A jet plane at takeoff can produce sound of intensity 10.0 W/m² at 30.0 m away. But you prefer the tranquil sound of normal conversation, which is 1.0 μW/m². Assume that the plane behaves like a point source of sound. (a) What is the closest dis-tance you should live from the airport runway to preserve your peace of mind? (b) What intensity from the jet does your friend experience if she lives twice as far from the runway as you do? (c) What power of sound does the jet produce at takeoff?

Verified step by step guidance

Step 1: Understand the concept of sound intensity and how it relates to distance from a point source. The intensity of sound from a point source decreases with the square of the distance from the source. This is known as the inverse square law for sound intensity.

Step 2: Use the inverse square law to find the closest distance you should live from the airport runway. The formula for sound intensity is I = P / (4πr²), where I is the intensity, P is the power of the source, and r is the distance from the source. You need to solve for r when I is 1.0 μW/m².

Step 3: Calculate the intensity your friend experiences if she lives twice as far from the runway as you do. Use the inverse square law again, knowing that if the distance is doubled, the intensity will be reduced by a factor of four.

Step 4: Determine the power of sound the jet produces at takeoff. Use the given intensity at 30.0 m (10.0 W/m²) and the formula I = P / (4πr²) to solve for P, the power of the sound source.

Step 5: Review the calculations and ensure that the units are consistent throughout the problem. Make sure to convert units where necessary, such as converting μW to W for consistency in calculations.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sound Intensity

Sound intensity is the power per unit area carried by a sound wave, measured in watts per square meter (W/m^2). It quantifies how much sound energy passes through a given area and is crucial for understanding the loudness of sound at different distances from the source.

Inverse Square Law

The inverse square law states that the intensity of sound from a point source decreases with the square of the distance from the source. This means that as you move away from the sound source, the intensity diminishes rapidly, which is essential for calculating sound levels at various distances.

Sound Power

Sound power is the total energy emitted by a sound source per unit time, measured in watts (W). Unlike intensity, sound power is independent of distance and describes the source's ability to produce sound, which is vital for determining the total acoustic output of the jet plane.

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Textbook Question

The speed of sound in air at 20°C is 344 m/s. (a) What is the wavelength of a sound wave with a frequency of 784 Hz, corresponding to the note G₅ on a piano, and how many milliseconds does each vibration take? (b) What is the wavelength of a sound wave one octave higher (twice the frequency) than the note in part (a)?

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Textbook Question

Energy Output. By measurement you determine that sound waves are spreading out equally in all directions from a point source and that the intensity is 0.026 W/m² at a distance of 4.3 m from the source. What is the intensity at a distance of 3.1 m from the source?

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Textbook Question

Energy Output. By measurement you determine that sound waves are spreading out equally in all directions from a point source and that the intensity is 0.026 W/m² at a distance of 4.3 m from the source. How much sound energy does the source emit in one hour if its power output remains constant?

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