Problem 20 - University Physics with Modern Physics

Question

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 dB; (b) 12 more babies are needed.

Step 1: Understand the Relationship Between Intensity and Decibels

The formula for the sound intensity level in decibels is given by:\[ L = 10 \log_{10} \left(\frac{I}{I_0}\right) \]where \( I \) is the intensity of the sound, and \( I_0 \) is the reference intensity. We need to find the increase in decibels, \( \Delta L \), when the number of sources increases.

Step 2: Calculate Increase in Decibels for Quadruplets

When one baby cries, the intensity is \( I \), so the level is \( L_1 = 10 \log_{10}\left(\frac{I}{I_0}\right) \). When four cry, the intensity is \( 4I \), so the new level is \( L_4 = 10 \log_{10}\left(\frac{4I}{I_0}\right) \). The change in decibels, \( \Delta L \), is calculated as:\[ \Delta L = L_4 - L_1 = 10 \log_{10}(4) \approx 6.02 \text{ dB} \]

Step 3: Determine New Intensity to Increase Level by Another 6.02 dB

We need to find the total number of babies causing a 6.02 dB increase again. The initial intensity was \( 4I \). If we denote \( n \) as the new total number of babies, the problem becomes to find \( n \) such that:\[ 10 \log_{10}\left(\frac{nI}{I_0}\right) - 10 \log_{10}\left(\frac{4I}{I_0}\right) = 6.02 \]Simplifying, we get:\[ 10 \log_{10}\left(\frac{n}{4}\right) = 6.02 \]\[ \log_{10}\left(\frac{n}{4}\right) = 0.602 \]\[ \frac{n}{4} = 10^{0.602} \approx 4 \]\[ n \approx 16 \]

Step 4: Calculate Additional Babies Needed

Since we had 4 babies initially, to reach the new count of approximately 16 babies, we need additional:\[ n - 4 = 16 - 4 = 12 \] extra babies to increase the sound intensity level again by approximately 6.02 dB.

Key Concepts

Decibels

Sound intensity is often measured in decibels (dB), which is a logarithmic unit. Decibels quantify how intense a sound is compared to a reference level.
Understanding decibels is crucial when discussing sound intensity because it allows us to express very large or small ratios in a more manageable way.

• One of the key points about decibels is that they express ratios, not absolute values. That means a specific decibel level indicates how much more intense one sound is relative to another.
• For example, a 10 dB increase represents a tenfold increase in intensity, a 20 dB increase means a hundredfold increase, and so forth.

In the original exercise, the change of 6.02 dB was calculated when multiple sound sources—crying babies in this case—contributed to overall intensity. Understanding this concept helps grasp how intensity levels can vary significantly even with small changes in decibel values.

Logarithms

Logarithms play a pivotal role in the calculation of sound intensity levels because these calculations heavily rely on logarithmic expressions.
The formula for sound intensity level is given as:\[ L = 10 \log_{10}\left(\frac{I}{I_0}\right) \]where \( L \) represents the level in decibels, \( I \) is the measured intensity, and \( I_0 \) is a reference value.

• Using a logarithm to calculate sound intensity level allows us to compare greatly different intensities in a systematic way. It compresses the scale of loudness, making it easier to manage.
• The logarithmic nature means each increase by 10 dB translates to a tenfold increase in the actual intensity. For the calculations in the solution, understanding how to manipulate logarithms was key, especially in finding the change from multiple sound sources.

Reference Intensity

Reference intensity, often denoted as \( I_0 \), is a crucial component when calculating sound intensity levels in decibels.
This reference value represents the baseline level of sound intensity, typically \( 10^{-12} \text{ W/m}^2 \) for air, which is the quietest sound that can be typically heard by humans.

• By using a standard reference intensity, it ensures our calculations for sound levels are consistent and comparable.
• In the context of the exercise, reference intensity is used to establish the sound level of both single and multiple jeopardizing sound sources, enabling the calculation of increases in intensity level by comparison.

As a result, when you see or hear that something is measured in decibels, it's often being measured against this standard level or another contextual reference value.