Chapter 12, Problem 74P

(a)

To determine

The sound intensity level right in front of the trumpet.

Answer to Problem 74P

The sound intensity level right in front of the trumpet is $138\text{dB}$.

Explanation of Solution

The energy rate of the trumpet is $0.800\text{W}$. The diameter of the bell opening is $12.7\text{cm}$.

Write the formula for the sound intensity level in decibels.

$\beta =\left(10\text{dB}\right){\log }_{10}\frac{I}{I_0}$ (I)

Here, $\beta$ is the intensity in decibels, $I$ is the intensity, $I_0$ is a constant.

Write the formula for intensity.

$I=\frac{P}{A}$ (II)

Here, $P$ is the power, $A$ is the area.

Write the formula for area.

$A=\frac{\pi d^2}{4}$ (III)

Here, $d$ is the diameter.

Substitute equation (III) in (II).

$\beta =\left(10\text{dB}\right){\log }_{10}\frac{4P}{I_0\pi d^2}$ (IV)

Conclusion:

Substitute $0.800\text{W}$ for $P$, $12.7\text{cm}$ for $d$, ${10}^{-12}{\text{W/m}}^{\text{2}}$ for $I_0$

$\begin{array}{c}\beta =\left(10\text{dB}\right){\log }_{10}\frac{4\left(0.800\text{W}\right)}{\left({10}^{-12}{\text{W/m}}^{\text{2}}\right)\pi {\left(12.7\text{cm}\right)}^2}\\ =\left(10\text{dB}\right){\log }_{10}\frac{4\left(0.800\text{W}\right)}{\left({10}^{-12}{\text{W/m}}^{\text{2}}\right)\pi {\left(12.7\times {10}^{-2}\text{m}\right)}^2}\\ =138\text{dB}\end{array}$

The sound intensity level right in front of the trumpet is $138\text{dB}$.

(b)

To determine

The sound intensity level at a distance of $10.0\text{m}$ the trumpet if it radiates sound uniformly in all direction.

Answer to Problem 74P

The sound intensity level at a distance of $10.0\text{m}$ the trumpet if it radiates sound uniformly in all direction is $88\text{dB}$.

Explanation of Solution

The energy rate of the trumpet is $0.800\text{W}$. The diameter of the bell opening is $12.7\text{cm}$.

Write the formula for the sound intensity level in decibels.

$\beta =\left(10\text{dB}\right){\log }_{10}\frac{I}{I_0}$ (I)

Here, $\beta$ is the intensity in decibels, $I$ is the intensity, $I_0$ is a constant.

Write the formula for intensity.

$I=\frac{P}{A}$ (II)

Here, $P$ is the power, $A$ is the area.

When the trumpet is giving out waves uniformly in all directions, the sound wave is spread out in uniform spherical shells.

Write the formula for area.

$A=4\pi d^2$ (V)

Here, $d$ is the distance from the trumpet.

Substitute equation (IV) in (II).

$\beta =\left(10\text{dB}\right){\log }_{10}\frac{P}{I_0\pi d^2}$ (VI)

Conclusion:

Substitute $0.800\text{W}$ for $P$, $10.0\text{m}$ for $d$, ${10}^{-12}{\text{W/m}}^{\text{2}}$ for $I_0$.

$\begin{array}{c}\beta =\left(10\text{dB}\right){\log }_{10}\frac{\left(0.800\text{W}\right)}{\left({10}^{-12}{\text{W/m}}^{\text{2}}\right)4\pi {\left(10.0\text{m}\right)}^2}\\ =88\text{dB}\end{array}$

The sound intensity level at a distance of $10.0\text{m}$ the trumpet if it radiates sound uniformly in all direction is $88\text{dB}$.