Question

During a rehearsal, all eight members of the first violin section of an orchestra play a very soft passage. The sound intensity level at a certain point in the concert hall is \(38.0 \mathrm{dB} .\) What is the sound intensity level at the same point if only one of the violinists plays the same passage? [Hint: When playing together, the violins are incoherent sources of sound.]

Short answer

The sound intensity level for one violin is approximately 28.97 dB.

Step-by-step solution

Understanding Sound Intensity Formula

Sound intensity level in decibels is given by the formula: \( L = 10 \log_{10}(I/I_0) \), where \( L \) is the sound level in decibels, \( I \) is the intensity of the sound, and \( I_0 \) is the reference intensity, typically \( 10^{-12} \text{ W/m}^2 \).

Calculate Intensity for Eight Violins

Given the sound intensity level of 8 violins is \( 38.0 \, \text{dB} \). Calculate the intensity \( I_8 \) using: \( 38 = 10 \log_{10}(I_8/I_0) \). Solving gives \( I_8 = 10^{(38/10)} \cdot I_0 = 10^{3.8} \cdot I_0 \).

Calculate Intensity for One Violin

The intensity from one violin, \( I_1 \), is one-eighth of the total intensity from 8 violins (since the sources are incoherent): \( I_1 = I_8 / 8 = (10^{3.8} I_0) / 8 \).

Calculate Sound Intensity Level for One Violin

Using the formula for sound intensity level, calculate \( L_1 \) for one violin: \( L_1 = 10 \log_{10}(I_1/I_0) = 10 \log_{10}( (10^{3.8} I_0) / (8I_0)) \). Simplifying gives \( L_1 = 10 \log_{10}(10^{3.8}/8) \).

Compute the Final Sound Intensity Level

Simplify \( L_1 = 10 (\log_{10}(10^{3.8}) - \log_{10}(8)) = 10(3.8 - \log_{10}(8)) \). Compute \( \log_{10}(8) \approx 0.903 \), so \( L_1 = 10(3.8 - 0.903) \approx 10 \times 2.897 \approx 28.97 \, \text{dB} \).

Key concepts

Decibels

Decibels are a unit of measurement used to express the intensity or power level of sound. Because human ears perceive sound on a logarithmic scale, decibels (dB) reflect this perceptual way of hearing rather than a linear scale. The decibel scale is relative and uses a reference intensity, typically represented as

• 10^{-12} ext{ W/m}^2

to make comparisons. A sound intensity level is derived from the equation\[L = 10 \, \log_{10}(I/I_0)\]where \(L\) is the sound level in decibels, \(I\) is the sound intensity, and \(I_0\) is the reference intensity. This scale allows smaller increases in measurement to reflect large increases in perceived sound, simplifying calculations and interpretations. The logarithmic aspect means that a sound at 40 dB is not twice as loud as 20 dB, but rather 100 times more intense.

Incoherent Sources

Incoherent sources describe multiple sound sources that are not in phase with each other. In the context of the exercise involving violins, when these instruments play together, they act as incoherent sources of sound. This means that their sound waves do not align perfectly, and thus their combined sound intensity does not simply add linearly. Instead, the total sound intensity when all sources operate together is the sum of their individual intensities, not their power levels. For example, if each violin contributes a certain intensity \(I_1\), the total intensity for eight violins is represented by 8\(I_1\). This concept helps us understand scenarios like the exercise, where reducing the number of violins to one affects overall sound intensity significantly and requires calculation to determine the new intensity in decibels.

Logarithmic Scale

A logarithmic scale is employed for expressing quantities that range over a wide band of values, like sound intensities. In sound measurement, a logarithmic scale is particularly useful because it matches the human ear's response. The perception of loudness is logarithmic rather than linear. Under a logarithmic scale, as applied to sound, an increase of 10 decibels represents a tenfold increase in sound intensity. So, the formula \[L = 10 \, \log_{10}(I/I_0)\]translates changes in a physical quantity into a scale that mirrors our auditory capacity more closely. The adoption of this scale means large variations in physical intensity correlate to manageable numerical values, allowing for both practical computation and perception analysis.

Reference Intensity

Reference intensity is the baseline level of intensity against which all other sound intensities are measured in decibels. It is universally agreed upon to be \[10^{-12} ext{ W/m}^2\]which is considered the threshold of hearing, or the quietest sound that the average human ear can detect. The formula \[L = 10 \, \log_{10}(I/I_0)\]is employed here, where \(I_0\) is the reference intensity, allowing us to calculate the decibel level. Without this standard point of reference, comparing different sound intensity levels would become challenging. It's especially pivotal in exercises that involve calculating sound levels resulting from multiple sources, such as the collective sound of many violins versus one. Understanding the concept of reference intensity aids in deciphering how much louder or softer a sound is compared to the quietest sound detectable by humans.