Problem 53 - Calculus with Concepts in Calculus

Question

Determine the number of decibels that corresponds to cach of the following intensities. a. \(10^{-12}\) (threshold of hearing) b. \(10^{-11}\) (rustling leaves) c. \(10^{-2}\) (power mower) d. 10 (jackhammer)

Step-by-Step Solution

Answer

a: 0 dB, b: 10 dB, c: 100 dB, d: 130 dB.

Step 1: Understanding the Decibel Formula

The decibel level of a sound is determined using the formula: \( L = 10 \log_{10}\left(\frac{I}{I_0}\right) \), where \( I \) is the intensity of the sound in watts per square meter, and \( I_0 = 10^{-12} \) W/m² is the intensity of the threshold of hearing.

Step 2: Calculating Decibels for Intensity a

For intensity \( I = 10^{-12} \):\[L = 10 \log_{10}\left(\frac{10^{-12}}{10^{-12}}\right) = 10 \log_{10}(1) = 10 \times 0 = 0 \, \text{dB}\]The number of decibels for \( I = 10^{-12} \) is 0 dB.

Step 3: Calculating Decibels for Intensity b

For intensity \( I = 10^{-11} \):\[L = 10 \log_{10}\left(\frac{10^{-11}}{10^{-12}}\right) = 10 \log_{10}(10) = 10 \times 1 = 10 \, \text{dB}\]The number of decibels for \( I = 10^{-11} \) is 10 dB.

Step 4: Calculating Decibels for Intensity c

For intensity \( I = 10^{-2} \):\[L = 10 \log_{10}\left(\frac{10^{-2}}{10^{-12}}\right) = 10 \log_{10}(10^{10}) = 10 \times 10 = 100 \, \text{dB}\]The number of decibels for \( I = 10^{-2} \) is 100 dB.

Step 5: Calculating Decibels for Intensity d

For intensity \( I = 10 \):\[L = 10 \log_{10}\left(\frac{10}{10^{-12}}\right) = 10 \log_{10}(10^{13}) = 10 \times 13 = 130 \, \text{dB}\]The number of decibels for \( I = 10 \) is 130 dB.

Key Concepts

logarithms

Logarithms play a crucial role in the calculation of decibels, a unit that measures the intensity level of sound. At its core, a logarithm is the inverse operation to exponentiation. This means it 'undoes' what exponentiation does. To put it simply, if you start with a number, take its logarithm, and then exponentiate the result, you will get back to your original number. In the context of sound, logarithms help compress the wide range of sound intensities into a more manageable scale.
In decibel calculations, specifically, a base 10 logarithm is used. This is often denoted as \(\log_{10}\). When you're dealing with sound intensity and decibels, the formula you'd encounter is:

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

Where:

• \( L \) is the sound level in decibels (dB)
• \( I \) is the intensity of the sound
• \( I_0 \) is the reference intensity, typically the threshold of hearing (\(10^{-12}\) W/m²).

The logarithm helps convert the intense variations in sound intensity into a linear scale that humans can better understand. Remember that each increase by 10 in the decibel scale represents a tenfold increase in sound intensity.

sound intensity

Sound intensity is a measure of the power of a sound wave passing through a particular area. It's essentially the amount of energy that the wave carries through a unit area in a unit time. Scientifically, it's measured in watts per square meter (W/m²). What makes sound intensity significant is how it helps in expressing the loudness of a sound.
In real-world applications, the range of sound intensity that humans can hear is vast. That's why we use decibels. We take the logarithm of the sound intensity level relative to a reference value, which is typically the threshold of hearing, to get the decibel value. This relative comparison expresses how much louder one sound is compared to another. As sound intensity increases, so does the decibel level indicating sound loudness. For example:

• At the threshold of hearing, sound intensity is \(10^{-12}\) W/m², equating to 0 dB.
• Normal conversation might be around \(10^{-6}\) W/m², leading to about 60 dB.
• A power mower at \(10^{-2}\) W/m² reaches 100 dB.

Utilizing sound intensity and decibels helps us practically gauge and manage sound levels in our day-to-day life.

threshold of hearing

The threshold of hearing is the quietest sound that the average human ear can understand in terms of sound intensity. Scientifically speaking, this threshold is commonly cited as an intensity level of \(10^{-12}\) W/m². This is often used as a reference point when measuring sound levels.
The human ear is remarkably sensitive to sound, capable of distinguishing between incredibly subtle differences in sound pressure. The lowest pressure change one can detect translates into this threshold intensity of \(10^{-12}\) W/m².
In practical terms, when we say a sound has 0 dB, it means the sound intensity is equal to the threshold of hearing. Even though it might seem like 0 dB would be silent, it simply denotes the faintest sound detectable by nearly all people. Getting beyond that point into positive values in decibels signifies an increase in volume or loudness.