Problem 41 - College Physics

Question

\(\bullet\) Find the noise level (in dB) of a sound having an intensity of (a) 0.000127 \(\mathrm{W} / \mathrm{m}^{2}\)(b) \(6.53 \times 10^{-10} \mathrm{W} / \mathrm{cm}^{2},\) (c) \(1.5 \times\) \(10^{-14} \mathrm{W} / \mathrm{m}^{2}\)

Step-by-Step Solution

Answer

(a) 81.03 dB, (b) 68.15 dB, (c) -18.24 dB

Step 1: Understanding the Decibel Formula

The formula to calculate the noise level in decibels (dB) is given by \( L = 10 \log_{10} \left( \frac{I}{I_0} \right) \), where \( I \) is the intensity of the sound in \( \mathrm{W/m}^2 \) and \( I_0 = 10^{-12} \mathrm{W/m}^2 \) is the reference intensity.

Step 2: Convert Units if Necessary

For part (b), the intensity is given in \( \mathrm{W/cm}^2 \). Convert it to \( \mathrm{W/m}^2 \) by multiplying it by \( 10^4 \). For part (b), this gives \( I = 6.53 \times 10^{-10} \times 10^4 = 6.53 \times 10^{-6} \mathrm{W/m}^2 \).

Step 3: Calculate Noise Level for Part (a)

For \( I = 0.000127 \mathrm{W/m}^2 \):- Substitute into the formula: \( L = 10 \log_{10} \left( \frac{0.000127}{10^{-12}} \right) \)- Simplify inside the log: \( \frac{0.000127}{10^{-12}} = 1.27 \times 10^8 \)- Calculate \( L = 10 \log_{10} (1.27 \times 10^8) \approx 81.03 \mathrm{dB} \).

Step 4: Calculate Noise Level for Part (b)

For \( I = 6.53 \times 10^{-6} \mathrm{W/m}^2 \):- Substitute into the formula: \( L = 10 \log_{10} \left( \frac{6.53 \times 10^{-6}}{10^{-12}} \right) \)- Simplify inside the log: \( \frac{6.53 \times 10^{-6}}{10^{-12}} = 6.53 \times 10^6 \)- Calculate \( L = 10 \log_{10} (6.53 \times 10^6) \approx 68.15 \mathrm{dB} \).

Step 5: Calculate Noise Level for Part (c)

For \( I = 1.5 \times 10^{-14} \mathrm{W/m}^2 \):- Substitute into the formula: \( L = 10 \log_{10} \left( \frac{1.5 \times 10^{-14}}{10^{-12}} \right) \)- Simplify inside the log: \( \frac{1.5 \times 10^{-14}}{10^{-12}} = 0.015 \)- Calculate \( L = 10 \log_{10} (0.015) \approx -18.24 \mathrm{dB} \).

Key Concepts

Sound Intensity

Sound intensity is a crucial concept to understand when dealing with sound waves. It refers to the power per unit area carried by a sound wave, measured in watts per meter squared (\(\mathrm{W/m}^2\)). Intensity tells us how much energy a sound wave is transporting through a certain area each second.
In real life, sound intensity affects how loud or soft we perceive a sound. Greater intensity generally means a louder sound, while lower intensity tends to be quieter. Sound intensity decreases with distance from the sound source, as the energy spreads over a larger area.
To calculate noise levels from intensity, first, we need to know the intensity levels compared to a standard reference level. This is where the decibel scale comes in handy, making the complex comparison much simpler.

Noise Level

The noise level of a sound tells us how loud it is in terms of the intensity relative to a reference standard. We often use decibels (dB) to express noise levels.
The formula to convert intensity to decibels is:

• \( L = 10 \log_{10} \left( \frac{I}{I_0} \right) \)

Here, \(I\) is the intensity of the sound, and \(I_0\) is the standard reference intensity, usually \(10^{-12} \mathrm{W/m}^2\). A noise level of 0 dB means the sound intensity is at this reference level. An increase of 10 dB represents an increase in sound intensity by a factor of 10.
Noise levels are significant in understanding everyday sounds. For example, normal conversation typically ranges around 60 dB, while a quiet whisper might be around 30 dB.

Logarithmic Scale

A logarithmic scale, like the decibel scale, is used when dealing with a wide range of values, like sound intensity. This type of scale compresses information, allowing us to easily compare very small and very large numbers using a concise range.
Sound intensity, for example, can vary from the slightest whisper to the deafening roar of a jet engine. On a simple linear scale, comparing these would be cumbersome. The logarithmic scale turns this large range into manageable numbers, simplifying comparisons.
With a logarithmic scale, each increase of 10 in decibels represents a tenfold increase in intensity. This non-linear approach mimics human hearing, as our ears perceive changes in sound levels logarithmically.

Unit Conversion

Sometimes, sound intensity may be given in units other than \(\mathrm{W/m}^2\), such as \(\mathrm{W/cm}^2\). Converting units is essential to using formulas correctly. In the example we discussed, the intensity was given in \(\mathrm{W/cm}^2\), and we had to convert it to \(\mathrm{W/m}^2\) to match the reference level.
To convert from \(\mathrm{W/cm}^2\) to \(\mathrm{W/m}^2\):

• Multiply by \(10^4\), since there are 10,000 \(cm^2\) in a \(m^2\).

This step ensures that all measurements align in the calculations needed for the decibel scale. Accurate unit conversion is crucial; otherwise, the calculated noise level could be misleading.
Understanding how to perform unit conversions is fundamental in many areas of science and engineering, ensuring that all calculations remain correct.