Problem 46 - College Physics

Question

\(\bullet\) The intensity due to a number of independent sound sources is the sum of the individual intensities. (a) When four quadruplets cry simultaneously, how many decibels greater is the sound intensity level than when a single one cries? (b) To increase the sound intensity level again by the same number of decibels as in part (a), how many more crying babies are required?

Step-by-Step Solution

Answer

(a) 6.02 decibels. (b) 12 more babies.

Step 1: Understand Sound Intensity and Decibels

Sound intensity level in decibels is calculated using the formula \( L = 10 \cdot \log_{10}(\frac{I}{I_0}) \), where \( I \) is the intensity and \( I_0 \) is a reference intensity. For this problem, we want to compare the sound intensity levels of one crying baby to four crying babies.

Step 2: Calculate Intensity for One Crying Baby

Let's denote the intensity of one crying baby as \( I_1 \). The sound level for one crying baby is \( L_1 = 10 \cdot \log_{10}(\frac{I_1}{I_0}) \).

Step 3: Calculate Intensity for Four Crying Babies

When four babies cry, the total intensity is \( I_4 = 4 \times I_1 \). The sound level for four babies is \( L_4 = 10 \cdot \log_{10}(\frac{4I_1}{I_0}) \).

Step 4: Calculate the Increase in Decibels from One to Four Babies

The increase in decibels when going from one to four babies crying is \( \Delta L = L_4 - L_1 = 10 \cdot \log_{10}(4) \). Simplifying gives \( \Delta L = 10 \cdot 0.602 = 6.02 \) decibels.

Step 5: Determine How Many Babies Increase by Another 6.02 Decibels

To increase the sound level by another 6.02 decibels, we need the new intensity to be \( 4 \times I_1 \times 4 = 16 \times I_1 \). Thus, you need \( 16 \) crying babies in total.

Step 6: Calculate Additional Number of Babies Required

Since we already have four babies and we need 16, the additional number of crying babies required is \( 16 - 4 = 12 \).

Key Concepts

Decibel Calculation

Decibel calculation is a method used to express the ratio of a particular sound intensity to a reference intensity. It simplifies the way we understand and compare different sound levels. The formula to calculate the sound level in decibels (dB) is given by:\[ L = 10 \cdot \log_{10}\left(\frac{I}{I_0}\right) \]

• Here, \( L \) represents the sound level in decibels.
• \( I \) is the sound intensity, which is the power per unit area measured in watts per square meter (W/m²).
• \( I_0 \) is the reference intensity, typically \( 10^{-12} \) W/m², which is the threshold of hearing for the average human ear.

This formula allows us to easily compare sound intensities by converting them into a more manageable logarithmic scale. Since decibels are logarithmic, small increases in decibel level correspond to significant increases in power.

Sound Level

Sound level refers to the loudness of a sound, which is measured in decibels (dB). It is an important concept in understanding how sound intensity affects our auditory perception. The sound level is determined by the sound intensity and the reference intensity using the decibel formula.When multiple sound sources are present, like in the scenario with crying babies, each source contributes to the total intensity. The overall sound level can be calculated using the cumulative intensity of all sources. For instance, if one baby produces an intensity \( I_1 \), then four babies contribute \( 4 \times I_1 \). The sound level increases with each additional source contributing sound.

Reference Intensity

Reference intensity is a critical component in calculating sound levels using decibels. It serves as the baseline in the decibel formula to compare sound intensities efficiently.

• The standard reference intensity \( I_0 \) is set at \( 10^{-12} \) watts per square meter. This is used because it represents the softest sound a typical human ear can detect, known as the threshold of hearing.
• Any sound intensity \( I \) that is greater than this threshold will have a positive decibel level, reflecting a sound that is audible to us.

Using a reference intensity helps in creating a uniform system to evaluate and communicate sound levels across different contexts and environments.

Independent Sound Sources

When discussing sound from multiple sources, especially in calculations, the idea of independent sound sources becomes significant. Each sound source contributes individually to the total sound intensity you experience.

• The total sound intensity from multiple sources is simply the sum of each independent intensity. For example, if each baby generates an intensity of \( I_1 \), the total for four crying babies is \( 4 \times I_1 \).
• This independence means that each source affects the overall sound level without being influenced by the closeness or interaction of others, aside from overlapping sounds potentially boosting combined acoustic pressure.

Understanding independent sound sources is essential when calculating the increase in decibels because each added source increases the total intensity logarithmically, as seen in the prompt's exercises.