Waves: the fundamental relation v = f·λ

All waves — light, sound, radio, water, seismic — obey the same simple relation: v = f·λ, where v is propagation speed (m/s), f is frequency (Hz, cycles per second), and λ is wavelength (m, distance between two crests). This calculator solves v, f, or λ given the other two, plus period T, angular frequency ω, wavenumber k, photon energy (light), and real-world context (AM radio, FM, 5G, visible, X-ray, etc.).

Fundamental formulas

• v = f · λ — speed = frequency × wavelength
• f = v / λ
• λ = v / f
• T = 1 / f — period (s)
• ω = 2π·f — angular frequency (rad/s)
• k = 2π/λ — wavenumber (rad/m)
• E = h·f — photon energy (light), h = 6.626×10⁻³⁴ J·s

Typical propagation speeds

Practical numerical examples

Example 1: A4 musical note

A4 = 440 Hz in air at 20°C (v=343):

• λ = v/f = 343/440 = 0.780 m = 78 cm
• Period T = 2.27 ms
• Order of size of a guitar or transverse flute

Example 2: FM radio

FM station at 100 MHz in vacuum:

• λ = c/f = 3×10⁸ / 10⁸ = 3 m
• FM antennas typically 75 cm (λ/4) or 1.5 m (λ/2)
• Period T = 10 ns

Example 3: 2.4 GHz WiFi

• λ = 3×10⁸ / 2.4×10⁹ = 0.125 m = 12.5 cm
• WiFi antennas ~3-6 cm (λ/4 or λ/2)
• Passes thin walls well, blocked by metal or thick walls

Example 4: 5G mmWave

5G at 28 GHz:

• λ = 3×10⁸ / 2.8×10¹⁰ = 1.07 cm
• Hence "mmWave" name
• High bandwidth but short range (100-300 m), blocked by walls

Example 5: green visible light

Green at λ = 550 nm in vacuum:

• f = c/λ = 3×10⁸ / 5.5×10⁻⁷ = 5.45×10¹⁴ Hz = 545 THz
• Photon energy E = h·f = 3.6×10⁻¹⁹ J ≈ 2.25 eV
• Typical visible photon energy

Example 6: ocean wave

8 s period wave in deep water (v ≈ 12.5 m/s):

• f = 1/T = 0.125 Hz
• λ = v·T = 12.5 × 8 = 100 m
• Ocean waves have very long λ

The complete EM spectrum

All EM radiation travels at the same speed in vacuum: c = 299,792,458 m/s (exact value, meter definition since 1983).

Acoustic spectrum: human ear

Human hearing declines with age: children to 20 kHz, adults ~15 kHz, elderly 8-10 kHz. Animals: dolphins 150 kHz, bats 200 kHz.

Musical notes and frequencies

One octave = doubling of frequency. Guitar string length halves per octave.

Physical derivation

A wave is a periodic disturbance propagating in space. In one period T the wave advances one wavelength λ. Speed = distance/time:

v = λ / T = λ · (1/T) = λ · f

Universal, applies to mechanical (sound, strings, water) and electromagnetic (light, radio, X) waves.

The wave equation

Traveling wave mathematical form:

y(x,t) = A·sin(k·x − ω·t + φ)

A amplitude, k = 2π/λ wavenumber, ω = 2π·f angular frequency, φ initial phase. Phase velocity v = ω/k = f·λ.

Applications

Music and acoustics

• Musical instrument design (organ pipe length, strings)
• Tuning: A4 440 Hz standard
• Concert halls: room length vs λ for unwanted reflections
• Microphones and speakers: bandwidth

Telecommunications

• Antenna sizing: L ≈ λ/4 or λ/2
• Band choice: low f penetrates obstacles; high f more bandwidth
• Propagation: ionosphere reflects HF, UHF is line-of-sight
• 5G mmWave: small λ → compact antennas but short range

Medicine

• Ultrasound: 2-15 MHz; higher f = better resolution but lower penetration
• X-rays (f ~10¹⁸ Hz): radiography, CT
• MRI: radio frequencies 60-300 MHz in magnetic field

Radar and sonar

• Radar: microwaves 1-40 GHz, aircraft and ships
• Sonar: acoustic 10-100 kHz in water
• Distance: t=2d/v → d=v·t/2

Astronomy

• Radio telescopes: detect stellar emissions at various λ
• Cosmological Doppler: galaxy redshift
• Gravitational waves (LIGO): f ~ 100 Hz, detected 2015

Seismology

• P waves (compression): faster, arrive first
• S waves (shear): slower, don't cross liquids
• P-S time difference → epicenter distance

Sound speed and temperature

Sound speed in air depends on temperature:

v ≈ 331.3 + 0.606·T(°C) m/s

At 0°C: 331 m/s. At 20°C: 343 m/s. At 40°C: 355 m/s. Kinetic theory: v ∝ √T. Why musical instruments go out of tune with temperature.

Doppler effect

When source and observer move relative to each other, perceived frequency changes:

f' = f · (v ± v_obs) / (v ∓ v_source)

Example: ambulance at 40 m/s with 700 Hz siren approaching stationary observer:

f' = 700 · 343/(343-40) = 792 Hz (higher)

After passing: f' = 626 Hz (lower). 166 Hz jump.

Refraction: waves crossing media

When a wave enters a new medium, speed changes but frequency stays constant (set by source). Wavelength changes:

λ₂/λ₁ = v₂/v₁ = n₁/n₂

with n = c/v refractive index. Vacuum (n=1) to water (n=1.33): λ reduced by 1.33×. Green light 550 nm in vacuum → 414 nm in water (still appears green!).

Standing waves and harmonics

On a string fixed at both ends, standing waves form at discrete wavelengths:

• λ_n = 2L/n for n = 1, 2, 3, ...
• f_n = n·v/(2L)

n=1 fundamental, n=2 first harmonic (octave), n=3 second harmonic. Instrument timbre comes from harmonic mixture.

History of wave theory

Huygens (1678, Netherlands): wave theory of light. Huygens' principle.

Thomas Young (1801): double-slit experiment proves wave nature of light.

Maxwell (1865): electromagnetism equations predict EM waves at light speed — light = EM wave!

Heinrich Hertz (1887): generates and detects radio waves, confirming Maxwell.

Planck (1900): E = h·f explains blackbody radiation.

Einstein (1905): photon as energy quantum h·f. Wave-particle duality.

De Broglie (1924): matter has wave properties λ = h/p.

FAQ

What is the fundamental wave formula?

v = f·λ, where v is speed (m/s), f frequency (Hz), λ wavelength (m). Universal for all waves.

Why is light speed constant?

From Maxwell's equations, c = 1/√(ε₀·μ₀), universal constants. Einstein (1905) deduced it's the same for all observers, founding special relativity.

Does frequency change when wave enters new medium?

No. Frequency stays constant (set by source). Speed v changes, so wavelength λ changes. Light vacuum → glass: f same, λ reduced.

What is Doppler effect?

Frequency change when source and observer move. Approaching: higher frequency (sound higher, light blue-shift); receding: lower (red-shift). f' = f·(v ± v_obs)/(v ∓ v_source).

How to calculate antenna length?

Typically L = λ/4 or λ/2. WiFi 2.4 GHz (λ=12.5 cm): L = 3.1 cm. FM 100 MHz (λ=3 m): L = 75 cm. AM 1 MHz (λ=300 m): large towers needed.

Difference longitudinal vs transverse waves?

Longitudinal: oscillation parallel to propagation (sound in air). Transverse: perpendicular (guitar string, light). EM waves are transverse; sound in fluids is longitudinal.

What is a photon?

Energy quantum of EM wave. Energy E = h·f (h = Planck constant). Wave-like (interference) and particle-like (photoelectric effect) — wave-particle duality.

Gravitational waves: new frontier

Predicted by Einstein (1916), first detected by LIGO (2015): oscillations of spacetime curvature from accelerating masses. Observed frequencies 10-1000 Hz, spacetime deformations 10⁻²¹ — equivalent to measuring a hair's width over Earth-Sun distance.

Common errors

• Forgetting frequency units: Hz = cycles/s
• Confusing f and ω: ω = 2π·f, differ by 6.28
• Applying v=fλ to relativistic particles: phase vs group velocity
• Using c in non-vacuum media: light in water travels at c/n
• Confusing vacuum λ with medium λ: light entering water: λ → λ/n

Updated July 2026