Wave Speed, Frequency & Wavelength — v = fλ Explained
Every wave in the universe — sound travelling through air, light crossing the vacuum of space, a ripple spreading across a pond — obeys the same fundamental relationship between speed, frequency, and wavelength. The wave equation v = fλ is simple enough to state in one line, yet it connects phenomena across the entire observable universe: from 50 Hz mains electricity with a 6,000 km wavelength, to gamma rays with wavelengths smaller than an atomic nucleus.
1. The Three Key Quantities — Defined Precisely
| Quantity | Symbol | Definition | SI Unit | Common Range |
|---|---|---|---|---|
| Wave speed | v | The speed at which the wave pattern moves through a medium (or vacuum) | m/s | 343 m/s (sound) to 3×10⁸ m/s (light) |
| Frequency | f | The number of complete wave cycles passing a fixed point per second | Hertz (Hz) | 20 Hz (low bass) to 10²⁰ Hz (gamma rays) |
| Wavelength | λ (lambda) | The distance between two adjacent points in the same phase — crest to crest, or trough to trough | metres (m) | 10⁻¹⁵ m (gamma) to thousands of km (radio) |
| Period | T | The time taken for one complete wave cycle: T = 1/f | seconds (s) | Inversely related to frequency |
Key Distinction: Wave speed is determined by the medium — not by frequency or amplitude. Frequency is determined by the source. Wavelength then adjusts accordingly (via v = fλ) to accommodate both. When light passes from air into glass, its frequency stays the same (set by the source), but its speed decreases — so its wavelength decreases proportionally.
2. Deriving v = fλ from First Principles
The derivation is beautifully simple. In one period T (the time for one complete cycle), the wave pattern moves forward by exactly one wavelength λ (by definition of wavelength). The definition of speed is distance divided by time:
For a given medium (fixed wave speed v): frequency and wavelength are inversely proportional.
Double the frequency → halve the wavelength. Halve the frequency → double the wavelength. Their product always equals the (fixed) wave speed.
This is why high-pitched sounds (high frequency) have short wavelengths, and low bass notes have long wavelengths.
3. Sound Waves — Speed in Different Media
Sound is a longitudinal mechanical wave. Its speed depends on the medium’s elasticity and density. More elastic and less dense → faster sound. This is why sound travels much faster in solids and liquids than in gases.
| Medium | Speed of Sound | Why Fast or Slow? |
|---|---|---|
| Air at 0°C | 331 m/s | Low density, moderate elasticity |
| Air at 20°C | 343 m/s | Higher temperature → faster molecules → faster propagation |
| Water (20°C) | ~1,480 m/s | Much higher bulk modulus (less compressible) than air |
| Steel | ~5,960 m/s | Very high elasticity (Young’s modulus), molecules tightly bound |
| Vacuum | 0 m/s | No medium — sound cannot travel in vacuum |
The human hearing range spans roughly 20 Hz to 20,000 Hz. In air at 20°C, using v = fλ:
- 20 Hz (lowest bass): λ = 343/20 = 17.15 m — longer than a bus
- 1,000 Hz (speech frequency): λ = 343/1000 = 0.343 m — about the length of a ruler
- 20,000 Hz (highest treble): λ = 343/20,000 = 0.017 m = 1.7 cm — about the width of a finger
4. Electromagnetic Waves and the Speed of Light
All electromagnetic radiation — radio, microwave, infrared, visible light, ultraviolet, X-ray, gamma — travels at c = 2.998 × 10⁸ m/s in vacuum. They differ only in frequency (and therefore wavelength). The electromagnetic spectrum spans over 20 orders of magnitude in frequency:
| Type | Wavelength | Frequency | Key Applications |
|---|---|---|---|
| Radio | > 1 mm | < 300 GHz | FM/AM radio, TV, Wi-Fi, 4G/5G |
| Microwave | 1 mm – 1 m | 300 MHz – 300 GHz | Microwave ovens, radar, satellite communication |
| Infrared | 700 nm – 1 mm | 430 THz – 300 GHz | Thermal cameras, TV remotes, fibre optics |
| Visible light | 380 – 700 nm | 430 – 790 THz | Human vision, photography, lasers, solar energy |
| Ultraviolet | 10 – 380 nm | 790 THz – 30 PHz | Sterilisation, sunburn, fluorescence |
| X-ray | 0.01 – 10 nm | 30 PHz – 30 EHz | Medical imaging, CT scans, airport security |
| Gamma ray | < 0.01 nm | > 30 EHz | Cancer radiotherapy, nuclear physics, PET scans |
5. The Doppler Effect
The Doppler effect is the change in observed frequency when the source or observer is in motion relative to the medium. When source and observer approach each other, observed frequency is higher (shorter wavelength). When they move apart, observed frequency is lower (longer wavelength).
Ambulance Siren
The siren sounds higher-pitched as the ambulance approaches (compressed wavefronts → shorter λ → higher f) and lower as it recedes (stretched wavefronts → longer λ → lower f).
Redshift
Light from distant galaxies is redshifted (lower frequency) — they are moving away from us. The degree of redshift tells us how fast they recede. This is key evidence for the expanding universe.
Speed Cameras
Police radar guns emit microwaves. The frequency of the reflected wave shifts by an amount proportional to the car’s speed — calculated by the Doppler formula.
Medical Ultrasound
Doppler ultrasound measures blood flow velocity by detecting the frequency shift of ultrasound reflected from moving blood cells.
6. Worked Examples
Problem: A guitar string vibrates at 440 Hz (concert A). The speed of sound in air is 343 m/s. What is the wavelength of the sound wave produced?
Problem: An FM radio station broadcasts at a wavelength of 2.78 m. What is its frequency? (c = 3.0 × 10⁸ m/s)
Problem: Blue light has a wavelength of 450 nm. What is its frequency?
Problem: An ambulance siren emits at 800 Hz and approaches at 30 m/s. What frequency does a stationary observer hear? (Speed of sound = 343 m/s)
7. Common Misconceptions
“Higher frequency waves travel faster.” For a given medium, wave speed is fixed regardless of frequency. All visible light frequencies travel at exactly c = 3 × 10⁸ m/s in vacuum. All audible frequencies travel at the same speed of sound in air. Speed is a property of the medium, not the wave’s frequency.
“Wavelength and amplitude are the same thing.” Wavelength (λ) is the spatial period — the distance from crest to crest, measured along the direction of wave travel. Amplitude is the maximum displacement from equilibrium, measured perpendicular to travel (for transverse waves). They describe completely different properties and are measured in different directions.
“The Doppler effect only applies to sound.” The Doppler effect applies to all waves — sound, light, radio, and even water waves. Cosmological redshift is the Doppler effect applied to light from distant galaxies. Radar speed guns use the Doppler effect with microwaves. Medical Doppler ultrasound uses it with sound waves in the MHz range.
8. Frequently Asked Questions
Conclusion
The wave equation v = fλ is one of the most elegantly simple and universally powerful equations in physics. From radio waves kilometres long to gamma rays smaller than a proton, from the bass notes of an organ pipe to the ultrasound used in medical imaging — all are described by this single relationship between speed, frequency, and wavelength.
Remember: wave speed depends on the medium. Frequency is set by the source. Wavelength is the consequence — it adjusts to satisfy v = fλ. And when source and observer are in relative motion, the Doppler effect shifts the observed frequency in a way that reveals their relative velocity — a principle exploited in everything from speed cameras to the discovery that our universe is expanding.