Question

If a sound source can be heard by a human at a distance of 5 km, the acoustic power of the source is

A. 0.001 W
B. 0.01 W
C. 0.1 W
D. 1 W

Answer 1

The sound's audible range at 5 km translates to an acoustic power of 0.1 W, aligning with the inverse square law. This law governs the rapid decrease in sound power with distance, calculated through the surface area of a sphere. Therefore, option C—0.1 W—is the accurate acoustic power of the source.

Explanation:

The sound power P decreases with the square of the distance from the source according to the inverse square law formula:
`\(P \propto (1)/(r^2)\)`, where r is the distance. Given that the sound can be heard at 5 km, let's calculate the power. The threshold of human hearing is about
`\(10^(-12)\) W/m\(^2\)` at a distance of 1 meter. So, at 5 km (5000 m), the power would be:

`\[P = \text{threshold power} * 4\pi r^2 = 10^(-12) * 4\pi * (5000)^2 = 0.1 \, \text{W}\]`

Therefore, the acoustic power of the source is 0.1 W, making option C the correct answer.

The inverse square law governs how sound propagates in a medium, showing that sound power diminishes rapidly with distance. In this case, the threshold power at 1 meter serves as the reference point to calculate the power at 5 km using the formula for surface area of a sphere.

The calculation involves using the surface area of a sphere equation
`\(4\pi r^2\)` to find the total power dissipated over the larger distance. By multiplying the threshold power by this surface area at 5 km, we arrive at the acoustic power of the source, which confirms that it's 0.1 W, falling within the range of human audibility at that distance.

Understanding the relationship between sound power and distance is crucial in determining how sound propagates in the environment. The inverse square law not only applies to sound but is a fundamental principle in various fields, illustrating how quantities such as intensity, power, or radiation flux decrease as the distance from the source increases.