Wavelength

Prof. Dr. Thorsten Jungmann (last updated 2025-09-03)

The wavelength (German: Wellenlänge) describes the spatial distance between two successive points of equal phase of a wave, such as two adjacent wave crests.

In the fundamentals of electrical engineering (EE1) only for completeness

This quantity appears in the EE1 script only for the sake of completeness, because it belongs to the components inductor and capacitor. Calculations and a deeper understanding are part of alternating current theory (EE2).

1 Quantity symbol and unit

The quantity symbol of the wavelength is $\lambda$ (‘lambda’).
Its unit is the metre with the unit symbol $\mathrm{m}$:

$$[\lambda ]=\mathrm{m}$$

2 Practical and worked examples

The wavelength results from the relationship between propagation velocity and frequency:

$$\lambda =\frac{v_p}{f}$$

Unit check:

$$[\lambda ]=\frac{[v_p]}{[f]}=\frac{m/s}{\mathrm{Hz}}=\frac{m/s}{1/\mathrm{s}}=\mathrm{m}$$

2.1 Wi-Fi at 2.4 GHz (electromagnetic wave)

Electromagnetic waves propagate in air approximately at the speed of light:

$$\lambda =\frac{c}{f}=\frac{3.0\cdot 10^{8}m/s}{2.4\cdot 10^9\mathrm{Hz}}=\underline{0.125\mathrm{m}}=12.5\mathrm{cm}$$

2.2 Visible light in glass

Green light (${\lambda }_0=550\mathrm{nm}$) in glass with refractive index $n=1.5$:

$${\lambda }_{glass}=\frac{{\lambda }_0}{n}=\frac{550\mathrm{nm}}{1.5}=\underline{367\mathrm{nm}}$$

2.3 Sound in air at 1.0 kHz

Speed of sound in air at 20 °C: $v_{sound}=343m/s$

$$\lambda =\frac{v_{sound}}{f}=\frac{343m/s}{1.0\cdot 10^3\mathrm{Hz}}=\underline{0.343\mathrm{m}}=34.3\mathrm{cm}$$

2.4 Mains frequency 50 Hz (electromagnetic wave)

$$\lambda =\frac{c}{f}=\frac{3.0\cdot 10^{8}m/s}{50\mathrm{Hz}}=\underline{6.0\cdot 10^6\mathrm{m}}=6000\mathrm{km}$$

3 Typical orders of magnitude

Electromagnetic waves:

• Mains frequency (50 Hz): $\lambda \approx 6000\mathrm{km}$
• Long wave (150 kHz): $\lambda \approx 2\mathrm{km}$
• Medium wave (1 MHz): $\lambda \approx 300\mathrm{m}$
• VHF (100 MHz): $\lambda \approx 3\mathrm{m}$
• Wi-Fi 2.4 GHz: $\lambda \approx 12.5\mathrm{cm}$
• Microwave (10 GHz): $\lambda \approx 3\mathrm{cm}$
• Visible light: $\lambda \approx 400$ to $700\mathrm{nm}$

Sound waves (air, 20 °C):

• Low tones (100 Hz): $\lambda \approx 3.4\mathrm{m}$
• Speech (1 kHz): $\lambda \approx 34\mathrm{cm}$
• High tones (10 kHz): $\lambda \approx 3.4\mathrm{cm}$

4 Example phrasings

• ‘Wi-Fi at 2.4 GHz has a wavelength of about $12.5\mathrm{cm}$.’
• ‘The wavelength is inversely proportional to the frequency.’
• ‘In denser media, the wavelength becomes shorter at the same frequency.’
• ‘The mains frequency of 50 Hz corresponds to a wavelength of 6 000 km.’

5 Related quantities

• Frequency and phase velocity:

$$\lambda =\frac{v_p}{f}$$

• Angular frequency:

$$\lambda =\frac{2\pi \cdot v_p}{\omega }$$

• Wavenumber:

$$k=\frac{2\pi }{\lambda }$$

• Refractive index for electromagnetic waves:

$${\lambda }_{medium}=\frac{{\lambda }_0}{n}$$

• Propagation velocity:

$$v_p=\lambda \cdot f$$

6 Related components

None within the fundamentals of electrical engineering.

• Antennas: length often $\lambda /4$ or $\lambda /2$ for optimum radiation
• Waveguides: dimensions depend on the wavelength
• Resonators: resonance frequency determined by dimensions and wavelength