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(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.

Short Answer

Expert verified
The sound intensity must be increased by a factor of 20. The original intensity is unnecessary as the increase depends only on the change in dB level.

Step by step solution

01

Understanding the Problem

We need to determine by how many times the sound intensity must be increased to increase the sound intensity level by 13.0 dB. This involves understanding the relationship between intensity levels in decibels and sound intensity.
02

The Decibel Formula

The sound intensity level in decibels (dB) is calculated using the formula: \( L = 10 \log_{10} \left( \frac{I}{I_0} \right) \), where \( L \) is the sound level in decibels, \( I \) is the sound intensity, and \( I_0 \) is the reference intensity, typically \( 10^{-12} \text{ W/m}^2 \).
03

Relating Change in Levels to Intensity

When the intensity level increases by \( \Delta L = 13.0 \text{ dB} \), it can be expressed as: \[ \Delta L = 10 \log_{10} \left( \frac{I_{new}}{I_{old}} \right) = 13.0 \].We need to solve for \( \frac{I_{new}}{I_{old}} \).
04

Isolating the Intensity Ratio

Rearrange the equation to solve for the intensity ratio:\[ \log_{10} \left( \frac{I_{new}}{I_{old}} \right) = \frac{13.0}{10} \],which simplifies to\[ \log_{10} \left( \frac{I_{new}}{I_{old}} \right) = 1.3 \].
05

Calculating the Intensity Factor

To solve for \( \frac{I_{new}}{I_{old}} \), exponentiate both sides:\[ \frac{I_{new}}{I_{old}} = 10^{1.3} \].Calculate this value,\[ \frac{I_{new}}{I_{old}} \approx 19.95 \].The sound intensity must be increased by a factor of approximately 20.
06

Explanation of Independence from Original Intensity

The formula for decibels involves a factor where sound intensity is expressed as a ratio (\( \frac{I}{I_0} \)), meaning that any proportional increase in \( I \) results in the same change in decibels regardless of the starting intensity. Hence, we do not need to know the original sound intensity for this calculation.

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Key Concepts

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

Decibel Formula
The decibel formula is a key concept for understanding sound intensity levels. This formula allows you to calculate the sound level in decibels (dB) using the intensity of the sound compared to a reference intensity.

The formula is expressed as:
  • \( L = 10 \log_{10} \left( \frac{I}{I_0} \right) \)
where:
  • \( L \) is the sound level in decibels
  • \( I \) is the sound intensity
  • \( I_0 \) is the reference intensity, typically \( 10^{-12} \text{ W/m}^2 \)
This logarithmic formula indicates that sound intensity is measured relative to the quietest sound the average human ear can hear. Understanding this formula is crucial for solving problems related to sound intensity changes.
Sound Intensity Increase
Increasing sound intensity can significantly affect the sound intensity level measured in decibels. When you aim to increase the sound intensity level, like in our exercise where a 13.0 dB increase is required, this involves a substantial change in the energy or power of the sound.

But, how do we calculate this increase? By referring to the decibel formula, we identify the intensity increase necessary to achieve a specific dB level change. For a 13.0 dB increase, as calculated in the example:
  • The intensity must be increased by approximately a factor of 20.
This calculation emphasizes how even small changes in decibels can relate to large changes in actual sound intensity, illustrating the "amplifying" effect of decibels in everyday experiences.
Intensity Ratio
The intensity ratio is significant when determining how much a sound's intensity needs to change to achieve a different decibel level. In our example, we examined how the intensity ratio, \( \frac{I_{new}}{I_{old}} \), helps solve for the sound intensity increase needed.

To find this ratio:
  • We start from the equation derived from the decibel difference: \( \log_{10} \left( \frac{I_{new}}{I_{old}} \right) = 1.3 \).
  • Exponentiating both sides, the intensity ratio is calculated as \( 10^{1.3} \), resulting in about 19.95.
This means, to increase the decibel level by 13.0 dB, the sound intensity must be multiplied nearly 20 times.
Logarithmic Scale
A logarithmic scale is used in the decibel system to quantify sound levels. This scale is not linear, meaning that equal steps on the scale represent equal ratios rather than equal increments.

Here's why it's helpful:
  • It allows us to manage very large ranges of values in a compact format.
  • A small increase in decibels represents a large increase in sound intensity.
  • Helps to better reflect how humans perceive sound, as our ears are sensitive to logarithmic changes.
Understanding the logarithmic scale clarifies why sound can feel exponentially louder despite small increments in decibel measurements.

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Most popular questions from this chapter

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?

The human vocal tract is a pipe that extends about 17 cm from the lips to the vocal folds (also called "vocal cords") near the middle of your throat. The vocal folds behave rather like the reed of a clarinet, and the vocal tract acts like a stopped pipe. Estimate the first three standing-wave frequencies of the vocal tract. Use \(v =\) 344 m/s. (The answers are only an estimate, since the position of lips and tongue affects the motion of air in the vocal tract.)

The longest pipe found in most medium-size pipe organs is 4.88 m (16 ft) long. What is the frequency of the note corresponding to the fundamental mode if the pipe is (a) open at both ends, (b) open at one end and closed at the other?

A police siren of frequency \(f_{siren}\) is attached to a vibrating platform. The platform and siren oscillate up and down in simple harmonic motion with amplitude \(A_p\) and frequency \(f_p\). (a) Find the maximum and minimum sound frequencies that you would hear at a position directly above the siren. (b) At what point in the motion of the platform is the maximum frequency heard? The minimum frequency? Explain.

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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