The Physics
Hypertextbook
Opus in profectus

# Intensity

## Summary

• Amplitude, intensity, and loudness are often used interchangeably, but the three terms have different meanings.
• Amplitude is a measure of the maximal change in whatever quantity is varying in a wave.
• For sound waves, the varying quantity [and its SI unit] could be…
• position [m]
• velocity [m/s]
• acceleration [m/s2]
• pressure [Pa]
• density [kg/m3]
• Position and density variations caused by sound waves are difficult to measure directly since…
• their magnitudes are small and
• the period of sound waves are brief.
• Pressure variations caused by sound waves can be detected…
• by animals with their ears and
• by machines with microphones
• Intensity is an objective measure of the time-averaged power density of a wave at a particular location.
• The SI unit of intensity is the watt per square meter.
• As an equation, intensity is defined as…  I = ⟨P⟩ A
Where…  I = intensity ⟨P⟩ = time-averaged power A = area through which the wave is propagating in a plane perpendicular to the direction of propagation
• When the amplitude of a sound wave is measured by the maximum displacement of the particles that make up the medium, its intensity is equal to…

I = 2π2ρf2vs2

Where…  I = intensity Δs = maximum displacement of particles in the medium (displacement amplitude) ρ = density of medium f = wave frequency v = wave speed
• When the amplitude of a sound wave is measured by the maximum gauge pressure of the medium in bulk, its intensity is equal to…  I = ∆P2 2ρv
Where…  I = intensity ∆P = maximum gauge pressure (pressure amplitude) ρ = density of medium v = wave speed
• When the amplitude of a sound wave is measured by the maximum change in density of the medium in bulk, its intensity is equal to…  I = ∆ρ2v3 2ρ
Where…  I = wave intensity Δρ = maximum density change (density amplitude) ρ = density of medium v = wave speed
• The level (L) of a quantity is the logarithm of the quantity compared to a reference value for the quantity.
• When the logarithm used is base ten, the SI unit is the bel (B).
• Although the bel is computed from quantities with units, the bel is itself unitless.
• The bel is too big for most purposes so the decibel (dB) is more common.

1 bel = 10 decibel

• A level increase of 3 dB is approximately equal to a doubling of intensity (since log 2 ≈ 0.3).
• Sound level values are typically computed using intensity or pressure amplitude.
• The equation for sound intensity level (often abbreviated SIL) in decibels is…  LI = 10 log ⎛⎝ I ⎞⎠ I0
where… LI = sound intensity level [dB] I = intensity [W/m2] I0 = reference intensity [W/m2]
• The equation for sound pressure level (often abbreviated SPL) in decibels is…  LP = 20 log ⎛⎝ ∆P ⎞⎠ ∆P0
where… LP = sound pressure level [dB] ∆P = maximum gauge pressure or pressure amplitude [W/m2] ∆P0 = reference gauge pressure [W/m2]
• The standard reference values for sounds are defined from pressure amplitudes.
• By definition, a pressure amplitude of 20 µPa in air corresponds to a sound pressure level of 0 dB.
• By definition, a pressure amplitude of 1 µPa in water corresponds to a sound pressure level of 0 dB.
• Levels are also commonly used for quantities in audio electronics, radio transmission, and other branches of science and engineering.
• Loudness is…
• FINISH THIS!