Problem 77 - Algebra 2

Question

Acoustics In Exercises \(76-78\) , the loudness measured in decibels \((d B)\) is defined by loudness \(=10\) log \(\frac{I}{I_{0}},\) where \(I\) is the intensity and \(I_{0}=10^{-12} \mathrm{W} / \mathrm{m}^{2}\) . The noise level inside a convertible driving along the freeway with its top up is 70 \(\mathrm{dB}\) . With the top down, the noise level is 99 \(\mathrm{dB}\) . a. Find the intenity of the sound with the top up and with the top down. b. By what percent does leaving the top up reduce the intensity of the sound?

Step-by-Step Solution

Answer

The sound intensity is approximately \(10^{-5} \) W/m² with the top up, and approximately \(7.94 \cdot 10^{-3} \) W/m² with the top down. Leaving the top up reduces the intensity by approximately 99.87%.

Step 1: Intensity calculation with the top up

First, use the formula for loudness given, \( loudness =10 \cdot \log \left( \frac{I}{I_{0}} \right) \), where \(I_{0}=10^{-12} \) W/m², and the loudness with the top up of 70 dB to find the intensity, as follows: \n Re-arranging, we have \(I=I_{0} \cdot 10^{\frac{loudness}{10}}\), \n So, \(I =10^{-12} \cdot 10^{\frac{70}{10}}=10^{-12} \cdot 10^{7} = 10^{-5} \) W/m².

Step 2: Intensity calculation with the top down

Now, perform the same calculation but using the loudness with the top down of 99 dB, as follows: \n \(I=10^{-12} \cdot 10^{\frac{99}{10}}=10^{-12} \cdot 10^{9.9} \) \nThis results in \(I \approx 7.94 \cdot 10^{-3} \) W/m²

Step 3: Percentage reduction calculation

For the last part of the exercise, calculate how much leaving the top up reduces the sound intensity. The reduction in intensity can be computed as a percetange by using the formula for percent change as follows: \n Percentage reduction \(= \frac{7.94 \cdot 10^{-3} - 10^{-5}}{7.94 \cdot 10^{-3}} \times 100\% \) \n This results in a percentage reduction of approximately 99.87%.

Key Concepts

Understanding Loudness

Loudness is how we perceive the intensity of sound. Think of it like how bright or dim a light feels to your eyes. It's not a straight measure; it considers human perception.

Our ears are sensitive to a wide range of sound intensities. This is where the concept of loudness comes into play. It helps us gauge how loud a sound feels, not just how strong it actually is in terms of energy. Loudness is usually measured in decibels (dB). This unit captures how loud something seems to an average person. It's important to note that loudness doesn't grow in a simple, linear way. This is why loudness uses a logarithmic scale, making it easier to understand sound intensity changes over a vast range.

Decibels Explained

Decibels are the units used to measure the perception of sound pressure or loudness. One decibel equals one-tenth of a bel, which is the base unit, though rarely used. In acoustics, decibels quantify the level of sound pressure.

The decibel scale is a comparative scale. It compares two quantities: the intensity of the sound in question and a reference intensity. In acoustics, the reference intensity is usually taken as the threshold of hearing:

• For typical sounds, 0 dB is the threshold of hearing.
• A normal conversation registers around 60 to 70 dB.
• Sounds above 85 dB can potentially damage hearing over prolonged exposure.

This measure tells us how loud a sound is compared to a reference sound. The formula for determining sound intensity in decibels ( extit{dB}) involves a proportional relationship with the logarithmic scale.

Sound Intensity Demystified

Sound intensity is the sound power per unit area. Imagine it as how powerful sound is when it hits your ears. The more intense the sound, the stronger its perceived loudness.

The intensity is expressed in watts per square meter (W/m²). It measures how much sound energy passes through, or seems to come from, a particular area. Intensity is crucial because it's part of the formula that determines loudness, measured in decibels. A key fact is that sound intensity can vary drastically. For instance, the background noise of a quiet room might be much less intense than a concert. Sound intensity is essential to comprehend the calculations involved in how we perceive loudness.

Logarithmic Scale Simplified

A logarithmic scale is a non-linear scale used for a large range of values. It helps manage very large or very small numbers, spacing them in a way that is more human-friendly. In acoustics, it is mainly used to measure sound loudness due to how our hearing system perceives sound intensities.

Logarithms allow us to transform multiplication into addition, which helps simplify the process of comparing sound intensities. Instead of thinking directly in terms of huge sound power numbers, the logarithmic scale focuses on ratios.
The formula for loudness \[L = 10 \log_{10} \left( \frac{I}{I_0} \right)\]involves logs because it fits better with human hearing. Here, \( I \) is the intensity in question and \( I_0 \) is the reference intensity, frequently taken as the threshold of hearing.