Ch 16: Traveling Waves

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경

Knight Calc 5th Edition

Ch 16: Traveling Waves

문제 82

16장, 문제 82

A loudspeaker, mounted on a tall pole, is engineered to emit 75% of its sound energy into the forward hemisphere, 25% toward the back. You measure an 85 dB sound intensity level when standing 3.5 m in front of and 2.5 m below the speaker. What is the speaker's power output?

검증된 단계별 안내

Determine the sound intensity (I) at the measurement point using the given sound intensity level (L) of 85 dB. Use the formula:

L = 10 \log (I / I_{0})

, where

I_{0}

is the reference intensity (1 × 10⁻¹² W/m²). Rearrange to solve for

I

I = I_{0} × 10

^(L/10).

Calculate the total distance (r) from the speaker to the measurement point using the Pythagorean theorem. The horizontal distance is 3.5 m, and the vertical distance is 2.5 m. Use the formula:

r = \sqrt{{3.5}^{2} + {2.5}^{2}}

Relate the sound intensity (I) to the power output (P) of the speaker using the formula:

I = P × f / (4 π r^{2})

, where

f

is the fraction of the power directed into the forward hemisphere (0.75 in this case). Rearrange to solve for

P

P = I × (4 π r^{2}) / f

Substitute the calculated values of

I

r

, and

f

into the formula for

P

to find the speaker's power output.

Verify the units of the final result to ensure the power output is expressed in watts (W), and confirm that the assumptions (e.g., uniform sound distribution in the forward hemisphere) are consistent with the problem's conditions.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Sound Intensity Level

The sound intensity level, measured in decibels (dB), quantifies the power per unit area of sound waves. It is a logarithmic scale, meaning that an increase of 10 dB represents a tenfold increase in intensity. Understanding this concept is crucial for relating the measured sound level to the actual power output of the loudspeaker.

Inverse Square Law

The inverse square law states that the intensity of sound decreases with the square of the distance from the source. This principle is essential for calculating how sound intensity changes as you move away from the loudspeaker, allowing you to determine the effective power output based on the distance at which the sound level is measured.

Acoustic Power Output

Acoustic power output refers to the total sound energy emitted by the loudspeaker per unit time, typically measured in watts. To find the power output, one must consider the sound intensity level at a specific distance and the distribution of sound energy in different directions, as indicated by the speaker's design.

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교과서 질문

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교과서 질문

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