Problem 88 - Algebra 2

Question

Sound The loudness in decibels (dB) of a sound is defined as 10 \(\log \frac{I}{I_{0}}\) I is the intensity of the sound. I \(I_{0}\) is \(10^{-12}\) , the intensity of a barely audible sound. Complete the table. Whisper \(10^{-10}\)

Step-by-Step Solution

Answer

The loudness of a whisper of intensity \(10^{-10} \text{ W/m}^2\) is 20 dB.

Step 1: Understanding the Decibel Formula

The loudness of a sound in decibels (dB) is given by the formula: \(L = 10 \times \text{log}\left(\frac{I}{I_0}\right)\), where \(L\) is the loudness in decibels, \(I\) is the intensity of the sound, and \(I_0\) is the intensity of the barely audible sound at \(10^{-12} \text{ W/m}^2\).

Step 2: Calculating the Loudness of a Whisper

Substitute the given intensity for the whisper \(I = 10^{-10} \text{ W/m}^2\) into the formula and solve for \(L\).

Step 3: Applying the Formula

Calculate the loudness using the formula: \(L = 10 \times \text{log}\left(\frac{10^{-10}}{10^{-12}}\right) = 10 \times \text{log}(100) = 10 \times 2 = 20 \text{ dB}\).

Key Concepts

Logarithmic Functions

Logarithmic functions are essential in understanding how decibels work to measure sound intensity. A logarithmic function is the inverse of an exponential function, represented by the equation: \( y = \log_b(x) \), where \( b \) is the base, \( x \) is the argument, and \( y \) is the logarithm of \( x \) to the base \( b \). In the case of decibels, we typically use a base of 10.

Constants and variables are significant in logarithms. For example, when calculating the loudness of a whisper, the intensity level is a variable that changes depending on the sound, while the reference intensity level \( I_0 \) is a constant that represents the threshold of human hearing. Logarithmic scales convert multiplicative relationships into additive ones, which is why they handle huge ranges of values, like those seen in sound intensities, which can vary from the faintest whisper to the roar of a jet engine.

Decibel Formula

The decibel formula is a mathematical representation that relates the intensity of a sound to its perceived loudness. The decibel unit is one-tenth of a 'Bel', named after Alexander Graham Bell. The formula for loudness in decibels is given by \( L = 10 \times \log\left(\frac{I}{I_0}\right) \), where \( L \) signifies the loudness in decibels, \( I \) is the sound intensity (in watts per square meter), and \( I_0 \) is the reference sound intensity, typically \( 10^{-12} \text{ W/m}^2 \). This reference is the quietest sound that the average human ear can hear.

Hearing a range of intensities simplifies complex numbers into manageable figures using the logarithmic function. The multiplication of the logarithmic result by 10 ensures that the decibel scale is both intuitive and practical for daily use.

Sound Intensity Level

Sound intensity level is a measure of how powerful a sound wave is per unit area and is quantified in watts per square meter (W/m^2). It's a key aspect when calculating loudness in decibels. Sound intensity is a physical, objective quantity that can be measured with instruments, and it's different from loudness, which is a subjective perception and can differ from person to person.

Changes in sound intensity level are often not linear; a tenfold increase in intensity doesn't lead to a tenfold increase in perceived loudness. Instead, our perception of loudness grows logarithmically, and that is why the decibel scale is used to represent intensity levels in a way that more closely aligns with our hearing perception.

Loudness Calculation

Loudness calculation is the process of translating the physical intensity of sound into a value that reflects the human perception of its 'loudness', which is not directly proportional. To calculate the loudness in decibels, you apply the decibel formula. As seen in the example with a whisper, which has an intensity of \( 10^{-10} \text{ W/m}^2 \), you compare it to the reference intensity \( I_0 = 10^{-12} \text{ W/m}^2 \) through the formula: \( L = 10 \times \log\left(\frac{I}{I_0}\right) \).

It's necessary to use a calculator with a logarithmic function to compute the value needed. The outcome shows that every sound has its intensity level, translating complex science into an easy-to-interpret number, indicating how loud a sound is compared to the threshold of hearing.