The Wave Equation
All waves — sound, light, water — follow the same basic relationship between speed, frequency, and wavelength.
Wave Equation
v = f × λ
v = wave speed (m/s)
f = frequency (Hz)
λ = wavelength (m)
Rearranged:
f = v / λ
λ = v / fWorked Examples
Sound (air, 20°C): v = 343 m/s
Middle C (262 Hz): λ = 343/262 = 1.31 m
Ultrasound (40 kHz): λ = 343/40,000 = 8.6 mm
Light in vacuum: v = 3×10⁸ m/s
Green light (550 nm): f = 3×10⁸/550×10⁻⁹ = 5.45×10¹⁴ Hz
Wi-Fi (2.4 GHz): λ = 3×10⁸/2.4×10⁹ = 0.125 m (12.5 cm)Electromagnetic Spectrum
- Radio: λ > 1 mm, f < 300 GHz
- Microwave: 1mm–1m, 300 MHz–300 GHz
- Infrared: 700 nm–1 mm
- Visible light: 380–700 nm
- Ultraviolet: 10–380 nm
- X-ray: 0.01–10 nm
- Gamma: < 0.01 nm
Calculate wave properties: Free Wave Frequency Calculator
Wave Frequency Quick-Reference Table
| Wave Type | Typical Frequency | Wavelength (approx.) | Speed |
|---|---|---|---|
| Audible sound (low bass) | 20 Hz | 17 m | 340 m/s (air) |
| Audible sound (high treble) | 20,000 Hz | 17 mm | 340 m/s (air) |
| FM radio | 88–108 MHz | 2.8–3.4 m | 3×10⁸ m/s |
| Visible light (red) | 430 THz | 700 nm | 3×10⁸ m/s |
| Visible light (violet) | 750 THz | 400 nm | 3×10⁸ m/s |
| X-rays | 10¹⁷–10¹⁹ Hz | 0.01–10 nm | 3×10⁸ m/s |
How Wave Frequency Works
Frequency (f) is the number of complete wave cycles passing a fixed point per second, measured in hertz (Hz). It relates to wavelength (λ) and wave speed (v) through v = fλ. For electromagnetic waves in a vacuum, v = c ≈ 3×10⁸ m/s. For sound in air at 20°C, v ≈ 343 m/s. Frequency and wavelength are inversely proportional: higher frequency means shorter wavelength at the same wave speed.
Period (T = 1/f) is the time for one complete cycle. A 50 Hz mains supply completes one cycle every 20 milliseconds. A 440 Hz concert A note completes one cycle every 2.27 ms. Understanding frequency is essential in acoustics, radio communication, optics, medical imaging, and seismology.
Common Mistakes
- Confusing frequency and period: Frequency = cycles per second; period = seconds per cycle. They are reciprocals (T = 1/f).
- Using the wrong wave speed: Sound speed varies with medium and temperature; light slows in glass or water. Always use the correct v for the medium.
- Mixing prefix scales: 1 MHz = 10⁶ Hz; 1 GHz = 10⁹ Hz. Converting carelessly between kHz, MHz, and GHz is a common error.
Frequently Asked Questions
Pitch is the auditory perception of frequency. Higher-frequency sound waves strike the ear's basilar membrane at a position tuned to that frequency, which the brain interprets as a higher pitch. The relationship is roughly logarithmic: doubling frequency raises pitch by exactly one octave.
When a wave source moves toward an observer, each successive wavefront is compressed — increasing the observed frequency. When it moves away, wavefronts are stretched — decreasing frequency. This is why an ambulance siren sounds higher-pitched approaching than receding, and why astronomers use redshift (Doppler shift of light) to measure how fast galaxies are moving away from us.
For electromagnetic radiation, photon energy E = hf, where h is Planck's constant (6.626×10⁻³⁴ J·s). Higher-frequency photons (X-rays, gamma rays) carry more energy per photon than lower-frequency ones (radio waves, infrared), which is why high-frequency radiation can damage biological tissue.