Wave Equation Calculator
Enter any two values below and click Calculate to find the third. Leave the unknown field blank.
Electromagnetic Spectrum Reference
All electromagnetic waves travel at the speed of light in a vacuum: c = 3 × 108 m/s
| Wave Type | Wavelength Range | Frequency Range | Common Uses |
|---|---|---|---|
| Radio Waves | > 1 mm (up to km) | < 300 GHz | AM/FM radio, TV broadcasting |
| Microwaves | 1 mm – 1 m | 300 MHz – 300 GHz | Microwave ovens, radar, 5G |
| Infrared (IR) | 700 nm – 1 mm | 300 GHz – 430 THz | Thermal imaging, remote controls |
| Visible Light | 380 nm – 700 nm | 430 THz – 790 THz | Human vision, photography |
| Ultraviolet (UV) | 10 nm – 380 nm | 790 THz – 30 PHz | Sterilisation, fluorescence |
| X-rays | 0.01 nm – 10 nm | 30 PHz – 30 EHz | Medical imaging, security scanning |
| Gamma Rays | < 0.01 nm | > 30 EHz | Cancer treatment, nuclear physics |
nm = nanometres (10-9 m) | THz = terahertz (1012 Hz) | PHz = petahertz (1015 Hz) | EHz = exahertz (1018 Hz)
Key Wave Speed Values
Understanding the Wave Equation
The wave equation v = fλ is one of the most fundamental relationships in physics. It connects three properties of any wave: the speed at which it travels, the frequency of oscillation, and the spatial period (wavelength). This equation applies equally to mechanical waves such as sound and water waves, and to electromagnetic waves including light, radio, and X-rays.
Breaking Down the Variables
Wave Speed (v) is the rate at which the wave pattern propagates through a medium, measured in metres per second (m/s). For electromagnetic waves in a vacuum, this is always the speed of light, c = 299,792,458 m/s (approximately 3 × 108 m/s). For sound waves, the speed depends on the medium and temperature.
Frequency (f) is the number of complete oscillations (cycles) that pass a fixed point each second. It is measured in Hertz (Hz), where 1 Hz = 1 cycle per second. Higher frequency means more energy per photon (for light) and a higher-pitched sound. Everyday frequencies range from 20 Hz (lowest audible sound) to 1018 Hz (gamma rays).
Wavelength (λ) is the distance between two adjacent points that are in phase — for example, from one crest to the next, or from one compression to the next in a sound wave. It is measured in metres (m), although nanometres (nm = 10-9 m) are commonly used for light. Frequency and wavelength are inversely proportional: if frequency doubles, wavelength halves (at constant speed).
Rearranging the Wave Equation
Depending on which quantity you need to find, you can rearrange v = fλ in three ways:
- v = f × λ — Find wave speed when frequency and wavelength are known
- f = v / λ — Find frequency when speed and wavelength are known
- λ = v / f — Find wavelength when speed and frequency are known
Worked Examples
Example 1: A sound wave has a frequency of 440 Hz (concert A). Find its wavelength in air at 20°C (v = 343 m/s).
λ = v / f = 343 / 440 = 0.780 m
Example 2: Green light has a wavelength of 550 nm. Find its frequency.
f = v / λ = (3 × 108) / (550 × 10-9) = 5.45 × 1014 Hz
Example 3: A wave has frequency 200 Hz and wavelength 1.7 m. Find wave speed.
v = f × λ = 200 × 1.7 = 340 m/s
Units and Conversions for A-Level Physics
When working with wave problems, you frequently need to convert between units. Keep these conversions in mind for UK A-Level and GCSE exams:
- 1 nm (nanometre) = 10-9 m
- 1 μm (micrometre) = 10-6 m
- 1 MHz (megahertz) = 106 Hz
- 1 GHz (gigahertz) = 109 Hz
- 1 THz (terahertz) = 1012 Hz
Waves in the UK Curriculum
The wave equation appears throughout both GCSE and A-Level Physics syllabuses in the UK. At GCSE (AQA, OCR, Edexcel), students need to understand transverse and longitudinal waves, the electromagnetic spectrum, and be able to apply v = fλ in calculations. At A-Level, this extends to standing waves, diffraction, interference, the Doppler effect, and photon energy E = hf, where the frequency calculated from the wave equation links directly to quantum physics.
Transverse vs Longitudinal Waves
Transverse waves oscillate perpendicular to the direction of propagation. All electromagnetic waves are transverse, as are water surface waves and seismic S-waves. Longitudinal waves oscillate parallel to the direction of travel, creating regions of compression and rarefaction. Sound waves and seismic P-waves are longitudinal. Both types obey v = fλ equally.
The Doppler Effect and Frequency
The Doppler effect occurs when a wave source moves relative to an observer. The observed frequency changes even though the source frequency remains constant: a moving source approaching the observer produces a higher observed frequency (shorter apparent wavelength), while a receding source produces a lower observed frequency. This principle is used in speed cameras, medical ultrasound, and astronomical redshift measurements.