Question

A jet plane at take-off can produce sound of intensity 9.20 W/m^2 at 30.2 m away. But you prefer the tranquil sound of normal conversation, which is 1.0 m0W/m^2. Assume that the plane behaves like a point source of sound.
Part A: What is the closest distance you should live from the airport runway to preserve your peace of mind? r=?km
Part B: What intensity from the jet does your friend experience if she lives twice as far from the runway as you do? I= ? m0W/m^2
Part C: What power of sound does the jet produce at take-off? P =? W

Answer 1

Concepts and reason

The concept required to solve the given problem is intensity of sound wave.

First obtain the relation between intensity and distance by using expression for intensity of sound wave. Then find the closest distance from the run way and intensity of sound from the jet by using relation between intensity and distance. Finally, find the power of sound by using relation between power and intensity.

Fundamentals

The intensity I of sound wave is the ratio of power P and area A of the surface perceived the sound.

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

For a point source, the power distributed in concentric spheres with an increase in radius and therefore an increase in area $4\pi {r^2}.$

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

Here, P is the power of the sound, r is the radius of the sphere, and I is the intensity of sound wave.

(A)

The relation between distance and intensity is given as follows:

$\frac{{{I_1}}}{{{I_2}}} = \frac{{r_2^2}}{{r_1^2}}$

Here, I₁ and I₂ are the intensity of sound wave at distance r₁ and r₂ respectively.

Rearrange the above equation for ${r_2}.$

${r_2} = {r_1}\sqrt {\left( {\frac{{{I_1}}}{{{I_2}}}} \right)}$

Substitute 30.2 m for r₁, $9.20{\rm{ W/}}{{\rm{m}}^{\rm{2}}}$ for I₁, and $1 \times {10^{ - 6}}{\rm{ W/}}{{\rm{m}}^{\rm{2}}}$ for I₂

$\begin{array}{c}\\{r_2} = {r_1}\sqrt {\left( {\frac{{{I_1}}}{{{I_2}}}} \right)} \\\\ = \left( {30.2{\rm{ m}}} \right)\sqrt {\left( {\frac{{9.20{\rm{ W/}}{{\rm{m}}^{\rm{2}}}}}{{1.0 \times {{10}^{ - 6}}{\rm{ W/}}{{\rm{m}}^{\rm{2}}}}}} \right)} \\\\ = 9.16 \times {10^4}{\rm{ m}}\\\\{\rm{ = 91}}{\rm{.6 km}}\\\end{array}$

(B)

The expression for the intensity experienced by the person from the jet plane is,

$\frac{{{I_1}}}{{{I_2}}} = \frac{{r_2^2}}{{r_1^2}}$

Here, I₁ and I₂ are the intensity of sound wave and r₁ and r₂ are the distances upto which sound wave perceived.

Rearrange the above equation for ${I_2}.$

${I_2} = \left( {\frac{{r_1^2}}{{r_2^2}}} \right){I_1}$

Substitute 91.6 km for r₁, 183.2 km for r₂, and $1.0 \times {10^{ - 6}}{\rm{ W/}}{{\rm{m}}^{\rm{2}}}$ for I₁.

$\begin{array}{c}\\{I_2} = \left( {\frac{{r_1^2}}{{r_2^2}}} \right){I_1}\\\\ = {\left( {\frac{{91.6{\rm{ km}}}}{{183.2{\rm{ km}}}}} \right)^2}\left( {1.0 \times {{10}^{ - 6}}{\rm{ W/}}{{\rm{m}}^{\rm{2}}}} \right)\\\\ = 0.25 \times {10^{ - 6}}{\rm{ W/}}{{\rm{m}}^{\rm{2}}}\left( {\frac{{{{10}^6}\,\mu {\rm{W}}}}{{1\,{\rm{W}}}}} \right)\\\\ = 0.25{\rm{ }}\mu {\rm{W/}}{{\rm{m}}^{\rm{2}}}\\\end{array}$

(C)

The expression for the power produced by the jet plane is,

$\begin{array}{c}\\P = I\left( {4\pi {r^2}} \right)\\\\ = \left( {9.20{\rm{ W/}}{{\rm{m}}^{\rm{2}}}} \right)\left( {4\pi {{\left( {30.2{\rm{ m}}} \right)}^2}} \right)\\\\ = 1.05 \times {10^5}{\rm{ W}}\\\end{array}$

Ans: Part A

The closest distance from the airport you should live peacefully is 91.6 km.

Part B

The intensity from the jet that your friend experience if she lives twice as far from the runway is $0.25{\rm{ }}\mu {\rm{W/}}{{\rm{m}}^{\rm{2}}}$ .

Part C

The power produced by the jet plane is $1.05 \times {10^5}{\rm{ W}}$ .