Wave Number Calculator

k = 2π/λ (rad/m) for general waves — mechanical, sound, water. Five modes: k from λ, λ from k, k from f & v, v from ω & k, and spectroscopy wavenumber σ̃.

Domain boundary. This tool computes the angular wave number k = 2π/λ for general waves (mechanical, sound, water — not the light spectrum). Spectroscopy wavenumber σ̃ = 1/λ (cm⁻¹) for light/EM spectrum is in mode 5 as a bonus; for the full light-spectrum calculator (color presets, photon energy E=hf, EM band classifier) see the light frequency calculator. The two quantities differ by exactly a factor of 2π — don’t confuse them.

Mode 1 — angular k from wavelength λ

Formula: k = 2π/λ. Result unit: rad/m.

Wavelength λ

λ unit

Mode 2 — Wavelength λ from k (inverse)

Formula: λ = 2π/k.

Wave number k (rad/m)

Mode 3 — k from frequency f and propagation speed v

Formula: k = 2πf/v (equivalent to k = ω/v).

Frequency f

f unit

Propagation speed v (m/s)

Mode 4 — Propagation speed v from ω and k (dispersion)

Formula: v = ω/k. Equivalent to v = fλ.

Angular frequency ω (rad/s)

Wave number k (rad/m)

Mode 5 — Spectroscopy wavenumber σ̃ = 1/λ

Formula: σ̃ = 1/λ. Spectroscopy convention for light/EM spectrum — not angular k (no 2π factor). For the full light spectrum see the light frequency calculator.

Wavelength λ

λ unit

Frequently asked questions

What is the (angular) wave number k?

Angular wave number k measures how densely a wave oscillates in space: k = 2π/λ in radians per meter (rad/m). The value of k states how many radians of phase the wave covers per meter. Verified example: λ = 500 nm gives k = 2π/(5×10⁻⁷ m) ≈ 12,566,370.6144 rad/m.

What is the difference between angular wave number k and spectroscopy wavenumber σ̃?

Angular k = 2π/λ (rad/m) is used for general waves (mechanical, sound, water), while spectroscopy wavenumber σ̃ = 1/λ (cm⁻¹) is the spectroscopy convention for light/EM spectrum. The two differ by exactly a factor of 2π: for λ = 500 nm, k ≈ 12,566,370.6144 rad/m whereas σ̃ = 20,000 cm⁻¹. For the full light-spectrum calculator (color presets, photon energy E=hf, EM band classifier) see the light frequency calculator.

How do I compute wave number k from frequency f and propagation speed v?

Use k = 2πf/v. Since ω = 2πf, this formula is equivalent to k = ω/v. Verified example: sound at f = 440 Hz in air with v = 343 m/s gives k = 2π×440/343 ≈ 8.0601 rad/m.

How are v = ω/k and v = fλ related?

They are the same formula. Substituting ω = 2πf and k = 2π/λ into v = ω/k yields v = (2πf)/(2π/λ) = f·λ. The formula v = ω/k is more useful for the dispersion relation ω(k) in dispersive media, while v = fλ is more practical for waves with a directly measured wavelength.

Which units are used for wave number k?

The SI unit is radians per meter (rad/m). In optical lab practice rad/cm is sometimes used, while spectroscopy uses cm⁻¹ for σ̃ (a different convention, without the 2π factor). If a problem uses other angular units, convert to radians first.

For k = 3 rad/m, what is the wavelength?

Use the inverse λ = 2π/k. For k = 3 rad/m: λ = 2π/3 ≈ 2.0944 m. As a consistency check, entering λ = 2.0944 m back into k mode gives k = 3 rad/m.

When should I use the v-from-ω-and-k mode?

The v = ω/k mode is used to analyze the dispersion relation: how phase velocity depends on wave number in a medium. Example: ω = 10 rad/s and k = 2 rad/m gives v = 5 m/s. If v changes with k, the medium is dispersive.

Verified examples

Example 1. λ = 500 nm → k = 2π/(5×10⁻⁷ m) ≈ 12,566,370.6144 rad/m; spectroscopy wavenumber σ̃ = 20,000 cm⁻¹.

Example 2. Sound f = 440 Hz, v = 343 m/s → k = 2π×440/343 ≈ 8.0601 rad/m.

Example 3. k = 3 rad/m → λ = 2π/3 ≈ 2.0944 m; ω = 10 rad/s, k = 2 rad/m → v = 5 m/s.

All examples were recomputed with 50-digit (Decimal) arithmetic and match the calculator engine above.

Quantity	Value
k in rad/cm	—
k in rad/mm	—
λ in meters (input check)	—

Quantity	Value
λ in nm	—
λ in μm	—
λ in mm	—

Wave Number Calculator

Mode 1 — angular k from wavelength λ

Mode 2 — Wavelength λ from k (inverse)

Mode 3 — k from frequency f and propagation speed v

Mode 4 — Propagation speed v from ω and k (dispersion)

Mode 5 — Spectroscopy wavenumber σ̃ = 1/λ

Frequently asked questions

Verified examples

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