Problem 99 - Pearson Physics

Question

. Think \& Calculate Ten violins playing simultaneously with the same intensity combine to give an intensity level of \(70 \mathrm{~dB}\). (a) What is the intensity level of each violin? (b) If the number of violins is increased to 100 , will the intensity level be more than, less than, or equal to \(80 \mathrm{~dB}\) ? Explain.

Step-by-Step Solution

Answer

(a) Each violin has an intensity level of 60 dB. (b) The intensity level for 100 violins is equal to 80 dB.

Step 1: Understanding the Relation Between Intensity and Decibels

The intensity level in decibels (dB) is given by the formula \( L = 10 \log_{10} \left( \frac{I}{I_0} \right) \), where \( L \) is the intensity level, \( I \) is the intensity, and \( I_0 \) is the reference intensity. We must first relate individual intensities of violins to the total intensity level by applying this formula.

Step 2: Calculating the Total Intensity

Knowing that ten violins together give an intensity level of 70 dB, we calculate the total intensity \( I_T \) using \( I_T = 10^{\frac{L}{10}}I_0 \). Thus, the total intensity \( I_T = 10^{\frac{70}{10}}I_0 = 10^7 I_0 \).

Step 3: Finding the Intensity of Each Violin

Since ten violins combine to have total intensity \( I_T \), each violin has an intensity \( I_V = \frac{I_T}{10} = \frac{10^7 I_0}{10} = 10^6 I_0 \). We can find the intensity level of each violin by plugging \( I_V \) back into the formula: \( L_V = 10 \log_{10} \left( \frac{10^6 I_0}{I_0} \right) = 60 \, \text{dB} \).

Step 4: Increasing the Number of Violins to 100

If the number of violins is increased to 100, the total intensity \( I'_T \) becomes \( I'_T = 100 \times 10^6 I_0 = 10^8 I_0 \).

Step 5: Calculate New Intensity Level for 100 Violins

The new intensity level for 100 violins is found using \( L' = 10 \log_{10} \left( \frac{10^8 I_0}{I_0} \right) = 10 \log_{10} (10^8) = 80 \, \text{dB} \).

Step 6: Conclusion on Intensity Level Comparison

By calculating the new intensity level, we confirm that the intensity level of 100 violins is exactly 80 dB, which means it is equal to the expected value.

Key Concepts

Decibel Scale

The decibel scale is a logarithmic scale used to measure the intensity of sound. It translates large variations in acoustic power into a scale that is easier to understand and use. This is particularly useful in assessing sound levels because our ears perceive the sound intensity logarithmically rather than linearly.

The decibel scale is expressed using the formula:

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

where:

• \( L \) is the intensity level in decibels (dB)
• \( I \) is the intensity of the sound
• \( I_0 \) is the reference intensity, which is typically set at the threshold of hearing, approximately \( 1 \times 10^{-12} \) watts per square meter

The decibel formula allows us to understand how loud a sound is compared to the quietest sound a typical human ear can hear.

Using this scale, an increase of 10 dB represents a sound that is perceived to be about twice as loud because it reflects a tenfold increase in acoustic intensity. In the context of the original exercise, understanding this concept allows us to see how the sound intensity level changes when more violins are added to the ensemble.

Logarithmic Relationship

The logarithmic relationship in sound measurement explains why the decibel scale is not linear. Instead of adding intensities linearly, as with regular numbers, they are combined using logarithmic operations. This reflects how human perception itself is logarithmic. Thus, combining sound pressures requires multiplying the intensity rather than direct addition.

For example, if one violin at an intensity level of 60 dB produces a certain loudness, then two violins do not simply add to make 120 dB. Instead, their combined intensity is calculated using the logarithmic rule:

• 
  The formula for each intensity value is used:
  \( I_T = 10^{\frac{L}{10}}I_0 \)

This accounts for the way large increments in the measured scale translate into smaller increments of perceived loudness.

In practical applications, such as in determining the intensity levels of multiple violins from the original problem, the logarithmic nature of the scale simplifies these calculations by allowing additive properties along the scale's base: base 10 in this case.

Acoustic Intensity

Acoustic intensity is the measure of sound power per unit area. Typically measured in watts per square meter, it defines how much sound energy passes through the air or any medium. The higher the intensity, the louder the sound.

This concept is crucial in understanding how individual sound sources combine to create a higher intensity. For instance, in our exercise, each violin contributes its own acoustic intensity to the total sound heard. With more violins, the resultant acoustic intensity is the sum of individual intensities expressed through a logarithmic scale.

To illustrate:

• Each violin has its acoustic intensity \( I_V \)
• The total intensity of ten violins \( I_T \) = 10 \( I_V \)

When the number of violins increases to 100, the acoustic intensity increases dramatically. This direct multiplication of individual intensities yields a new total intensity, from which the new decibel level can be calculated. The formula again bridges this back to our decibel scale.

Thus, understanding acoustic intensity helps clarify why scaling up the number of sound sources leads to significant changes on the decibel scale.