Wavelength, Frequency & Wave Number — Complete Reference

Frequency and wavelength are two ways of describing the same electromagnetic wave. Every RF engineer needs to convert between them fluently — from choosing antenna dimensions to designing resonant structures, filters, and transmission lines. Adding a dielectric medium changes the picture significantly: the wave slows down, the guided wavelength shortens, and antenna dimensions must be recalculated.

The Fundamental Relationship

In free space (vacuum or air, εr = 1), the speed of light c = 299,792,458 m/s connects frequency and wavelength directly:

Free-space wavelength

λ₀ = c f = 299,792,458 f [Hz] [m]

f = c λ₀

Quick estimate: λ₀ (mm) ≈ 300 / f (GHz) | at 2.4 GHz: λ₀ = 300/2.4 = 125 mm

Wavelength in a Dielectric Medium

When an electromagnetic wave travels through a dielectric material with relative permittivity εr > 1, it slows down. The phase velocity in the medium becomes:

Phase velocity and guided wavelength in dielectric

vp = c √εr

λg = vp f = λ₀ √εr

FR-4 (εr=4.4): λg = λ₀ / √4.4 = λ₀ / 2.098 — wavelength is halved compared to free space.

Practical impact: A patch antenna on FR-4 at 2.4 GHz has patch length L ≈ λg/2 ≈ 29.8 mm, compared to λ₀/2 = 62.5 mm in free space. The dielectric reduces the antenna size by a factor of √εr — extremely useful for compact PCB antenna design.

Wave Number k

The wave number (also called the propagation constant in lossless media) describes how many radians of phase accumulate per unit length:

Wave number

k₀ = 2π λ₀ = 2πf c [rad/m]

k = k₀ · √εr = 2π λg (in dielectric)

k is used in field equations, S-parameter analysis, and when computing electrical length of transmission line sections.

Electrical Length

The electrical length of a physical structure is how long it appears in wavelengths. A λ/4 transformer at 2.4 GHz on FR-4 is physically shorter than λ₀/4 by exactly √εr. This is why electrical length matters more than physical length in RF design:

Electrical length

θ = k · L = 2π · L λg [radians]

L = physical length of structure. θ = π/2 (90°) for a quarter-wave element.

Common RF Frequencies — Wavelength Reference

Band / Application	Frequency	λ₀ (free space)	λg on FR-4 (εr=4.4)	λ/4 on FR-4
LoRa / ISM 433 MHz	433 MHz	692 mm	330 mm	82.4 mm
LoRa EU 868 MHz	868 MHz	346 mm	165 mm	41.2 mm
LoRa US / ISM 915	915 MHz	328 mm	156 mm	39.1 mm
GPS L1	1575 MHz	190 mm	90.7 mm	22.7 mm
Wi-Fi / BT 2.4 GHz	2400 MHz	124.9 mm	59.5 mm	14.9 mm
5G NR Sub-6 GHz	3500 MHz	85.7 mm	40.8 mm	10.2 mm
Wi-Fi 5 GHz	5800 MHz	51.7 mm	24.6 mm	6.2 mm
Wi-Fi 6E / 60 GHz	60 GHz	5.00 mm	2.38 mm	0.60 mm
Automotive radar	77 GHz	3.89 mm	1.85 mm	0.46 mm

Why Wavelength Matters in Antenna Design

Antenna dimensions are fundamentally tied to wavelength. The most common relationships:

Half-wave dipole: total length ≈ 0.95 × λ/2 (with end effect correction)
Quarter-wave monopole: length ≈ λ/4 above a ground plane
Patch antenna length: L ≈ λg/2 in the substrate (reduced by fringing fields)
Quarter-wave transformer: length = λg/4 of the transmission line medium
Electrical spacing in arrays: typically λ/2 between elements for no grating lobes

Rule of thumb: λ₀ (mm) ≈ 300 / f (GHz). On FR-4, divide by 2.1 (≈√4.4). On Rogers 5880, divide by 1.48 (≈√2.2). Keep this in your head and you can estimate dimensions at any design stage without a calculator.

Velocity Factor

The velocity factor (VF) is the ratio of the phase velocity in the medium to the speed of light in vacuum. It is the reciprocal of √εr and is frequently quoted for coaxial cables and transmission lines:

Velocity factor

VF = vp c = 1 √εr = λg λ₀

Air-filled coax: VF ≈ 0.98 | Solid PTFE coax: VF ≈ 0.69 | FR-4 microstrip: VF ≈ 0.48 (effective εr ≈ 3.0)

Wavelength & Frequency Calculator

Wavelength, Frequency & Wave Number — Complete Reference

The Fundamental Relationship

Wavelength in a Dielectric Medium

Wave Number k

Electrical Length

Common RF Frequencies — Wavelength Reference

Why Wavelength Matters in Antenna Design

Velocity Factor

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