Wave Speed, Frequency, and Wavelength: The Equation That Governs Every Wave

Every wave in the universe — sound travelling through air, light crossing the vacuum of space, a ripple spreading across still water — obeys a single equation: v = fλ. Wave speed equals frequency multiplied by wavelength. Three quantities, one relationship, and it holds without exception for every wave that has ever been measured. Understanding this equation from first principles, not just as a formula to plug numbers into, is the foundation of all wave physics.

Below, v = fλ is derived from the definitions of each quantity, then applied across sound, light, and mechanical waves — with the Doppler effect, dispersion, and six worked examples showing exactly how the equation behaves in every context.

The Wave Equation: v = fλ

Every wave — a ripple on water, sound in air, a transverse wave on a string, light crossing a vacuum — obeys one fundamental relationship:

Wave speed equals frequency multiplied by wavelength. This applies to every wave that has ever been measured, in any medium, of any type. It is not an empirical coincidence — it follows directly from the definitions of the quantities involved.

Where it comes from:

Frequency $f$ is the number of complete wave cycles passing a fixed point per second, measured in hertz (Hz). Wavelength $\lambda$ is the length of one complete cycle — crest to crest, or trough to trough — measured in metres. If $f$ complete cycles pass a point every second, and each cycle occupies $\lambda$ metres of space, the wavefront advances $f \times \lambda$ metres every second. That advance per second is wave speed. The equation is a geometric identity, not a rule to memorise.

Rearrangements — use whichever the problem requires:

If you know any two of the three quantities, the third follows immediately.

Period: The Reciprocal of Frequency

Period $T$ is the time for one complete wave cycle to pass a fixed point, measured in seconds. Frequency and period are exact reciprocals:

A wave with frequency 200 Hz has period $T = 1/200 = 0.005$ s — each complete cycle takes 5 milliseconds. A pendulum with period 2 s has frequency $f = 0.5$ Hz — half a cycle per second.

The wave equation can also be written using period:

This form is useful when period is given directly rather than frequency. In one period $T$, the wave travels exactly one wavelength $\lambda$ — so speed equals wavelength divided by period.

What Determines Wave Speed?

The most important subtlety in wave physics is this: wave speed is determined by the medium, not by the frequency or wavelength of the wave.

For any given medium under given conditions, wave speed is fixed. When the source changes its frequency — vibrating faster or slower — the wavelength adjusts proportionally to keep $v = f\lambda$ satisfied. Speed does not change. Wavelength does.

This means:

• In a given medium, high frequency = short wavelength
• In a given medium, low frequency = long wavelength
• The product $f \times \lambda$ always equals the fixed wave speed of that medium

For mechanical waves, speed depends on the elastic and inertial properties of the medium:

Wave type	Speed formula	Physical meaning
Wave on a string	v = √(T / μ)	T = tension, μ = mass per unit length
Sound in a gas	v = √(γP / ρ)	γ = adiabatic index, P = pressure, ρ = density
Sound in a solid	v = √(E / ρ)	E = Young’s modulus, ρ = density

For electromagnetic waves in vacuum, speed is a universal constant:

All electromagnetic waves — radio, light, X-rays, gamma rays — travel at exactly this speed in vacuum, regardless of frequency.

Wave Speed in Different Media

Wave speed changes when a wave moves from one medium to another. This has major physical consequences — it is the cause of refraction, dispersion, and the bending of light through lenses and prisms.

Wave	Medium	Speed
Sound	Air (20°C)	343 m/s
Sound	Water (20°C)	1,480 m/s
Sound	Steel	5,100 m/s
Light	Vacuum	3.00 × 10⁸ m/s
Light	Water	2.25 × 10⁸ m/s
Light	Glass (typical)	2.00 × 10⁸ m/s
Light	Diamond	1.24 × 10⁸ m/s

Sound travels faster in denser solids because the stronger inter-atomic bonds transmit the disturbance more rapidly. Light slows in transparent materials because it interacts with the atoms of the medium — the denser the material optically, the slower light travels through it.

When a wave enters a new medium, its frequency does not change — it is set by the source and stays constant. But its speed changes, so its wavelength must also change to satisfy $v = f\lambda$. This change in wavelength (and speed) at a boundary is the direct cause of refraction — the bending of waves at interfaces — which is why a straw appears bent in a glass of water and why lenses focus light.

The Electromagnetic Spectrum

Every form of electromagnetic radiation obeys $c = f\lambda$ with the same speed $c = 3 \times 10^8$ m/s in vacuum. The entire electromagnetic spectrum is simply this one equation applied across an enormous range of frequencies:

Type	Frequency	Wavelength	Common uses
Radio waves	3 Hz – 300 MHz	1 mm – 100,000 km	AM/FM radio, MRI scanners
Microwaves	300 MHz – 300 GHz	1 mm – 1 m	WiFi, radar, microwave ovens
Infrared	300 GHz – 430 THz	700 nm – 1 mm	Thermal imaging, TV remotes
Visible light	430 – 750 THz	400 – 700 nm	Human vision
Ultraviolet	750 THz – 30 PHz	10 – 400 nm	Sterilisation, sunburn
X-rays	30 PHz – 30 EHz	0.01 – 10 nm	Medical imaging
Gamma rays	Above 30 EHz	Below 0.01 nm	Cancer radiotherapy, nuclear medicine

The only difference between a radio wave and a gamma ray is frequency — and therefore wavelength. Both are electromagnetic waves. Both travel at $c$ in vacuum. The wave equation $v = f\lambda$ is the single relationship that organises all of them.

Visible light occupies only a narrow band of this spectrum. Within visible light, wavelength determines colour:

Colour	Wavelength	Frequency
Red	620–700 nm	4.3–4.8 × 10¹⁴ Hz
Green	490–560 nm	5.4–6.1 × 10¹⁴ Hz
Violet	380–450 nm	6.7–7.9 × 10¹⁴ Hz

The Doppler Effect

The Doppler effect is the change in observed frequency — and therefore wavelength — that occurs when the source of a wave and the observer are moving relative to each other. Wave speed in the medium does not change. What changes is how many wave cycles per second reach the observer.

When source and observer approach each other: wavefronts are compressed. The observer receives more cycles per second than the source emits — observed frequency is higher, wavelength is shorter.

When source and observer move apart: wavefronts are stretched. The observer receives fewer cycles per second — observed frequency is lower, wavelength is longer.

The observed frequency is:

Use + for observer moving toward source, − for moving away. Use − for source moving toward observer, + for moving away.

Doppler example: An ambulance siren emits at 700 Hz. The ambulance approaches at 30 m/s. Speed of sound = 343 m/s. What frequency does a stationary observer hear?

The observer hears a pitch about 10% higher than the emitted frequency.

The Doppler effect in astronomy: Light from distant galaxies is redshifted — its frequency is lower and wavelength longer than the emitted values. This is Doppler-like evidence that galaxies are moving away from us, and it underpins the discovery that the universe is expanding. The wave equation $v = f\lambda$ is the starting point for this cosmological measurement.

Dispersion: When Wave Speed Varies with Frequency

In most real media, wave speed varies slightly with frequency. This is called dispersion. When different frequencies travel at different speeds, a wave pulse containing multiple frequencies spreads out over time — the pulse disperses.

The most familiar example is white light through a glass prism. In vacuum, all frequencies of light travel at $c$ — no dispersion. In glass, the speed depends on frequency: violet light (high frequency) slows more than red light (low frequency). Since $\lambda = v/f$ and $v$ changes while $f$ stays constant, different colours have different wavelengths in glass — and refract at different angles when entering or leaving the glass surface. The result is the familiar rainbow separation of colours.

This is also why fibre-optic communication systems must carefully manage dispersion: a pulse of light containing a range of frequencies will broaden as it travels, limiting data transmission rates over long distances.

Dispersion does not violate $v = f\lambda$. It simply means that in a dispersive medium, the value of $v$ depends on $f$ — so the equation must be applied separately at each frequency.

Wave Speed and Energy

Wave speed, frequency, and wavelength all relate to the energy a wave carries, though in different ways depending on the wave type.

Electromagnetic waves: The energy of a single photon is:

Where $h = 6.626 \times 10^{-34}$ J·s is Planck’s constant. Energy depends on frequency alone, not on wave speed or wavelength directly. Using $f = c/\lambda$:

Higher frequency (shorter wavelength) means higher energy per photon. X-rays have frequencies around $10^{18}$ Hz — roughly a million times higher than visible light — and carry correspondingly more energy per photon, which is why they penetrate tissue and why high doses are dangerous.

Mechanical transverse waves: Energy depends on both frequency and amplitude A:

Doubling the frequency quadruples the energy. Doubling the amplitude also quadruples the energy. These are independent contributions — a high-frequency wave of small amplitude and a low-frequency wave of large amplitude can carry equal energy if the products $f^2A^2$ are equal.

Worked Examples

Example 1 — Finding Wavelength from Frequency

Problem: A sound wave in air has frequency 440 Hz (concert A). Speed of sound = 343 m/s. Find the wavelength.

Concert A has a wavelength of about 78 cm — roughly the width of an outstretched arm.

Example 2 — Finding Frequency of Visible Light

Problem: Green light has wavelength 550 nm in vacuum. Find its frequency.

Example 3 — Period and Wavelength

Problem: A wave has period 0.008 s and travels at 320 m/s. Find its frequency and wavelength.

Example 4 — Wave on a String

Problem: A string has tension 90 N and linear mass density 0.004 kg/m. A wave of frequency 60 Hz travels along it. Find the wave speed and wavelength.

Example 5 — Doppler Effect

Problem: A train horn emits at 520 Hz and approaches a stationary observer at 25 m/s. Speed of sound = 343 m/s. What frequency does the observer hear?

The observer hears the horn about 41 Hz higher than its emitted frequency.

Example 6 — Wavelength of an FM Radio Wave

Problem: An FM station broadcasts at 101.5 MHz. Find the wavelength.

FM radio waves are roughly 3 metres long — comparable to the height of a room.

Frequently Asked Questions