v = wave speed (m/s) • f = frequency (Hz) • λ = wavelength (m)
Understanding the Wave Equation v = fλ
The wave equation v = fλ (pronounced "v equals f lambda") is one of the most fundamental relationships in physics. It connects three properties of any wave: its speed, its frequency, and its wavelength. The equation holds for all wave types — mechanical waves (sound, water), electromagnetic waves (light, radio), and even quantum mechanical matter waves.
The symbol λ (lambda) is the Greek letter used to represent wavelength — the distance between two successive identical points on a wave (for example, peak to peak). Frequency (f) is the number of complete wave cycles that pass a point each second, measured in Hertz (Hz). Wave speed (v) is how fast the disturbance propagates through the medium.
Wave Properties: A Complete Guide
Every wave has several key properties:
- Wavelength (λ): The distance between two consecutive corresponding points in phase. Measured in metres (m), nanometres (nm) for light, or kilometres (km) for radio.
- Frequency (f): The number of oscillations per second. Measured in Hertz (Hz). 1 Hz = 1 cycle per second.
- Period (T): The time for one complete oscillation. T = 1/f. If f = 50 Hz (UK mains frequency), T = 0.02 s = 20 ms.
- Amplitude: The maximum displacement from equilibrium. Related to energy but not to wave speed or frequency in the wave equation.
- Wave speed (v): How fast the wavefront moves. Depends on the medium (for sound) or is constant at c in a vacuum (for light).
Speed of Light: c = 299,792,458 m/s
The speed of light in a vacuum is one of the fundamental constants of nature, denoted c = 299,792,458 m/s (exactly, by definition). This is approximately 3 × 10&sup8; m/s. Light takes:
- About 8 minutes 20 seconds to travel from the Sun to Earth
- About 1.3 seconds to travel from the Moon to Earth
- About 4.2 years to reach the nearest star (Proxima Centauri)
When light passes through a transparent medium (glass, water), it slows down by a factor equal to the medium's refractive index (n): v = c/n. Glass has n ≈ 1.5, so light travels at about 2×10&sup8; m/s inside glass. This slowing causes refraction (bending of light at interfaces).
Speed of Sound in Different Media
| Medium | Speed of Sound | Temperature/Conditions |
|---|---|---|
| Air | 343 m/s | 20°C, sea level |
| Air | 331 m/s | 0°C |
| Water (fresh) | 1,481 m/s | 25°C |
| Seawater | 1,531 m/s | 25°C |
| Steel | 5,960 m/s | Room temperature |
| Aluminium | 6,320 m/s | Room temperature |
| Wood (oak) | 3,850 m/s | Room temperature |
| Vacuum | 0 m/s | Sound cannot travel |
Sound travels faster in stiffer materials and at higher temperatures (temperature increases the average speed of air molecules). The speed of sound in air increases by approximately 0.6 m/s per °C rise in temperature.
The Electromagnetic Spectrum
All electromagnetic (EM) waves travel at c = 3×10&sup8; m/s in a vacuum and obey v = fλ with v = c. They differ only in frequency (and therefore wavelength). Here is the complete EM spectrum:
| Wave Type | Frequency Range | Wavelength Range | Applications |
|---|---|---|---|
| Radio | 30 Hz – 300 MHz | > 1 m | Broadcasting, communications |
| Microwave | 300 MHz – 300 GHz | 1 mm – 1 m | Wi-Fi, radar, microwave ovens |
| Infrared (IR) | 300 GHz – 430 THz | 700 nm – 1 mm | Thermal imaging, remote controls |
| Visible Light | 430 – 770 THz | 390 – 700 nm | Human vision, photography |
| Ultraviolet (UV) | 770 THz – 30 PHz | 10 – 390 nm | Sterilisation, fluorescence, sunburn |
| X-Ray | 30 PHz – 30 EHz | 0.01 – 10 nm | Medical imaging, security scanning |
| Gamma Ray | > 30 EHz | < 0.01 nm | Cancer treatment, nuclear medicine |
Visible Light: Wavelengths and Colours
The narrow band of EM radiation visible to the human eye spans approximately 390–700 nm. Different wavelengths correspond to different perceived colours:
| Colour | Wavelength (nm) | Frequency (THz) |
|---|---|---|
| Violet | 390–450 | 667–769 |
| Blue | 450–495 | 606–667 |
| Green | 495–570 | 526–606 |
| Yellow | 570–590 | 508–526 |
| Orange | 590–620 | 484–508 |
| Red | 620–700 | 428–484 |
Radio Waves and Broadcasting
Radio waves used in broadcasting have wavelengths from millimetres to kilometres. FM radio in the UK operates between 87.5 and 108 MHz. At 100 MHz:
FM Radio Wavelength Example
Frequency: 100 MHz = 1 × 10&sup8; Hz. Wave speed: c = 3 × 10&sup8; m/s
λ = v/f = 3 × 10&sup8; / 1 × 10&sup8; = 3 m
FM radio wavelengths are around 3 metres, which is why FM aerials are approximately this length (or a fraction/multiple thereof). AM radio at 1 MHz has wavelength = 300 m, making it capable of diffracting around hills and buildings.
Worked Examples
Example 1: Green Light Wavelength
Green light has frequency approximately 545 THz = 5.45 × 10¹&sup4; Hz. Speed = c = 2.998 × 10&sup8; m/s.
λ = c/f = 2.998 × 10&sup8; / 5.45 × 10¹&sup4; = 5.50 × 10²⁻&sup7; m = 550 nm
Example 2: Period from Frequency
UK mains electricity frequency is 50 Hz. Period T = 1/f = 1/50 = 0.02 s = 20 ms. The voltage completes 50 full cycles each second. In the USA, the mains frequency is 60 Hz (T = 16.7 ms).
Example 3: Sound Wavelength in a Room
Middle A note: 440 Hz. Speed of sound in air: 343 m/s.
λ = v/f = 343 / 440 = 0.780 m = 78 cm
This wavelength is comparable to room dimensions, which is why low-frequency acoustic modes (room resonances) are important in studio design.
The Doppler Effect
When a wave source moves relative to an observer, the observed frequency changes — this is the Doppler effect. When an ambulance approaches with siren wailing, the sound waves are compressed (shorter wavelength, higher frequency). As it moves away, waves are stretched (longer wavelength, lower frequency). The Doppler effect applies to all waves and is used in speed cameras (radar guns), weather radar, and astronomy (measuring galaxy recession speeds via redshift).
Medical Applications of Waves
The EM spectrum has critical medical applications:
- X-rays (0.01–10 nm): Penetrate soft tissue but are absorbed by bone. Used in diagnostic imaging and CT scans.
- MRI: Uses radio waves (not ionising radiation) in conjunction with strong magnetic fields to image soft tissue with excellent contrast.
- Ultrasound: High-frequency sound waves (2–18 MHz) used in pregnancy scans and physiotherapy. Not EM radiation but follows the same wave equation.
- Gamma rays: Used in cancer radiotherapy (targeting tumours) and in PET scans (detection of gamma photons from radioactive tracers).
GCSE and A-Level Wave Topics
For GCSE Physics, you need to know: wave equation v = fλ, transverse vs longitudinal waves, the electromagnetic spectrum and its uses and hazards, reflection, refraction, and the Doppler effect (Higher tier).
For A-Level Physics: wave superposition, standing waves, diffraction and Young's double-slit experiment (λ = ax/D), polarisation, refraction and Snell's law (n&sub1;sinθ&sub1; = n&sub2;sinθ&sub2;), and photoelectric effect (linking wave and particle nature of light: E = hf).