Human perception of relative intensity levels and the need for a…

The idea

Human hearing responds to a very wide range of sound intensities, from about 1 × 10⁻¹² W m⁻² at the threshold to around 1 W m⁻² at the pain threshold, a ratio of about 10¹². A linear scale would compress most everyday sounds into a tiny part of the range and make comparisons awkward. The logarithmic decibel scale compresses this range: sound intensity level in dB is 10 log₁₀(I / I₀), where I₀ is 1 × 10⁻¹² W m⁻². Equal ratios of intensity correspond to equal differences in dB, which matches the way loudness is perceived: a sound must increase by roughly a factor of ten in intensity to be heard as about twice as loud. Thus the logarithmic scale reflects relative, rather than absolute, changes in intensity.

What earns the marks

• State that the ear detects sound intensities spanning many orders of magnitude, so a linear intensity scale would be unwieldy.
• Define sound intensity level using the logarithmic relationship 10 log₁₀(I / I₀), with I₀ equal to 1 × 10⁻¹² W m⁻².
• Explain that equal ratios of intensity produce equal differences in decibels, so the scale reflects relative changes in intensity.
• Relate the logarithmic scale to perception by noting that a tenfold increase in intensity corresponds to an increase of 10 dB and is heard as a modest change in loudness.
• Use the scale to compare sounds, for example recognising that 60 dB is 10³ times the intensity of 30 dB because the difference is 30 dB.

Common mistakes and how to put them right

• Treating the decibel value as directly proportional to intensity.

  Put it right by using the logarithmic relationship and noting that equal dB differences correspond to equal intensity ratios.

• Using the wrong reference intensity or omitting I₀ when calculating sound intensity level.

  Put it right by stating that I₀ is 1 × 10⁻¹² W m⁻².

• Assuming a 10 dB increase doubles loudness.

  Put it right by explaining that a tenfold intensity increase of 10 dB is perceived as only a moderate loudness change, often described as roughly a doubling of perceived loudness.

Exam tips

• Quote the logarithmic equation and the reference intensity when explaining why the decibel scale is used.
• Convert between intensity ratios and dB differences by using powers of ten, for example a 20 dB difference corresponds to an intensity ratio of 10².
• Link the mathematical compression of the scale to the everyday observation that very large intensity changes are heard as moderate loudness changes.