Question

In a test of a subsonic jet, the jet flies overhead at an altitude of room. The sound intensity on the
ground as the jet passes overhead is \[60{\text{ }}dB\]. At what altitude should the jet plane fly so that
the ground noise is not greater than \[120{\text{ }}dB\]?

Answer 1

Hint:
1. Loudness is given by the formula \[L = 10 \times {\log _{10}}\left( {\dfrac{I}{{{I_0}}}} \right)\]. It is measured in decibels.
2. The intensity I is given by \[I = \dfrac{{{I_0}}}{{{r^2}}}\]

Complete step by step Solution:

The relationship between loudness & the intensity is given by
     \[L = 10\log \left( {\dfrac{I}{{{I_0}}}} \right)\]
Where ‘L is the sound intensity level \[\left( {loudness} \right),\]I is the sound’s intensity & I0 is the intensity of the reference \[{I_0} = 1pw/{m^2}\]
By using this you can solve the problem.
Given attitude height \[ = {\text{ }}100{\text{ }}m\]
The sound intensity on the ground as the jet passes by is \[160{\text{ }}dB\]. Hence from the formula
\[L = 10\log \left( {\dfrac{I}{{{I_0}}}} \right)\]
\[ = 10\log \left( {\dfrac{{{I_0}}}{{{r^2}{I_0}}}} \right) = 10\log \left( {\dfrac{1}{{{r^2}}}} \right)\]
\[L = 160\,\,dB,\,\,r = 100\,m\]
\[160\,\,dB = 10\,\log \left( {\dfrac{1}{{{{\left( {\log } \right)}^2}}}} \right)\] ……….(1)
Let the destitute be x meter for the sound intensity of \[120{\text{ }}dB.\]
So, \[120 = 10{\log _{10}}\left( {\dfrac{1}{{{x^2}}}} \right)\] ………(2)
Subtraction equation (2) form (1)
\[\dfrac{{160 - 120}}{{10}} = {\log _{10}}\left( {\dfrac{1}{{10000}}} \right) - {\log _{10}}\left( {\dfrac{1}{{{x^2}}}} \right)\]
\[4 = {\log _{10}}\left( {\dfrac{{{x^2}}}{{10000}}} \right)\]
\[ \Rightarrow {x^2} = {\left( {10000} \right)^2}\]
\[{x^2} = 10000 = 10\,\,km\]
\[\therefore \] the jet should pass \[10{\text{ }}km\] above the ground so that noise 9 not greater than \[120{\text{ }}dB.\]

Note:
1. Intensity is also given as power used per unit surface area. \[\dfrac{\rho }{{4\pi {r^2}}}\]
2. Loudness also depends on the pressure. If the pressure increases, pitch increases, so loudness increases.