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?

Ch 15: Mechanical Waves

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 15: Mechanical Waves

Problem 27b

Chapter 15, Problem 27b

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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First, understand that the intensity of a sound wave is defined as the power per unit area. The formula for intensity $ I $ is $ I = \frac{P}{A} $, where $ P $ is the power and $ A $ is the area over which the power is distributed.

Since the sound waves are spreading out equally in all directions, they form a sphere around the point source. The area $ A $ of a sphere is given by $ A = 4\pi r^2 $, where $ r $ is the radius of the sphere.

Substitute the given values into the intensity formula to find the power $ P $. Use $ I = 0.026 \text{ W/m}^2 $ and $ r = 4.3 \text{ m} $ to calculate $ P = I \times A = 0.026 \times 4\pi (4.3)^2 $.

Once you have the power $ P $, calculate the total energy emitted in one hour. Energy $ E $ is the product of power and time, $ E = P \times t $. Convert one hour into seconds (3600 seconds) and use $ E = P \times 3600 $.

Ensure all units are consistent and check your calculations for any errors. The final expression for energy will give you the total sound energy emitted by the source in one hour.

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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 describes how much energy passes through a given area and is crucial for determining the energy output of a sound source. In this problem, the intensity at a specific distance helps calculate the total energy emitted.

Inverse Square Law

The inverse square law states that the intensity of a wave, such as sound, decreases with the square of the distance from the source. This principle is essential for understanding how sound spreads out in space, affecting the calculation of total energy emitted by the source over a given area.

Power and Energy Relationship

Power is the rate at which energy is emitted or transferred, measured in watts (W). Energy is the total amount of work done or heat transferred, measured in joules (J). The relationship between power and energy is given by the formula: Energy = Power × Time. This concept is key to calculating the total sound energy emitted over a specified time period.

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A fellow student with a mathematical bent tells you that the wave function of a traveling wave on a thin rope is $y(x,t)=$\left$(2.30$\operatorname{mm)}\]\cos$[$\left$(16.98$\text{ }$rad/m$\right$)x+(742$\text{ }$rad/s$\right$)t]$ . Being more practical, you measure the rope to have a length of $1.35$\text{ m}$$ and a mass of $0.00338$\operatorname{kg}$$ . You are then asked to determine the following: (d) wave speed; (e) direction the wave is traveling;

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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