Wave Speed Calculator

Period, Frequency, Angular Frequency

The cycle period T = 1/f and angular frequency ω = 2π f appear everywhere from AC circuits to simple harmonic motion. The wavelength is the spatial equivalent: λ = v T.

T = (1)/(f), \quad \omega = 2π f, \quad \lambda = v T = (v)/(f)

Speed in Specific Media

For sound in air at 20 °C, v ≈ 343 m/s (faster in warm air: v ≈ 331 + 0.6 T m/s). For sound in water, v ≈ 1480 m/s; in steel, v ≈ 5960 m/s. Seismic P-waves in earth crust move at ~6–13 km/s. In all mechanical media dp = √(K/ρ) where K is the relevant modulus (bulk for fluids, Young for solids along extension) and ρ is the density.

v_\textmech = √((\textmodulus))/(\rho)

Light in Vacuum and Matter

In vacuum light travels at exactly c = 299 792 458 m/s — the basis of the SI definition of the metre. In a medium the light slows: v = c / n, where n is the refractive index (e.g. n ≈ 1.0003 in air, ≈ 1.33 in water, ≈ 1.5 in typical glass, ≈ 2.42 in diamond).

v_\textlight = c / n, \quad c = 299\,792\,458\ \textm/s

Dispersion

For most media the speed is roughly the same for every frequency of the wave, but in dispersive media (water waves in deep ocean, optical glasses, wave guides) v depends on frequency. This is why prisms split white light and ocean waves sort themselves by period.

v = v(\omega)

Wavenumber and Wave Vector

The wave number k = 2π/λ counts radians of phase per metre. The angular-frequency/wave-number relationship ω(k) is called the dispersion relation. In vacuum light is linear: ω = c k. In matter the same form holds but with c replaced effectively by v.

k = (2π)/(\lambda), \quad \omega_\textvac = c k

What Sets the Speed: the Medium, Not the Source

A wave's speed is a property of the medium it travels through, not of whatever produced it. Shouting louder does not make sound arrive sooner. For mechanical waves the speed is set by a competition between a restoring force and an inertia term, and always takes the same general shape: stiffness over density, under a square root.

On a stretched string the speed is v = √(T/\mu) with tension T and mass per unit length \mu — which is why tightening a guitar string raises its pitch and why the thick bass strings are slower and lower. In a fluid the bulk modulus plays the role of stiffness. The practical consequence for the wave equation v = f\lambda is important: when a wave crosses into a new medium its *frequency* is unchanged, because the boundary oscillates at the driving rate, so the wavelength must change to accommodate the new speed.

v_\textstring = √(\fracT)\mu, \qquad v_\textfluid = √(\fracK)\rho

Typical Wave Speeds

Sound in dry air at 20 °C travels at about 343 m/s, rising roughly 0.6 m/s per degree Celsius — a temperature dependence, not a pressure one. In water sound moves at about 1480 m/s, and in steel at roughly 5900 m/s, because stiffness rises faster than density across those materials.

Electromagnetic waves in vacuum travel at exactly 299,792,458 m/s by definition of the metre. In glass with refractive index 1.5 that drops to about 200,000 km/s, and in a typical optical fibre a signal covers 1000 km in around 5 ms — a floor on latency that no amount of engineering can remove. Seismic P-waves run at 5–8 km/s through crust while S-waves manage 3–4.5 km/s, and it is exactly that difference in arrival times that lets a single seismograph station estimate its distance from an earthquake.

Dispersion, Phase and Group Velocity

When the speed depends on frequency the medium is dispersive, and a pulse made of many frequencies spreads out as it travels. Here two different speeds matter: the phase velocity v_p = \omega/k describes how an individual crest moves, while the group velocity v_g = d\omega/dk describes how the envelope — and therefore the energy and any information — actually travels.

In a non-dispersive medium the two coincide and v = f\lambda tells the whole story. In a dispersive one they differ, sometimes dramatically: this is why a prism splits white light, why deep-water waves of different wavelengths outrun each other so that swell arrives sorted by period, and why long-haul optical fibres need dispersion compensation to stop adjacent pulses smearing into each other. The relation v = f\lambda still holds at each individual frequency; what fails is the assumption that a single speed describes the whole signal.

v_p = (\omega)/(k), \qquad v_g = (d\omega)/(dk)

Worked Example

Inputs

• frequency: 440
• wavelength: 0.78

Result: v = 343.2 m/s — Recognition: speed of sound in air at 20 °C.

Explanation

A 440 Hz sound wave (concert-A pitch) has wavelength λ = v / f = 343 / 440 ≈ 0.78 m in 20 °C air. The wavelength of an audible tone is on the order of centimetres to metres, which is why sound bends easily around obstacles (diffraction). Light at the same 440 Hz would have λ = c/440 = 6.82 × 10⁵ m — far longer than the Earth–Moon distance, so "radio at 440 Hz" is essentially static.

Second Scenario

Inputs

• frequency: 330
• wavelength: 0.78

Result: v = 343.2 m/s — Recognition: speed of sound in air at 20 °C.

Explanation

This scenario uses different inputs (frequency = 330, wavelength = 0.78) to show how changing one variable affects the wave speed result. Run the calculator above with these values to get the exact updated output with step-by-step work.