Amplitude & Loudness

Sound is a mechanical wave — a travelling disturbance that compresses and rarefies the air (or any medium) as it propagates. Every sound wave has a characteristic amplitude, which measures how much the air pressure deviates from its undisturbed value at the peak of each compression. The larger that peak deviation, the more energetic the wave, and the louder the sound perceived by a listener. Amplitude is therefore the physical quantity that underlies our everyday experience of loudness, linking the mathematics of wave motion directly to human hearing. Think of dropping a pebble into a still pond. A small pebble creates gentle ripples with low amplitude; a boulder creates towering waves with high amplitude. Both sets of waves travel at the same speed across the pond's surface, but the boulder's waves carry far more energy outwards with each cycle. Sound waves in air behave analogously: a quiet whisper barely disturbs the air pressure (tiny amplitude), whereas the thunderclap of a bass drum creates large pressure swings (high amplitude). The frequency — how many oscillations occur per second — determines pitch, while amplitude determines loudness. Crucially, the energy transported by a wave is not simply proportional to its amplitude but to the square of the amplitude. This square relationship means that doubling the amplitude quadruples the energy (and the sound intensity). Acoustical engineers and physicists measure loudness on the decibel scale precisely because our ears respond to intensity over an enormous range — roughly twelve orders of magnitude from the threshold of hearing to the threshold of pain.

Key takeaways

What amplitude means for a sound wave
The amplitude–energy relationship
From intensity to the decibel scale
Amplitude versus frequency: independent properties

What amplitude means for a sound wave

For a sound wave travelling through air, amplitude A is the maximum displacement of air molecules from their equilibrium positions, or equivalently the maximum excess pressure ΔP_max above or below atmospheric pressure. In a sinusoidal wave, the pressure at any point varies as p(x,t) = ΔP_max sin(kx − ωt), where k = 2π/λ is the wave number and ω = 2πf is the angular frequency. The amplitude ΔP_max is measured in pascals (Pa); typical speech produces pressure amplitudes of order 0.02 Pa against a background atmospheric pressure of 101 325 Pa — an astonishingly small fraction.

The amplitude–energy relationship

The intensity I of a sound wave — the energy transported per second per unit area perpendicular to the direction of travel — is proportional to the square of the pressure amplitude: I ∝ A². This arises because wave energy has two contributions (kinetic and potential) that both scale as A², adding together. If you increase a speaker's output so that its amplitude doubles, the intensity quadruples (2² = 4). Intensity is measured in watts per square metre (W m⁻²). This square relationship is universal for mechanical and electromagnetic waves alike.

From intensity to the decibel scale

Because the human ear perceives loudness approximately logarithmically over a range from I₀ = 1 × 10⁻¹² W m⁻² (threshold of hearing) to about 1 W m⁻² (threshold of pain), physicists define the sound level β in decibels (dB) as β = 10 log₁₀(I / I₀). A doubling of intensity corresponds to an increase of roughly 3 dB; a tenfold increase corresponds to 10 dB. Normal conversation (~60 dB) has an intensity one million (10⁶) times greater than the threshold of hearing, yet only 60 dB above it — reflecting the logarithmic compression performed by your inner ear.

Amplitude versus frequency: independent properties

A common misconception is that louder sounds must have higher pitch. In reality, amplitude and frequency are independent properties of a wave. You can have a very loud (high-amplitude) low-pitched bass note, or a very quiet (low-amplitude) high-pitched tone. A graph of a loud low-frequency wave shows tall, widely spaced oscillations, while a quiet high-frequency wave shows short, closely spaced oscillations. The ear has separate receptor cells (hair cells in the cochlea) sensitive to different frequencies, allowing it to decode pitch and loudness independently.

The Key Equation

I = \frac{(\Delta P_{\max})^2}{2\rho v}

I — Intensity (W m⁻²): Power transported by the wave per unit area perpendicular to its direction of travel.
\Delta P_{\max} — Pressure amplitude (Pa): Maximum deviation of air pressure from its equilibrium (atmospheric) value; this is the amplitude of the sound wave.
\rho — Density of medium (kg m⁻³): Mass per unit volume of the medium through which sound travels (≈ 1.20 kg m⁻³ for air at 20 °C).
v — Speed of sound (m s⁻¹): Speed at which the wave propagates through the medium (≈ 343 m s⁻¹ in air at 20 °C).

Worked Example

Problem. A loudspeaker produces a sound wave in air with a pressure amplitude of 0.40 Pa. Air at 20 °C has a density of 1.20 kg m⁻³ and sound travels at 343 m s⁻¹. (a) Calculate the intensity of the sound wave. (b) Calculate the sound level in decibels, given that the threshold of hearing is I₀ = 1.0 × 10⁻¹² W m⁻².

I \approx 1.94 \times 10^{-4}\text{ W m}^{-2};\quad \beta \approx 82.9\text{ dB}

Answer. Intensity ≈ 1.94 × 10⁻⁴ W m⁻²; Sound level ≈ 82.9 dB

Figure

Diagram

Origins · 1 minAlexander Graham Bell and how this was discovered

Alexander Graham Bell

1847–1922 · Scottish-American

Bell conducted pioneering research on sound transmission and the nature of acoustic signals, leading to the invention of the telephone in 1876. His detailed study of how sound energy is carried and transduced gave scientists a practical framework for quantifying sound levels. The decibel unit — central to measuring loudness — was named in his honour, recognising that his work established the quantitative link between physical wave properties (amplitude, intensity) and perceived loudness. Bell considered the telephone his greatest intrusion on his real work and reportedly refused to have one in his study. He was also a dedicated teacher of the deaf and regarded that work as his most important contribution to humanity.

Wave Speed, Frequency & Wavelength — Frequency and wavelength determine pitch independently of amplitude; understanding both properties together gives a complete description of any sound wave.
The Decibel Scale — The decibel scale converts sound intensity (directly linked to amplitude squared) into a logarithmic measure of loudness that matches human hearing perception.
Transverse and Longitudinal Waves — Sound is a longitudinal wave; understanding how particles oscillate parallel to the wave's direction of travel clarifies what amplitude physically represents in a sound wave.
Reflection and Echoes — When sound reflects off a surface its amplitude (and therefore intensity) can decrease, illustrating how energy is distributed and absorbed as waves travel.