Ch 16: Sound & Hearing

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 16: Sound & Hearing

Problem 13a

Chapter 16, Problem 13a

Sound is detected when a sound wave causes the tympanic membrane (the eardrum) to vibrate. Typically, the diameter of this membrane is about 8.4 mm in humans. How much energy is delivered to the eardrum each second when someone whispers (20 dB) a secret in your ear?

Verified step by step guidance

First, understand that the sound intensity level in decibels (dB) is given by the formula:

L = 10 \log \frac{I}{I}

₀, where

I

is the intensity of the sound wave and

I

₀ is the reference intensity, typically

1.0 times 10^- 12 W / m^{2}

Next, rearrange the formula to solve for the intensity

I

I = I

₀⁢10L10. Substitute

L

with

20

dB to find the intensity of the whisper.

Calculate the area of the tympanic membrane using the formula for the area of a circle:

A = π r^{2}

, where

r

is the radius of the eardrum. Convert the diameter from millimeters to meters and divide by 2 to find the radius.

Once you have the intensity and the area, calculate the power delivered to the eardrum using the formula:

P = I A

, where

P

is the power,

I

is the intensity, and

A

is the area.

Finally, since power is the energy delivered per second, the calculated power is the energy delivered to the eardrum each second when someone whispers a secret in your ear.

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Key Concepts

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

Sound Intensity and Decibels

Sound intensity is the power per unit area carried by a wave, measured in watts per square meter (W/m²). Decibels (dB) are a logarithmic unit used to express the intensity of sound, where 0 dB is the threshold of hearing. A whisper at 20 dB indicates a very low sound intensity, which can be converted to power using the formula: I = 10^(dB/10) * I₀, where I₀ is the reference intensity (10⁻¹² W/m²).

Area of the Tympanic Membrane

The tympanic membrane, or eardrum, is a small, circular membrane with a typical diameter of 8.4 mm in humans. To calculate the area, use the formula for the area of a circle: A = πr², where r is the radius. For the eardrum, the radius is half the diameter, which is 4.2 mm or 0.0042 meters. This area is crucial for determining the total energy delivered to the eardrum.

Energy Transfer in Sound Waves

Energy transfer in sound waves is related to the intensity and area over which the sound is distributed. The energy delivered to the eardrum per second, or power, can be calculated using the formula: Power = Intensity × Area. This calculation helps determine how much energy is transferred to the eardrum when a sound wave, such as a whisper, causes it to vibrate.

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Related Practice

Textbook Question

You are trying to overhear a juicy conversation, but from your distance of 15.0 m, it sounds like only an average whisper of 20.0 dB. How close should you move to the chatterboxes for the sound level to be 60.0 dB?

Textbook Question

(a) By what factor must the sound intensity be increased to raise the sound intensity level by 13.0 dB? (b) Explain why you don't need to know the original sound intensity

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