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You live on a busy street, but as a music lover, you want to reduce the traffic noise. (a) If you install special soundreflecting windows that reduce the sound intensity level (in dB) by 30 dB, by what fraction have you lowered the sound intensity (in W\(/\)m\(^2\))? (b) If, instead, you reduce the intensity by half, what change (in dB) do you make in the sound intensity level?

Short Answer

Expert verified
(a) Intensity reduced by 1/1000. (b) Change is -3 dB.

Step by step solution

01

Understanding the dB Scale

The decibel (dB) scale is logarithmic and measures sound intensity levels. A change of 10 dB represents a tenfold change in intensity.
02

Converting dB Reduction to Intensity Reduction

A reduction of 30 dB means \(10^{ rac{-30}{10}} = 10^{-3}\).Thus, the intensity is reduced to \( \frac{1}{1000} \) of its original value.
03

Calculating Fraction of Intensity Reduction

The fraction by which the sound intensity is reduced is \( \frac{1}{1000} \), meaning the sound intensity is reduced to 0.1% of its original level.
04

Understanding Half Reduction in Intensity

A reduction by half means the new intensity is \( \frac{1}{2} \) of the original intensity.
05

Calculating dB Change for Half Intensity Reduction

Use the formula for intensity level difference: \(\Delta L = 10 \log_{10} \left( \frac{I_2}{I_1} \right)\),where \( I_2 = \frac{1}{2} I_1 \).Substitute and calculate:\(\Delta L = 10 \log_{10} \left( \frac{1}{2} \right) = -3 \) dB.

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Key Concepts

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

Decibel Scale
The decibel (dB) scale is a unit of measurement used to represent sound intensity levels. It is particularly useful because it translates large variations in sound intensity into a more manageable range of numbers. This scale is logarithmic, not linear, meaning each step on the scale represents a multiplicative change in intensity. For instance, an increase in sound intensity by 10 dB equates to a tenfold increase in intensity, allowing us to easily comprehend very large or very small numbers without losing precision or clarity.
When you hear that the sound level is reduced by 30 dB, you can imagine it as dropping down several levels of intensity, specifically reducing the sound power to one-thousandth (\(\frac{1}{1000}\)) of the original level. This little trick of the dB scale enables better communication of sound reductions or increases, especially in fields like acoustics, where sound intensity can vary drastically from quiet whispers to loud jet engines.
Logarithmic Scale
A logarithmic scale is different from a linear scale. Instead of increasing in equal steps, each step in a logarithmic scale is a constant factor, such as tenfold, which simplifies handling large ranges of values. This is the principle behind the decibel system for sound.
  • Logarithmic scales help represent hard-to-grasp phenomena like sound and earthquakes.
  • They convert multiplicative relationships into additive ones, providing clarity.

In sound measurements, this makes it much easier to handle the wide spectrum from the soft rustling of leaves to the roaring of a jet. By using a logarithm to express changes in intensity, the decibel scale offers a pleasant way to translate these differences into understandable terms. For example, a change of 10 dB means the intensity changes by a factor of 10. Thus, a 20 dB change represents a 100-fold difference due to the power of the logarithm being raised with each jump. This simplifies large multiplicative changes of sound into more manageable additive steps.
Sound Reduction
Sound reduction is the process of decreasing the intensity and therefore the impact of unwanted noise. Understanding sound reduction often involves grasping the concept of decibels and logarithmic scales, as these help us quantify how effective noise reduction measures are.
  • For instance, reducing sound intensity by 30 dB is a significant decrease as it represents a reduction by 99.9% to just 0.1% of the original intensity.
  • Alternatively, reducing the intensity by half (\(\frac{1}{2}\)) results in a reduction of about 3 dB, a far smaller scale of change.

Each method of sound reduction, whether through special windows, insulation, or soundproofing materials, has its specific level of efficacy. The decibel and logarithmic scales help us appreciate these differences and understand how effective each method might be. By learning how various sound reduction techniques are measured and what those measurements translate to in terms of real-world effect, one can make informed decisions on the best way to mitigate noise in various environments such as homes, offices, or urban spaces.

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Most popular questions from this chapter

Standing sound waves are produced in a pipe that is 1.20 m long. For the fundamental and first two overtones, determine the locations along the pipe (measured from the left end) of the displacement nodes and the pressure nodes if (a) the pipe is open at both ends and (b) the pipe is closed at the left end and open at the right end.

The sound from a trumpet radiates uniformly in all directions in 20\(^\circ\)C air. At a distance of 5.00 m from the trumpet the sound intensity level is 52.0 dB. The frequency is 587 Hz. (a) What is the pressure amplitude at this distance? (b) What is the displacement amplitude? (c) At what distance is the sound intensity level 30.0 dB?

Horseshoe bats (genus \(Rhinolophus\)) emit sounds from their nostrils and then listen to the frequency of the sound reflected from their prey to determine the prey's speed. (The "horseshoe" that gives the bat its name is a depression around the nostrils that acts like a focusing mirror, so that the bat emits sound in a narrow beam like a flashlight.) A \(Rhinolophus\) flying at speed \(v_{bat}\) emits sound of frequency \(f_{bat}\); the sound it hears reflected from an insect flying toward it has a higher frequency \(f_{refl}\). (a) Show that the speed of the insect is $$vinsect = v\Bigg[\frac{f_{refl}(v - v_{bat}) - f_{bat}(v + v_{bat})}{f_{refl}(v - v_{bat}) + f_{bat}(v + v_{bat})}\Bigg] $$ where \(v\) is the speed of sound. (b) If \(f_{bat} =\) 80.7 kHz, \(f_{refl} =\) 83.5 kHz, and \(v_{bat} =\) 3.9 m/s, calculate the speed of the insect.

A baby's mouth is 30 cm from her father's ear and 1.50 m from her mother's ear. What is the difference between the sound intensity levels heard by the father and by the mother?

A stationary police car emits a sound of frequency 1200 Hz that bounces off a car on the highway and returns with a frequency of 1250 Hz. The police car is right next to the highway, so the moving car is traveling directly toward or away from it. (a) How fast was the moving car going? Was it moving toward or away from the police car? (b) What frequency would the police car have received if it had been traveling toward the other car at 20.0 m/s?

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