Angular frequency relates to frequency by: ω = 2πf.
Wave speed and wavelength relate through: v = fλ.
Combining both gives the working equation: λ = 2πv / ω. For light in a medium, phase speed is often approximated as v = c / n.
- Enter angular frequency ω and choose its unit.
- Select a medium preset, or choose a custom refractive index or wave speed.
- Pick the wavelength output unit that matches your work.
- Optionally enable advanced outputs for frequency, wavenumber, period, and energy.
- Press Calculate. Results appear above the form with download buttons.
| Scenario | ω (rad/s) | n | v (m/s) | λ (m) | λ (nm) |
|---|---|---|---|---|---|
| Vacuum, high ω | 1 | 1 | 29979246 | 0.0000018836516 | 1883.6516 |
| Air, visible-range ω | 377 | 1.0003 | 29970255 | 0.00000049949248 | 499.49248 |
| Water, moderate ω | 2 | 1.333 | 22490057 | 0.0000070654597 | 7065.4597 |
| Glass, lower ω | 5 | 1.5 | 19986164 | 0.000025115354 | 25115.354 |
Angular frequency and frequency link
Angular frequency ω is measured in rad/s and connects directly to frequency by ω = 2πf. For a 500 THz optical wave, f ≈ 5×10^14 Hz, so ω ≈ 3.14×10^15 rad/s. This calculator accepts scaled units to reduce typing errors.
Wavelength from speed and ω
Wavelength depends on wave speed v through λ = 2πv/ω. In vacuum v equals c = 299,792,458 m/s. In a medium, v typically falls to c/n. With n = 1.5, the speed becomes about 2.00×10^8 m/s, increasing λ for the same ω.
Typical refractive index values
Refractive index is dimensionless and often ranges from 1.0003 (air) to about 1.33 (water) and 1.50 (common glass). Selecting a preset applies v = c/n automatically. Use a custom n when you know the material, temperature, or wavelength conditions.
Using measured wave speed
Many non‑electromagnetic waves are better described by measured speed instead of n. For sound in air near room temperature, v ≈ 343 m/s. If ω = 2000 rad/s, the wavelength is roughly 1.08 m. Choose the custom speed option for strings, fluids, and structures.
Wavenumber, period, and related outputs
The calculator can also report wavenumber k in rad/m using k = 2π/λ. Large k values indicate tightly spaced wavefronts, common in optics. It also provides the period T = 1/f, useful for timing, sampling rates, and resonant systems that respond to cycle duration. Engineers often compare k across designs to judge wave confinement, propagation loss, and mesh resolution in simulations at high frequencies.
Photon energy for radiation
For electromagnetic waves, photon energy can be estimated using E = h·f. With f = 5×10^14 Hz, E ≈ 3.31×10^-19 J, which is about 2.07 eV. This helps connect wavelength outputs to spectroscopy bands and detector sensitivity.
Practical accuracy and exports
To get reliable results, keep ω positive, choose consistent units, and confirm the medium model. If you enter a custom speed, n is not used. After calculating, use the download buttons to export a clean CSV table or a one‑page PDF summary.
FAQs
What is the difference between ω and f?
Angular frequency ω is measured in rad/s, while frequency f is in Hz. They are linked by ω = 2πf, so one cycle per second equals 2π rad/s.
Why does refractive index change wavelength?
For the same ω, wavelength depends on speed v. In many optical materials v ≈ c/n, so higher n lowers v and reduces λ compared with vacuum.
When should I use custom wave speed?
Use custom speed when the wave is not light or when you already measured v. Examples include sound in air, waves on strings, or vibration waves in solids.
Which wavelength unit should I select?
Pick the unit that matches your context: meters for long waves, millimeters for microwaves, micrometers for infrared, and nanometers for visible and ultraviolet work.
What does the calculator mean by wavenumber k?
Here k is the angular wavenumber in rad/m, defined as k = 2π/λ. It describes how quickly phase changes with distance and is common in wave equations.
Is photon energy relevant for all waves?
Photon energy E = h·f applies to electromagnetic radiation. Mechanical waves like sound do not have photons in the same sense, so treat the energy output as optics‑focused.