AcademySound
Academy
Sound Intensity
Level 1 - Physics topic page in Sound.
Principle
Sound intensity is average power transmitted per unit area.
Notation
\(I\)
sound intensity
\(\mathrm{W\,m^{-2}}\)
\(P\)
average sound power
\(\mathrm{W}\)
\(A\)
area crossed by the sound
\(\mathrm{m^{2}}\)
\(\Delta p_{\max}\)
pressure amplitude
\(\mathrm{Pa}\)
\(\rho\)
medium density
\(\mathrm{kg\,m^{-3}}\)
\(\beta\)
sound intensity level
\(\mathrm{dB}\)
Method
Intensity measures energy transport, not pressure itself. For a plane sinusoidal sound wave, intensity scales with the square of pressure amplitude.
Power per area
\[I=\frac{P}{A}\]
Pressure-amplitude form
\[I=\frac{(\Delta p_{\max})^2}{2\rho v}\]
Spherical spreading
\[I=\frac{P}{4\pi r^2}\]
Use this for an ideal point source radiating equally in all directions.
Decibel level
\[\beta=10\log_{10}\left(\frac{I}{I_0}\right)\]
The logarithmic level compares an intensity with a reference intensity \(I_0\), commonly \(10^\{-12\}\\,\\mathrm\{W\\,m^\{-2\}}\) in air acoustics.
Rules
These are the compact intensity relations.
Intensity
\[I=\frac{P}{A}\]
Pressure form
\[I=\frac{(\Delta p_{\max})^2}{2\rho v}\]
Point source
\[I=\frac{P}{4\pi r^2}\]
Decibel level
\[\beta=10\log_{10}\left(\frac{I}{I_0}\right)\]
Examples
Question
A source radiates
\[0.50\,\mathrm{W}\]
uniformly. Find the intensity \[4.0\,\mathrm{m}\]
away.Answer
\[I=\frac{P}{4\pi r^2}=\frac{0.50}{4\pi(4.0)^2}=2.5\times10^{-3}\,\mathrm{W\,m^{-2}}\]
Checks
- Doubling pressure amplitude quadruples intensity in the plane-wave model.
- Moving twice as far from an ideal point source reduces intensity by a factor of four.
- Decibels are logarithmic; adding \(10\\,\\mathrm\{dB\}\) multiplies intensity by \(10\).
- Intensity is always nonnegative.