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Chapter 16: Problem 102

An amplified guitar has a sound intensity level that is 14 dB greater than the same unamplified sound. What is the ratio of the amplified intensity to the unamplified intensity?

Short Answer

Expert verified
The intensity ratio is approximately 25.12.

Step by step solution

01

Understanding the Decibel Scale

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 dB, \( I \) is the intensity, and \( I_0 \) is the reference intensity. A difference of 14 dB means that we have \( L_{amp} = L_{unamp} + 14 \).
02

Formulate the Intensity Ratio Equation

The difference in sound levels can be expressed in terms of the intensities: \( 14 = 10 \, \log_{10} \left( \frac{I_{amp}}{I_{unamp}} \right) \), where \( I_{amp} \) is the intensity of the amplified sound and \( I_{unamp} \) is the intensity of the unamplified sound.
03

Solve for the Intensity Ratio

Divide both sides of the equation by 10 to isolate the logarithm: \( 1.4 = \log_{10} \left( \frac{I_{amp}}{I_{unamp}} \right) \). Exponentiate both sides with base 10 to solve for the ratio: \( \frac{I_{amp}}{I_{unamp}} = 10^{1.4} \).
04

Calculate the Final Ratio

Compute \( 10^{1.4} \) to find the ratio: \( 10^{1.4} \approx 25.12 \). Therefore, the intensity of the amplified sound is approximately 25.12 times greater than that of the unamplified sound.

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Key Concepts

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

Decibel Scale
Decibels are a unit of measurement used to express the intensity level of sounds. The decibel scale is logarithmic, meaning each increase of 10 dB represents a tenfold increase in sound intensity. This is important because our ears perceive sound intensity on a logarithmic scale. To compare two sound levels, we use the formula:
  • \[ L = 10 \log_{10} \left( \frac{I}{I_0} \right) \]
where \( L \) is the sound level in decibels, \( I \) is the intensity of the sound, and \( I_0 \) is a reference intensity, often taken as the threshold of hearing (e.g., \( 10^{-12} \, \text{W/m}^2 \)).
Understanding the decibel formula helps us find how many times more intense one sound is compared to another. In this exercise, a 14 dB increase means the amplified sound is significantly more intense than the unamplified sound, demonstrating the logarithmic nature of the scale.
The decibel scale is widely used in many fields due to its ability to concisely express large variations in sound intensities minimizing the complexity.
Intensity Ratio
The intensity ratio is a comparison of two sound intensities, often expressed as a ratio of one intensity to another. This can provide insight into how much stronger or weaker one sound is compared to another. In the case of the amplified and unamplified guitar sound, the exercise demonstrates calculating these differences using decibels.
Using the relationship from the original solution, we can write:
  • \[ 14 = 10 \log_{10} \left( \frac{I_{amp}}{I_{unamp}} \right) \]
This sets up an equation to find how the ratio relates to the decibel increase. By rearranging and solving the logarithmic equation, we find how many times \( I_{amp} \) is compared to \( I_{unamp} \).
Understanding intensity ratio is key in comparing sound levels, particularly in audio engineering and acoustics where precise control of sound is critical. Calculating intensity ratios reveals the magnitude impact of changes in dB levels.
Sound Intensity Calculation
Calculating sound intensity involves using the logarithmic formula from the decibel scale to find intensities from known sound levels. From the previous calculation, once we have the rearranged formula:
  • \[ \log_{10} \left( \frac{I_{amp}}{I_{unamp}} \right) = 1.4 \]
For exponential conversion, we exponentiate both sides with base 10:
  • \[ \frac{I_{amp}}{I_{unamp}} = 10^{1.4} \]
Calculating \( 10^{1.4} \) results in approximately 25.12, indicating the amplified intensity is 25.12 times greater than the unamplified.
This mathematical process helps visualize how an increase in decibels translates to significant changes in intensity, important for designing and assessing the acoustic impact in various settings. Understanding sound intensity calculations enhances accuracy when measuring sound energy in different environments.

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Most popular questions from this chapter

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