Frequency, wavelength, and the speed of sound

The relationship

A sound wave oscillates at a particular frequency (f), measured in hertz (Hz). At a given speed of sound (c), the distance the wave travels in one complete oscillation cycle is the wavelength (λ):

λ = c / f

Equivalently:

c = f × λ

The speed of sound in air at 20°C is approximately 343 m/s. This value appears throughout acoustic calculations and is worth memorising.

Wavelengths across the audio spectrum

The audio frequency range spans nearly three orders of magnitude in wavelength — from 17 m at 20 Hz to 17 mm at 20 kHz. This wide range is the reason acoustic behaviour varies so dramatically with frequency. A solution that works at 1 kHz may be physically irrelevant at 100 Hz.

Temperature dependence

The speed of sound in air is not constant — it depends primarily on temperature:

c ≈ 331.3 + 0.606 × T m/s

where T is temperature in degrees Celsius. At 0°C, c ≈ 331 m/s. At 20°C, c ≈ 343 m/s. At 35°C, c ≈ 352 m/s.

Humidity has a minor effect on the speed of sound (less than 0.5% across typical indoor conditions) and is generally neglected in room acoustics. Temperature gradients in large outdoor spaces or long ducts can cause sound to refract — bend toward the cooler medium — but this effect is rarely significant in small indoor environments.

Why wavelength matters

The physical significance of wavelength arises because acoustic behaviour changes character when the wavelength is comparable to the dimensions of objects or spaces involved.

Room modal behaviour. A room exhibits standing wave resonances — room modes — at frequencies where the room dimensions are integer multiples of a half-wavelength. A room 4.3 m long has an axial mode at c / (2 × 4.3) ≈ 40 Hz, where the wavelength is exactly twice the room length. Understanding room modes requires knowing the wavelengths involved. Understanding room modes — a practical guide covers this in detail.

Driver directivity. A loudspeaker cone radiates omnidirectionally when its circumference is much smaller than the wavelength. As frequency rises and wavelength shortens, radiation concentrates on-axis. The transition occurs when the cone circumference equals approximately one wavelength — i.e. when πd ≈ λ, or f ≈ c/(πd), where d is the effective cone diameter. This is why large-diameter woofers beam at relatively low frequencies.

Absorption effectiveness. Porous absorbers work by dissipating energy as air moves through the material. Maximum absorption occurs when the absorber is placed at a quarter wavelength from the wall, where particle velocity is highest. A 100 mm thick porous panel is effective down to approximately c / (4 × 0.1) ≈ 858 Hz. At 100 Hz (λ = 3.43 m), effective absorption requires either very thick material (approximately 860 mm for quarter-wave placement) or a resonant absorber tuned to that frequency.

Diffuser design. The performance of a quadratic residue diffuser (QRD) depends on the relationship between well depth and wavelength. The design frequency sets the minimum wavelength the diffuser can scatter; at lower frequencies, the diffuser dimensions are too small relative to the wavelength and it acts as a flat reflector.

Barrier and aperture effects. A solid barrier blocks sound effectively when its dimensions exceed the wavelength. At 100 Hz (λ = 3.43 m), a typical room partition provides modest isolation because diffraction around the edges is significant; at 4 kHz (λ = 86 mm) the same partition is highly effective. Similarly, sound diffracts around obstacles whose dimensions are comparable to or smaller than the wavelength, a phenomenon that limits the effectiveness of acoustic screens at low frequencies.

A practical reference

When working on any acoustic problem, compute the wavelength at the frequencies of interest first. It immediately reveals whether a geometric (ray-based) approach is valid (wavelength much smaller than room dimensions), whether a proposed absorber can work in the frequency range of concern, and whether a driver will be beaming at the intended crossover frequency.