Question

Use the following discussion: The loudness \(L(x),\) measured in decibels \((d B),\) of a sound of intensity \(x\), measured in watts per square meter, is defined as \(L(x)=10 \log \frac{x}{I_{0}},\) where \(I_{0}=10^{-12}\) watt per square meter is the least intense sound that a human ear can detect. Determine the loudness, in decibels, of each of the following sounds. Diesel truck traveling 40 miles per hour 50 feet away: intensity 10 times that of a passenger car traveling 50 miles per hour 50 feet away, whose loudness is 70 decibels.

Short answer

The loudness of the diesel truck is 80 decibels.

Step-by-step solution

Comprehend the loudness function

The loudness function is defined as \[ L(x) = 10 \, log \frac{x}{I_0}\] where \( I_0 = 10^{-12} \) watt per square meter.

Determine the intensity of the passenger car

Given that the loudness of the passenger car is 70 decibels, use the loudness function to find its intensity. First, set up the equation using the known loudness: \[70 = 10 \, log \frac{x_{car}}{I_0}\] Next, solve for \( x_{car} \): Divide both sides by 10: \[7 = log \frac{x_{car}}{I_0}\] Rewrite in exponential form: \[10^7 = \frac{x_{car}}{I_0}\] Solve for \( x_{car} \): \[x_{car} = 10^7 \times I_0 = 10^7 \times 10^{-12} = 10^{-5}\] watts per square meter.

Calculate the intensity of the diesel truck

The diesel truck has an intensity 10 times that of the passenger car. Therefore, \[ x_{truck} = 10 \times x_{car} = 10 \times 10^{-5} = 10^{-4}\] watts per square meter.

Determine the loudness of the diesel truck

Use the loudness function to find \( L_{truck} \): \[ L_{truck} = 10 \log \frac{x_{truck}}{I_0}=10 \log \frac{10^{-4}}{10^{-12}}=10 \log (10^8) \] Simplify: \[L_{truck} = 10 \times 8=80 \text{dB}\]

Key concepts

Loudness Function

To understand how we measure sound loudness, we start with the loudness function. This function connects the intensity of a sound, measured in watts per square meter, to its loudness measured in decibels (dB). It is mathematically expressed as:
\[L(x) = 10 \, \log \frac{x}{I_0}\]
Here, \(L(x)\) represents the loudness in decibels and \(x\) is the sound intensity. Notably, \(I_0\) is a reference intensity, specifically the least intense sound that a human can detect, which is 10^-12 watts per square meter.

Logarithms in Equations

Logarithms play a crucial role in transforming the multiplicative relationship between sound intensities into an additive one, which is easier to handle. In the equation \( L(x) = 10 \log \frac{x}{I_0} \), the 'log' refers to the logarithm base 10. This is why we often see '10' as the base for changes in decibel levels.
For example, if a sound's intensity doubles, the decibel level does not double but increases by about 3 dB, thanks to the logarithmic scale. This property simplifies the comparison of various sound intensities, making logarithms fundamental in sound level calculations.

Sound Intensity

Sound intensity, denoted as \(x\), is a measure of how much energy a sound wave brings to a unit area per unit time. It is measured in watts per square meter (W/m²). The intensity determines the loudness, but human perception of loudness is not linear; hence, the use of a logarithmic function to describe it in decibels.
In practice, we may compare sound intensities to determine their relative loudness. For instance, knowing that a diesel truck has an intensity 10 times that of a passenger car helps us calculate its decibel level efficiently.

Decibels

Decibels (dB) are the units used to measure sound loudness. They offer a way to express different sound intensities on a manageable scale. The core idea is that a tenfold increase in intensity leads to a 10 dB increase in loudness due to the logarithmic nature of the scale.
For example, if a passenger car's loudness is 70 dB, and a diesel truck's intensity is 10 times greater, the truck's loudness can be calculated as follows:
\(\ L_{truck} = 10 \log \frac{x_{truck}}{I_0} = 10 \log \frac{10^{-4}}{10^{-12}} = 10 \log (10^8) = 80 \text{dB}\)

Watts per Square Meter

Sound intensity is expressed in watts per square meter (W/m²). This unit tells us how much power (or energy per second) is passing through one square meter.
For instance, the reference intensity \(I_0\) is 10^-12 W/m², meaning it's the smallest detectable intensity by the human ear. This benchmark is critical in calculating the loudness of other sounds, using the formula
\( L(x) = 10 \log \frac{x}{I_0}\). By knowing the intensity of a sound and comparing it to this reference, we can find the sound's loudness in decibels. For example, an intensity of 10^-4 W/m² results in a loudness of 80 dB.