Wave Speed, Frequency and Wavelength — The Complete Physics Guide
The wave equation v = fλ connects three fundamental properties of any periodic wave: speed, frequency, and wavelength. It is one of the most versatile equations in physics, applying equally to sound waves in air, light waves in a vacuum, seismic waves through rock, electromagnetic waves across the spectrum, and ripples on a water surface. Understanding this relationship deeply gives you insight into acoustics, optics, telecommunications, and much more.
Understanding v = fλ
The wave speed equation can be understood intuitively. Frequency (f) measures how many complete wave cycles pass a fixed point per second — measured in hertz (Hz), where 1 Hz = 1 cycle per second. Wavelength (λ) is the distance between successive identical points on the wave — the distance from crest to crest or trough to trough, measured in metres.
The wave speed is simply the product: in each second, f cycles pass, each of length λ, so the wave front advances a distance f × λ. This is v = fλ. Rearranged: f = v/λ (frequency from speed and wavelength) and λ = v/f (wavelength from speed and frequency).
Crucially, wave speed is determined by the medium, not by the source. The speed of sound in air at 20°C is always 343 m/s regardless of whether the source is a whisper or a jet engine. Changing the frequency of a sound source changes the wavelength of the emitted sound, not the speed. This is why all musical notes in air at the same temperature travel at the same speed — only their wavelengths differ.
Wave Speed in Different Media
Wave speed depends on the physical properties of the medium through which it travels. For mechanical waves (sound, seismic), speed depends on the restoring force (stiffness) and inertia (density) of the medium. Stiffer and less dense media support faster wave propagation.
Sound travels at approximately 343 m/s in air at 20°C, 1,481 m/s in water (about 4× faster), and 5,960 m/s in steel (about 17× faster). The increase in speed with medium density might seem counterintuitive, but the stiffness effect dominates — steel is both denser and enormously stiffer than air. This is why earthquakes generate seismic waves that travel through rock at thousands of metres per second.
Light (and all electromagnetic radiation) travels at exactly 299,792,458 m/s in a vacuum — a universal constant denoted c. In a medium, light slows by a factor equal to the refractive index: v = c/n. In glass (n ≈ 1.5), light travels at about 200,000 km/s. In diamond (n ≈ 2.4), it slows to about 125,000 km/s. This slowing is what causes refraction — the bending of light at interfaces between media with different refractive indices.
Temperature affects wave speed significantly for gases. In air, v_sound ≈ 331 + 0.6T m/s, where T is temperature in Celsius. At 0°C, sound travels at 331 m/s; at 100°C, at 391 m/s. This is why musical instruments go out of tune as temperature changes — different speeds lead to different resonant frequencies in the instrument's air column.
Worked Example 1 — Sound Wave
Problem: A sound wave has frequency 440 Hz (concert A) and travels at 343 m/s in air. What is its wavelength?
λ = v/f = 343/440 = 0.780 m = 78.0 cm
Worked Example 2 — Light Wave
Problem: Red light has wavelength 650 nm in vacuum. What is its frequency?
f = v/λ = (3 × 10⁸) / (650 × 10⁻⁹) = 4.62 × 10¹⁴ Hz
Worked Example 3 — Period and Wavelength
Problem: Seismic P-waves travel through granite at 5,500 m/s with a period of 0.2 s. Find wavelength and frequency.
f = 1/T = 1/0.2 = 5 Hz
λ = v/f = 5500/5 = 1,100 m = 1.1 km
Applications of Wave Speed
Radio communications: Radio waves travel at the speed of light. AM radio (535–1,605 kHz) has wavelengths from 187 m to 560 m. FM radio (87.5–108 MHz) has wavelengths from 2.8 m to 3.4 m. Different wavelengths interact with the atmosphere and terrain differently, which is why AM radio travels further around the Earth's curvature while FM provides higher fidelity over shorter distances.
Medical ultrasound: Ultrasound imaging uses frequencies of 1–20 MHz. In soft tissue (v ≈ 1,540 m/s), this corresponds to wavelengths of 0.077–1.54 mm. Shorter wavelengths give better spatial resolution — one of the reasons higher frequency probes are used for imaging superficial structures while lower frequencies penetrate deeper tissues.
Sonar and echolocation: Sonar systems (ships, submarines) and animal echolocation (bats, dolphins) use the wave equation to determine distance. By measuring the time Δt between emitting a pulse and receiving its echo, the distance is d = v × Δt / 2. Dolphins use ultrasound at up to 200 kHz — wavelengths of about 7.5 mm in water — allowing them to resolve objects smaller than 1 cm.
Seismology: Earthquake waves travel through the Earth at different speeds in different rock types. By measuring arrival times at multiple seismographs, the epicentre and depth of an earthquake can be determined precisely. The different speeds of P-waves (longitudinal, faster) and S-waves (transverse, slower) through Earth's layers allowed seismologists to discover and map Earth's liquid outer core and solid inner core.
The Electromagnetic Spectrum and Wave Speed
All electromagnetic radiation — radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays — travels at the same speed in vacuum: c = 299,792,458 m/s. The only difference between these types of radiation is their frequency (and consequently their wavelength, since λ = c/f). Radio waves have frequencies from about 30 Hz to 300 GHz, wavelengths from 1 mm to 10,000 km. Gamma rays have frequencies above 10¹⁹ Hz and wavelengths smaller than 10 picometres — about the size of an atomic nucleus.
Visible light occupies a narrow band from about 380 nm (violet) to 700 nm (red). The frequency range is approximately 430–770 THz (terahertz). Human colour perception maps to this frequency range: our three types of colour-sensitive cones peak at wavelengths corresponding to blue, green, and red. The colours of the rainbow correspond to the dispersion of white light into its component wavelengths by raindrops acting as prisms.
The speed of light in a vacuum is one of the fundamental constants of nature. In Einstein's special relativity, it serves as the universal speed limit — no information or matter can travel faster. It also appears in E = mc², connecting mass and energy. The definition of the metre is now fixed in terms of the speed of light: 1 metre is the distance light travels in 1/299,792,458 of a second.
Dispersion — When Wave Speed Depends on Frequency
In a non-dispersive medium, all frequencies travel at the same speed and the wave equation v = fλ is straightforward. In a dispersive medium, speed varies with frequency — different colours of light travel at different speeds in glass, which is why prisms split white light into a spectrum. The relationship between wave speed and frequency in a dispersive medium is described by the dispersion relation.
Dispersion causes problems in telecommunications: a pulse of light containing many frequencies spreads out as it travels through a fibre optic cable, because different frequency components travel at slightly different speeds. This pulse spreading (chromatic dispersion) limits the data rate over long fibre optic links and is compensated by carefully designed dispersion-compensating fibres and signal processing.
The speed of sound in air is non-dispersive at audio frequencies — all audible frequencies travel at the same speed, which is why music sounds the same whether you are near or far from the source (just quieter with distance). If sound were dispersive, notes from an orchestra would reach different audience members in different orders depending on their seat — a chaotic listening experience.
Wave Speed and Supersonic Motion
When a source moves through a medium at speeds comparable to or exceeding the wave speed, interesting effects arise. The Mach number is the ratio of the source's speed to the wave speed in the medium: Ma = v_source/v_wave. At Ma = 1, the source is moving at exactly the wave speed — supersonic for sound.
As a supersonic aircraft exceeds Ma = 1, it outruns the pressure waves it generates. These waves pile up into a conical shock wave — a Mach cone — that trails behind the aircraft. When this cone of compressed air sweeps past a ground observer, they hear a sharp "sonic boom." The half-angle of the Mach cone is given by sin(θ) = v_wave/v_source = 1/Ma.
The analogous effect for light is Cherenkov radiation — the eerie blue glow seen in nuclear reactors. Particles travelling through water (or glass) move faster than light travels in that medium (which is c/n, where n ≈ 1.33 for water). Just as a supersonic aircraft creates a sonic boom, these faster-than-light-in-medium particles create an electromagnetic "light boom" — Cherenkov radiation. It does not violate special relativity because nothing exceeds c, the speed of light in vacuum.