Frequency, wavelength and period: λ = v / f, velocity factors and antenna lengths
A wave's frequency, its period and its wavelength are one fact written three ways. The period is how long one cycle takes, T = 1/f; the wavelength is how far the wave travels in that time, λ = v/f, where v is the speed of the wave in whatever it is travelling through: c for radio in air, a fraction of c inside a cable, and only 343 m/s for sound. From the wavelength come the sizes of things that interact with the wave: a dipole is half of it, a whip a quarter, and a PCB trace longer than a tenth of it stops being a wire and becomes a transmission line. This calculator converts any of the three into the others for a chosen medium and gives the antenna lengths and the band.
How to use the wavelength calculator
- Type the one figure you know and say which it is: a frequency (2.4G, 100M), a wavelength in metres (2, 125m for millimetres) or a period (20m for milliseconds, 1u). Switching the selector converts the value you typed.
- Pick the medium in the menu row: free space, a coax or twisted pair with its velocity factor, FR-4 microstrip, or sound in air, water or steel.
- The drawing shows two wavelengths with λ, λ/2 and λ/4 bracketed and a dipole and a whip drawn to them; the panel places the frequency on the spectrum and lists the figures. The Common bands tab tabulates familiar frequencies in the same medium.
The formulas
λ = v / f T = 1 / f ω = 2πf c = 299 792 458 m/s
In free space v = c, so a handy form is λ(m) = 300 / f(MHz): 100 MHz is 3 m, 1 GHz is 30 cm, 2.4 GHz is 12.5 cm. Light covers 30 cm in a nanosecond, which is why signal timing on a board and in a cable is measured in nanoseconds per metre. The angular frequency ω = 2πf is the same frequency in radians per second, the form that reactance and phasor sums use.
Velocity factor and sound
Inside a cable the wave travels through the dielectric, and its speed drops to a fraction of c called the velocity factor: 0.66 for solid-polyethylene coax such as RG-58 and RG-213, about 0.82–0.85 for foam-dielectric cable such as LMR-400, around 0.65 for twisted pair and roughly 0.5–0.6 for a microstrip on FR-4. Anything cut to a wavelength inside the cable, a quarter-wave stub, a half-wave balun, a delay line, is shorter by that factor. Sound is a different matter: 343 m/s in air at 20 °C, 1480 m/s in water, about 5960 m/s in steel, so a 1 kHz tone is 34 cm long in air and a 40 kHz ultrasonic ranging pulse 8.6 mm.
| Medium | Speed | λ at 100 MHz | λ/4 |
|---|---|---|---|
| Free space / air (radio, c) | 300 Mm/s (1 c) | 2.998 m | 74.9 cm |
| Solid-PE coax, RG-58 / RG-213 (0.66 c) | 198 Mm/s (0.66 c) | 1.979 m | 49.5 cm |
| Foam coax, LMR-400 (0.82–0.85 c) | 252 Mm/s (0.84 c) | 2.518 m | 63 cm |
| Twisted pair, Cat 5 (0.65 c) | 195 Mm/s (0.65 c) | 1.949 m | 48.7 cm |
| FR-4 microstrip (about 0.55 c) | 165 Mm/s (0.55 c) | 1.649 m | 41.2 cm |
Antenna lengths
dipole ≈ 0.95 × λ/2 whip ≈ 0.95 × λ/4 (143 / fMHz metres and 71.5 / fMHz metres)
A half-wave dipole resonates when each arm is a quarter wavelength, but a real wire is electrically a little longer than its physical length (its thickness, its ends, the insulation), so the cut length is about 5 % less than λ/2: the old formula 468 / f(MHz) feet, or 143 / f(MHz) metres. A quarter-wave whip is the same thing over a ground plane: half the dipole. The calculator's antenna figures use the free-space wavelength with the 0.95 factor; for a thick element or a loaded antenna trim to resonance with an analyser.
The λ/10 rule
When a wire, trace or lead is much shorter than the wavelength, the voltage is the same all along it and it behaves as a plain conductor. Once it is longer than about a tenth of the wavelength (in the board's material, so divide by the velocity factor), the voltage differs along its length and it is a transmission line: it needs a controlled impedance and a matched termination or it reflects. At 100 MHz that is about 15 cm on FR-4; at 2.4 GHz about 7 mm, which is why RF layouts look the way they do. The same rule says when a digital edge matters: a 1 ns rise time has energy up to about 350 MHz, so a trace over 4 cm or so wants termination.
The bands
The ITU names the radio spectrum by decades: VLF 3–30 kHz, LF 30–300 kHz (long wave), MF 300 kHz–3 MHz (medium wave), HF 3–30 MHz (short wave), VHF 30–300 MHz (FM, the 2 m band), UHF 300 MHz–3 GHz (TV, mobile, Wi-Fi 2.4 GHz, the 70 cm band), SHF 3–30 GHz (microwave, 5 GHz Wi-Fi, satellite), EHF 30–300 GHz (millimetre wave, 5G high bands). The amateur bands are named by wavelength (160, 80, 40, 20, 15, 10, 6, 2 m, 70 cm, 23 cm), which is the same thing the other way round.
Your wave, step by step
- Frequency and period: 2.4 GHz, T = 1/f = 416.7 ps, ω = 15.1 Grad/s.
- Wavelength: 300 Mm/s / 2.4 GHz = 12.49 cm; λ/2 6.25 cm, λ/4 3.12 cm.
- Antennas: wire dipole 5.93 cm, quarter-wave whip 2.97 cm; UHF (300 MHz–3 GHz): TV, Wi-Fi, 70 cm.
Worked example: 2.4 GHz Wi-Fi
At 2.4 GHz the period is 1 / 2.4 × 10⁹ = 416.7 ps and the free-space wavelength c / f = 299 792 458 / 2.4 × 10⁹ = 12.49 cm. A half wave is 6.246 cm and a quarter wave 3.123 cm: the little whip on a router or a module's PCB trace antenna is a quarter wave, cut to about 2.97 cm in wire, and a dipole would be 5.93 cm tip to tip. A tenth of the wavelength is 1.25 cm, so on FR-4 (velocity factor about 0.55) any trace longer than 6.87 mm to the antenna must be a 50 Ω controlled-impedance line. The band is UHF, between 300 MHz and 3 GHz. In RG-58 coax the same signal has a wavelength of only 8.244 cm.
Questions
Why is the dipole shorter than λ/2?
The end effect: a real wire is electrically about 5 % longer than its length, so it is cut 5 % short to resonate. Thick elements need a little more.
Does the wavelength change in a cable?
Yes: the frequency stays the same but the wave is slower, so the wavelength shrinks by the velocity factor. Stubs and matching sections are cut to the cable's wavelength.
What is the wavelength of the 50 Hz mains?
About 6000 km in free space, which is why power lines are not antennas and why mains wiring can be treated as lumped.
How is sound different?
Only the speed: 343 m/s in air instead of 300 000 km/s, so the same formula gives wavelengths a million times shorter for the same frequency.