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Chapter 17: Problem 19

Calculate the sound level in decibels of a sound wave that has an intensity of 4.00\(\mu \mathrm{W} / \mathrm{m}^{2}\) .

Short Answer

Expert verified
The sound level is 60.2 decibels.

Step by step solution

01

Understanding the Decibel Scale

The sound level in decibels (dB) can be calculated using the following formula: \[ L = 10 \log{\left(\frac{I}{I_0}\right)} \] where \( L \) is the sound level in decibels, \( I \) is the intensity of the sound wave in watts per square meter, and \( I_0 \) is the reference intensity, which is typically \( 1 \times 10^{-12} \) W/m\( ^2 \). This reference intensity corresponds to the threshold of hearing.
02

Convert the Intensity into Watts per Square Meter

We have the given intensity \( I = 4.00 \mu \mathrm{W} / \mathrm{m}^{2} \). Since \(1 \mu \mathrm{W} = 1 \times 10^{-6} \mathrm{W} \), we convert the intensity to watts per square meter: \[ I = 4.00 \times 10^{-6} \mathrm{W}/\mathrm{m}^2 \].
03

Apply the Formula to Find the Sound Level

Substitute the value of \( I \) and \( I_0 \) into the decibel formula: \[ L = 10 \log{\left(\frac{4.00 \times 10^{-6}}{1 \times 10^{-12}}\right)} \] Now use a calculator to evaluate the logarithm and find \( L \).
04

Calculate the Result

Calculate the result: \[ L = 10 \log{\left(4.00 \times 10^{6}\right)} \] Simplify within the logarithm to get \( L = 10 \times 6.02 \), since \( \log{4.00 \times 10^{6}} \approx 6.02 \). Finally, \( L = 60.2 \) dB.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Decibels
Decibels (dB) are a unit of measurement that quantifies the intensity of a sound or the power level of an electrical signal by comparing it with a given level on a logarithmic scale. It's named after Alexander Graham Bell, the inventor of the telephone. The dB scale is a convenient way to express both very large and very small values in a compact format. In the context of sound, the decibel level tells us how powerful a sound is relative to a standard reference level.

Typically, the softest sound a human ear can hear, also known as the threshold of hearing, is used as this reference point. It's important to note that the decibel scale is logarithmic, not linear. This means that an increase by 10 dB represents a sound that is 10 times more intense, or powerful. However, to our ears, it sounds roughly twice as loud. Understanding dB is crucial because it helps compare the physical intensity of different sounds and gauge their potential impact on hearing.
Intensity of Sound Wave
The intensity of a sound wave is a measure of the sound energy transmitted through a unit area in a specified direction per unit time. Measured in watts per square meter (W/ would be ten times louder than 60 dB. With this understanding, students can approach sound level calculations with a clearer grasp of the underlying principles.

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Most popular questions from this chapter

The ocean floor is underlain by a layer of basalt that constitutes the crust, or uppermost layer, of the Earth in that region. Below this crust is found denser periodotite rock, which forms the Earth鈥檚 mantle. The boundary between these two layers is called the Mohorovicic discontinuity (鈥淢oho鈥� for short). If an explosive charge is set off at the surface of the basalt, it generates a seismic wave that is reflected back out at the Moho. If the speed of this wave in basalt is 6.50 km/s and the two-way travel time is 1.85 s, what is the thickness of this oceanic crust?

A rescue plane flies horizontally at a constant speed searching for a disabled boat. When the plane is directly above the boat, the boat's crew blows a loud horn. By the time the plane's sound detector perceives the horn's sound, the plane has traveled a distance equal to half its altitude above the ocean. If it takes the sound 2.00 s to reach the plane, determine (a) the speed of the plane and (b) its altitude. Take the speed of sound to be 343 \(\mathrm{m} / \mathrm{s}\) .

To permit measurement of her speed, a skydiver carries a buzer emitting a steady tone at 1800 \(\mathrm{Hz}\) . A friend on the ground at the landing site directly below listens to the amplified sound he receives. Assume that the air is calm and that the sound speed is 343 \(\mathrm{m} / \mathrm{s}\) , independent of altitude. While the skydiver is falling at terminal speed, her friend on the ground receives waves of frequency 2150 \(\mathrm{Hz}\) . (a) What is the skydiver's speed of descent? (b) What If? Suppose the skydiver can hear the sound of the buzer reflected from the ground. What frequency does she receive?

Suppose that you hear a clap of thunder 16.2 s after seeing the associated lightning stroke. The speed of sound waves in air is 343 \(\mathrm{m} / \mathrm{s}\) , and the speed of light is \(8.00 \times 10^{8} \mathrm{m} / \mathrm{s}\) . How fare you from the lightning stroke?

The smallest wavelength possible for a sound wave in air is on the order of the separation distance between air molecules. Find the order of magnitude of the highest-frequency sound wave possible in air, assuming a wave speed of 343 \(\mathrm{m} / \mathrm{s}\) , density \(1.20 \mathrm{kg} / \mathrm{m}^{3},\) and an average molecular mass of \(4.82 \times 10^{-26} \mathrm{kg}\) .
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