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Ch 15: Mechanical Waves
Young & Freedman Calc - University Physics 15th Edition
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 15th EditionCh 15: Mechanical WavesProblem 27a
Chapter 15, Problem 27a

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?

Verified step by step guidance
1
Understand that intensity (I) of a sound wave is defined as the power (P) per unit area (A), and it decreases with distance from the source. The formula for intensity is I = P/A.
Recognize that for a point source emitting sound waves equally in all directions, the area (A) over which the sound spreads is the surface area of a sphere, given by A = 4πr², where r is the distance from the source.
Use the inverse square law for intensity, which states that intensity is inversely proportional to the square of the distance from the source: I₁/I₂ = (r₂/r₁)².
Substitute the known values into the inverse square law formula: I₁ = 0.026 W/m² at r₁ = 4.3 m, and find I₂ at r₂ = 3.1 m.
Solve for I₂ using the formula I₂ = I₁ * (r₁/r₂)², substituting the values to find the intensity at 3.1 m from the source.

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Key Concepts

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

Intensity of Sound Waves

Intensity is the power per unit area carried by a wave, measured in watts per square meter (W/m^2). It describes how much energy a sound wave transmits through a given area and decreases as the distance from the source increases due to the spreading of the wave.
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Sound Intensity Level and the Decibel Scale

Inverse Square Law

The inverse square law states that the intensity of a wave is inversely proportional to the square of the distance from the source. This means that as you move further from the source, the intensity decreases rapidly, specifically by a factor of the distance squared.
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Point Source of Sound

A point source is an idealized source of sound that emits waves equally in all directions. This simplification allows for easier calculations of wave properties like intensity, as it assumes uniform distribution of energy across a spherical surface centered on the source.
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Related Practice
Textbook Question

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)=(2.30mm)cos[(16.98 rad/m)x+(742 rad/s)t]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 m1.35\(\text{ m}\) and a mass of 0.00338kg0.00338\(\operatorname{kg}\). You are then asked to determine the following: (f) tension in the rope; (g) average power transmitted by the wave.

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Textbook Question

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)=(2.30mm)cos[(16.98 rad/m)x+(742 rad/s)t]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 m1.35\(\text{ m}\) and a mass of 0.00338kg0.00338\(\operatorname{kg}\). You are then asked to determine the following: (d) wave speed; (e) direction the wave is traveling;

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Textbook Question

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)=(2.30mm)cos[(16.98 rad/m)x+(742 rad/s)t]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 m1.35\(\text{ m}\) and a mass of 0.00338kg0.00338\(\operatorname{kg}\). You are then asked to determine the following: (a) amplitude; (b) frequency; (c) wavelength.

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Textbook Question

A 1.50-m-long rope is stretched between two supports with a tension that makes the speed of transverse waves 62.0 m/s.What are the wavelength and frequency of the second overtone?

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Textbook Question

A 1.50-m-long rope is stretched between two supports with a tension that makes the speed of transverse waves 62.0 m/s.What are the wavelength and frequency of the fundamental?

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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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