Chapter 12: Q24P (page 328)

Question: (II) Your ears can accommodate a huge range of sound levels. What is the ratio of highest to lowest intensity at (a) 100 Hz, (b) 5000 Hz? (See Fig. 12–6.)

Short Answer

Expert verified

(a) At \(100{\rm{ Hz}}\), highest to lowest intensity ratio is \({10^9}\) and (b) At \(5000{\rm{ Hz}}\), highest to lowest intensity at \(5000{\rm{ Hz}}\) is \({10^{12}}\).

Step by step solution

Step-1:-Loudness for difference frequencies

The graph is a representation of equal loudness for the different frequencies and sound levels.

Step-2:-Given data

The given frequencies are \(100\;{\rm{Hz}}\) and \(5000\;{\rm{Hz}}\).

Step-3:-Calculation of ratio of intensity at \({\bf{100}}\;{\bf{Hz}}\)

At \(100{\rm{ Hz}}\), the highest intensity is approximately around \(1{\rm{ W/}}{{\rm{m}}^2}\) and the lowest intensity is around \({10^{ - 9}}{\rm{ W/}}{{\rm{m}}^2}\). So the ratio is calculated as;

\(\begin{array}{c}\frac{{{\rm{highest intensity}}}}{{{\rm{lowest intensity}}}} = \frac{{1{\rm{ W/}}{{\rm{m}}^2}}}{{{{10}^{ - 9}}{\rm{ W/}}{{\rm{m}}^2}}}\\ = {10^9}\end{array}\)

The ratio of highest to lowest intensity at \(100{\rm{ Hz}}\) is \({10^9}\).

Step-4:-Calculation of ratio of intensity at \({\bf{5000}}\;{\bf{Hz}}\)

At \(5000{\rm{ Hz}}\) the highest intensity is around \({\rm{1}}{{\rm{0}}^{ - 1}}{\rm{ W/}}{{\rm{m}}^2}\) and the lowest intensity is around \({10^{ - 13}}{\rm{ W/}}{{\rm{m}}^2}\). So the ratio is calculated as;

\(\begin{array}{c}\frac{{{\rm{highest intensity}}}}{{{\rm{lowest intensity}}}} = \frac{{{\rm{1}}{{\rm{0}}^{ - 1}}{\rm{ W/}}{{\rm{m}}^2}}}{{{{10}^{ - 13}}{\rm{ W/}}{{\rm{m}}^2}}}\\ = {10^{12}}\end{array}\)

The ratio of highest to lowest intensity at \(5000{\rm{ Hz}}\) is \({10^{12}}\).