Problem 96 - Pearson Physics

Question

A bird watcher is hoping to add the white-throated sparrow to her life list of observed species. How far could she be from the bird and still be able to hear it? Assume that there is no reflection or absorption of the sparrow's sound and that the power of the sound output is \(3.15 \times 10^{-5} \mathrm{~W}\). (Recall that the minimum intensity of sound a human can hear is \(10^{-12} \mathrm{~W} / \mathrm{m}^{2}\).)

Step-by-Step Solution

Answer

The bird watcher can be about 1581 meters away and still hear the sparrow.

Step 1: Understanding the Problem

We need to find the maximum distance from which the sparrow's sound can be heard. Given the power of the sound is \(3.15 \times 10^{-5} \mathrm{~W}\) and the minimum audible intensity for humans is \(10^{-12} \mathrm{~W} / \mathrm{m}^2\).

Step 2: Applying the Intensity Formula

The intensity \(I\) of sound is defined as the power \(P\) spread over an area \(A\), specifically a sphere with radius \(r\) (distance). The formula is:\[ I = \frac{P}{A} = \frac{P}{4\pi r^2} \]Where \(4\pi r^2\) is the surface area of a sphere.

Step 3: Setting Up the Equation

Set the intensity \(I\) equal to the minimum audible intensity:\[ 10^{-12} = \frac{3.15 \times 10^{-5}}{4\pi r^2} \]

Step 4: Solving for Maximum Distance

Solve for \(r\) in the equation. First, multiply both sides by \(4\pi r^2\), then divide to isolate \(r^2\):\[ 10^{-12} \times 4\pi r^2 = 3.15 \times 10^{-5} \]\[ r^2 = \frac{3.15 \times 10^{-5}}{4\pi \times 10^{-12}} \]Calculate \(r^2\) and then find \(r\):\[ r^2 = \frac{3.15 \times 10^{-5}}{4\pi \times 10^{-12}} \approx 2.50 \times 10^6 \]\[ r \approx \sqrt{2.50 \times 10^6} \approx 1581 \text{ meters} \]

Step 5: Concluding the Results

The bird watcher can be approximately 1581 meters away from the sparrow and still be able to hear it, assuming ideal conditions with no sound absorption or reflection.

Key Concepts

Acoustic Power

Acoustic power is a measure of the total energy a sound source, like a sparrow, releases per second. This energy is distributed through the air as sound waves. In the given scenario, the sparrow emits sound with an acoustic power of \(3.15 \times 10^{-5} \text{ W}\). This value represents how much "work" the sparrow's chirping sound does over time to transmit energy into the environment.
To understand acoustic power, consider it as the "strength" of the sound source. Bigger power means a louder sound that can propagate further. If you've ever noticed a loudspeaker at a concert, its strong acoustic power is what allows us to hear it from a distance.

• Sound power is constant no matter the distance from the source.
• Measured in watts (W), indicating energy per unit time.
• Does not change unless the sound source emits more or less energy.

Acoustic power affects how far and clearly we can hear a sound based on its initial energy output.

Minimum Audible Intensity

Minimum audible intensity is the lowest level of sound intensity that the average human ear can detect. A standard value for this is \(10^{-12} \, \text{W/m}^2\). This level is the faintest sound the human ear can hear without any obstructions or amplifications. Think of a whisper or the ticking of a watch from a few meters away as being close to this threshold.
The concept of minimum audible intensity is important for understanding why certain sounds are noticeable from distances and others aren't. If a sound has an intensity below this threshold, it simply won't be picked up by the average listener.

• Intensity is dependent on both the power and area over which it spreads.
• Affected by background noise, which can mask faint sounds.
• Critical for designing devices like hearing aids.

In practical scenarios, achieving the minimum audible intensity enables one to determine how sound propagates through environments.

Spherical Wave Model

The spherical wave model is a mathematical way to consider how sound spreads through open spaces. When a sound source like a bird chirps, it emits sound waves that move outward in all directions, forming a sphere. This spherical spreading is what we account for when calculating how far away a sound can be heard.
Using the formula \(I = \frac{P}{4\pi r^2}\), where \(P\) is power, and \(r\) is the radius, you can determine the intensity at any point of the sphere. Here, intensity tells us how much energy per unit area reaches that point.

• Assumes ideal conditions without interference, reflection, or absorption.
• The surface area increases with distance, causing intensity to decrease.
• Useful for calculating distances in outdoor settings.

This model simplifies complex physical phenomena, allowing us to solve problems like determining the maximum distance for audible sound.