Sound intensity and decibels Knowledge-Based Study

Study notes

1. Definition of Sound Intensity

   Sound waves transmit mechanical energy through a medium via vibrations. Sound intensity, denoted as I, is defined as the average rate at which sound energy flows through a unit area perpendicular to the direction of propagation:

   $I = \frac{P}{A}$

   where P represents the acoustic power in watts (W) and A represents the surface area in square meters ($\text{m}^2$). The SI unit of sound intensity is watts per square meter ($\text{W/m}^2$).

   Spherical Waves and the Inverse-Square Law

   When an isotropic point source radiates sound uniformly in all directions into an open, lossless medium, the acoustic energy spreads over the surface of expanding spheres of radius r. Since the surface area of a sphere is $A = 4\pi r^2$, the intensity at distance r from the source is:

   $I = \frac{P}{4\pi r^2}$

   This relationship shows that sound intensity is inversely proportional to the square of the distance from a point source:

   $I \propto \frac{1}{r^2}$

   Doubling the distance from the sound source reduces the intensity to one-fourth of its initial value.

   Why a Logarithmic Scale?

   The human ear can detect an extraordinarily broad range of sound intensities, spanning from approximately $10^{-12}\text{ W/m}^2$ to over $1\text{ W/m}^2$. Because human perception of loudness is roughly logarithmic rather than linear, intensity is conventionally measured on a logarithmic scale known as the sound intensity level ($\beta$).

   Definition of Sound Intensity Level

   The sound intensity level in decibels (dB) is defined relative to the threshold of hearing ($I_0$):

   $\beta = 10 \log_{10}\left(\frac{I}{I_0}\right)$

   where:

   • $I$ is the sound intensity in $\text{W/m}^2$.
   • $I_0 = 1.0 \times 10^{-12}\text{ W/m}^2$ is the standard reference intensity, corresponding roughly to the quietest sound an average human ear can perceive at a frequency of 1000 Hz.
   • $\beta$ is expressed in decibels (dB), a dimensionless quantity.

   At the threshold of hearing, $I = I_0$, giving $\beta = 10 \log_{10}(1) = 0\text{ dB}$. At the threshold of pain, $I = 1.0\text{ W/m}^2$, yielding $\beta = 10 \log_{10}(10^{12}) = 120\text{ dB}$.

   Changes in Sound Intensity Level

   Because the decibel scale is logarithmic, multiplicative changes in intensity correspond to additive shifts in decibel level:

   • Factor of 10 increase: If intensity increases by a factor of 10, $\Delta \beta = 10 \log_{10}(10) = +10\text{ dB}$.
   • Doubling the intensity: If intensity doubles ($I_2 = 2I_1$), the increase in level is $\Delta \beta = 10 \log_{10}(2) \approx +3.01\text{ dB}$.

   Combining Independent Sound Sources

   Decibel levels cannot be added directly. When multiple incoherent sound sources emit sound simultaneously, their intensities add arithmetically, not their decibel ratings.

   For example, if one machine produces a level of 70 dB, two identical machines operating simultaneously will produce twice the intensity:

   $I_{\text{total}} = 2I_1$

   $\beta_{\text{total}} = 10 \log_{10}\left(\frac{2I_1}{I_0}\right) = 10 \log_{10}\left(\frac{I_1}{I_0}\right) + 10 \log_{10}(2) = 70\text{ dB} + 3.01\text{ dB} \approx 73\text{ dB}$

   Ten identical 70 dB sources operating together would raise the total level by 10 dB to 80 dB.