 / 2L) × √(T / μ). Higher harmonics are whole-number multiples of f₁.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.
Related tools
Wavelength Frequency CalculatorConvert frequency to wavelength and back for light, radio or sound, with photon energy and band.Frequency ConverterConvert Hz, kHz, MHz, GHz, RPM, rad/s, BPM and period, with wavelength in air or vacuum.Note Frequency CalculatorConvert notes to Hz and Hz to the nearest note with cents, MIDI number and adjustable A4.Speed Distance Time CalculatorSolve for speed, distance or travel time and get your arrival time.Half Life CalculatorSolve radioactive decay for amount left, half-life, time or initial amount, with a curve.Kinetic Energy CalculatorSolve KE = ½mv² and PE = mgh for energy, mass, speed or height, with unit conversions.Projectile Motion CalculatorRange, max height, flight time and impact speed of a projectile, with an animated trajectory.Free Fall CalculatorFall time and impact velocity from a height, or height from time or speed, with a chart and table.Meeting Cost CalculatorWork out what a meeting really costs from attendees and salaries, with a live cost ticker.Velocity CalculatorSolve v = d/t and a = Δv/t for any variable, plus a SUVAT solver for any three kinematics values.Torque CalculatorSolve τ = r·F·sin θ for torque, force, lever arm or angle, and convert torque to power at any rpm.Acceleration CalculatorAcceleration from velocity change and time, distance and time, or force and mass.Force CalculatorSolve F = ma for force, mass or acceleration, plus weight on every planet.Density CalculatorSolve density, mass or volume with unit conversion and a table of common material densities.Momentum CalculatorCalculate momentum (p = mv), mass, velocity, impulse and momentum change with unit conversion.Potential Energy CalculatorGravitational PE = mgh and spring PE = ½kx²: solve for energy, mass, height, k or stretch.Power CalculatorCalculate power from work and time, force and velocity, or volts and amps, in W, kW and hp.PPI CalculatorPixel density (PPI), dot pitch and retina viewing distance from resolution and screen size.Download Time CalculatorHow long a file takes to download at your internet speed, or the speed you need.Pressure CalculatorSolve P = F/A for pressure, force or area in Pa, kPa, bar, psi and atm.Work CalculatorSolve W = F·d·cos θ for work, force, distance or angle in J, kWh and ft-lb.Hooke's Law CalculatorSolve F = kx for spring force, spring constant or extension, plus stored spring energy.Centripetal Force CalculatorCentripetal force and acceleration from mass, radius and speed or RPM: F = mv²/r.Gravitational Force CalculatorNewton's law of gravitation F = Gm₁m₂/r² for any two masses, with planet presets.Speed of Sound CalculatorSpeed of sound from air temperature or medium, and distance from an echo or thunder delay.Doppler Effect CalculatorObserved frequency from source and observer speeds, or source speed from a pitch shift.Specific Heat CalculatorSolve Q = mcΔT for heat energy, mass, specific heat or temperature change, with materials.Decibel CalculatorAdd dB levels, convert power or voltage ratios to dB, and find sound level at a distance.Bernoulli Equation CalculatorSolve Bernoulli's equation for pressure, velocity or height, with pipe continuity and flow.Buoyancy CalculatorBuoyant force from fluid density, volume and gravity, with a float or sink check.Escape Velocity CalculatorEscape and orbital velocity for any planet, moon or custom mass and radius.Friction Force CalculatorStatic and kinetic friction from material presets, normal force or mass and incline angle.Inclined Plane CalculatorForces, acceleration and push force for a mass on a slope with friction.Light Travel Time CalculatorHow long light or radio takes to reach the Moon, Sun, Mars, Voyager 1 or any distance.E=mc2 CalculatorConvert mass to energy and back with E=mc², in joules, kWh, eV and tons of TNT.Orbital Period CalculatorOrbital period from central mass and semi-major axis with Kepler's third law and presets.Pendulum Period CalculatorPendulum period and frequency from length and gravity, or length from period.Photon Energy CalculatorPhoton energy in eV and joules from wavelength or frequency, with momentum and color.Spring Rate CalculatorCompression spring rate from wire size, coil diameter, coils and material, plus solid height.Standard Atmosphere (ISA) CalculatorISA 1976 temperature, pressure, density, speed of sound and viscosity at any altitude to 86 km.Steam Table CalculatorSaturated and superheated steam properties: enthalpy, entropy, volume and quality (IAPWS-IF97).Superheat and Subcooling CalculatorFind superheat, subcooling and target values from gauge pressures and line temperatures.Thermal Expansion CalculatorLinear, area or volume expansion from size, temperature change and material coefficient.Top of Descent CalculatorWhen to start descending: distance and time to top of descent, required fpm, 3-degree rule.True Airspeed CalculatorTrue airspeed, Mach number and density altitude from IAS/CAS, pressure altitude and OAT.Valve Cv CalculatorSize a valve: required Cv or Kv for liquid, gas or steam, or flow and pressure drop from Cv.Water Heating Time CalculatorHow long to heat water: time, energy and cost from volume, temperatures and heater power.Snell's Law CalculatorRefraction angle, critical angle and total internal reflection from n₁, n₂ and the angle.Speeds and Feeds CalculatorSpindle RPM and feed rate (mm/min, in/min) from tool size, flutes, surface speed and chip load.Terminal Velocity CalculatorTerminal velocity and time to 95% from mass, drag coefficient, area and air density.Thin Lens Equation CalculatorSolve 1/f = 1/do + 1/di for any value, with magnification, image type and a ray diagram.Time Dilation CalculatorLorentz factor, dilated time and length contraction from speed and proper time.Displacement CalculatorDisplacement from velocity and time, s = ut + ½at², or start and end coordinates, with steps.Kinematics Calculator (SUVAT)Enter any 3 of s, u, v, a and t to solve the other 2 with the SUVAT equations shown.3D Printer Filament Length CalculatorFilament weight to length and back for PLA, PETG, ABS and more, plus spool remaining.Fuel Injector Size CalculatorInjector size in lb/hr and cc/min from target horsepower, BSFC, cylinders, duty cycle and fuel.Hull Speed CalculatorTheoretical displacement hull speed from waterline length, in knots, mph and km/h.OEE CalculatorOverall equipment effectiveness from time, ideal cycle time and counts: A × P × Q.Scuba Air Consumption CalculatorSAC and RMV from a logged dive, then gas needed, turn pressure and bottom time for the next.Takt Time CalculatorTakt time from available time and demand, plus stations needed and a bottleneck check.Wind Turbine Power CalculatorWind turbine power output and annual kWh from rotor size, wind speed, air density and Cp.Air Density CalculatorMoist air density in kg/m³ and lb/ft³ from temperature, pressure or altitude, and humidity.Angular Velocity CalculatorFind angular velocity in rad/s, deg/s and RPM from angle and time, RPM or linear speed.Antipode MapFind the exact opposite point on Earth for any place, shown on a dual map with land or ocean.Average Velocity CalculatorAverage speed and average velocity over several trip segments, with displacement and direction.Barometric Pressure TodayCurrent air pressure near you with a rising or falling trend and a 48-hour chart in hPa or inHg.Carbon Dating CalculatorEstimate age from percent carbon-14 remaining or activity, or find C-14 left after any age.Coulomb's Law CalculatorElectrostatic force between two charges, attractive or repulsive; solve for charge or distance.