Problem 18 - University Physics with Modern Physics

Question

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

Step-by-Step Solution

Answer

(a) Intensity reduced by a factor of 1000. (b) 3.01 dB reduction.

Step 1: Understanding Sound Reduction in dB

Sound intensity level reduction can be calculated using the formula: \(\Delta L = 10 \log_{10}\left(\frac{I_1}{I_2}\right)\), where \(\Delta L\) is the change in intensity level in decibels, \(I_1\) is the initial intensity, and \(I_2\) is the final intensity. We need to find \(\frac{I_1}{I_2}\) given that \(\Delta L = 30\, \mathrm{dB}\).

Step 2: Solving for Fractional Reduction

Using the sound reduction formula: \[30 = 10 \log_{10}\left(\frac{I_1}{I_2}\right)\] Divide both sides by 10: \[3 = \log_{10}\left(\frac{I_1}{I_2}\right)\] Solve for \(\frac{I_1}{I_2}\) by converting from log base 10:\[\frac{I_1}{I_2} = 10^3 = 1000\] Thus, the sound intensity has been reduced by a factor of 1000.

Step 3: Determining Change in dB for Halved Intensity

When the intensity is reduced by half, the ratio \(\frac{I_1}{I_2} = 2\). Using the same formula:\[\Delta L = 10 \log_{10}\left(\frac{I_1}{I_2}\right)\]\[\Delta L = 10 \log_{10}(2)\]Calculate \(\log_{10}(2)\) which is approximately 0.301:\[\Delta L = 10 \times 0.301 = 3.01\, \mathrm{dB}\]Therefore, reducing the intensity by half changes the sound intensity level by approximately \(3.01 \, \mathrm{dB}\).

Key Concepts

Decibel Reduction

Decibels (\(\mathrm{dB}\)) are units that measure sound intensity levels. Reducing sound intensity in decibels might seem abstract, but it has practical implications in noise control. The decibel scale is logarithmic, which means a reduction of 30 decibels equates to a significant reduction in sound intensity. For instance, in our exercise, a 30 dB reduction reduces traffic noise by a factor of 1000. This huge drop results from the logarithmic nature of the decibel scale, where each 10 dB represents a tenfold change in intensity. Understanding decibel reduction is key to addressing noise pollution and enhancing acoustic comfort in everyday life.

Logarithmic Scale

The logarithmic scale is integral to understanding how sound intensity levels work. Unlike linear scales, where equal increments signify equal changes in value, the logarithmic scale is exponential. It gets its form from multiplying or dividing by powers of 10. In sound intensity, this means each 10 dB change is a tenfold increase or decrease in intensity. For our exercise, reducing the sound level by 30 dB means decreasing intensity to one-thousandth of its original value. This type of scale helps compress a wide range of sound intensities into a more manageable range, making it easier for us to understand and control sounds.

Sound Intensity Level

Sound intensity level determines how much sound energy passes through a certain area. Measured in watts per square meter (\(\mathrm{W/m^2}\)), it helps quantify how loud or quiet a sound is. When we refer to certain decibels reduced in intensity — like the 30 dB in our scenario — it specifically affects this intensity level. By decreasing the sound intensity level, less energy reaches our ears, producing less perceived noise. Understanding sound intensity levels can help set benchmarks for environments like homes or workplaces, promoting effective noise management.

Physics Problems

Physics problems involving sound, like those concerning sound intensity and decibels, teach practical applications of physics concepts. These problems often require understanding formulas that relate intensity and decibel changes. For this exercise, solving the physics problem involves using the logarithmic formula to relate changes in decibels to changes in intensity. This process illustrates how we can apply mathematical and physical principles to solve real-world problems, such as reducing unwanted noise in our environments. Tackling such physics problems equips students with problem-solving skills that transcend academic settings.

Noise Reduction Techniques

Noise reduction techniques play a crucial role in managing sound within our environments. Methods like installing sound-reflecting windows, as mentioned in the exercise, are practical solutions. Such techniques aim to lower sound intensity levels significantly — like the 30 dB reduction achieved in the problem scenario. Other techniques include soundproofing walls, using noise-canceling devices, or altering building materials. The key is to apply strategies that suit specific noise situations, whether it's reducing traffic noise or managing sound in a recording studio. Mastering noise reduction techniques improves quality of life and allows one to create spaces conducive to concentration and relaxation.