Question

Sound Intensity The level of sound \(\beta\) (in decibels) with an intensity of \(I\) is $$ \beta(I)=10 \log _{10}\left(\frac{I}{I_{0}}\right) $$ where \(I_{0}\) is an intensity of \(10^{-16}\) watt per square centimeter, corresponding roughly to the faintest sound that can be heard. Determine \(\beta(I)\) for the following. (a) \(I=10^{-14}\) watt per square centimeter (whisper) (b) \(I=10^{-9}\) watt per square centimeter (busy street corner) (c) \(I=10^{-6.5}\) watt per square centimeter (air hammer) (d) \(I=10^{-4}\) watt per square centimeter (threshold of pain)

Short answer

The decibel levels for the given sound intensities are: (a) 20 decibels, (b) 70 decibels, (c) 95 decibels, and (d) 120 decibels.

Step-by-step solution

Understand the formula

The formula \(\beta(I)=10 \log _{10}\left(\frac{I}{I_{0}}\right)\) is given, where \(\beta\) is the sound level in decibels, \(I\) is the intensity of the sound, and \(I_{0}\) is the faintest sound intensity that can be heard. The task involves solving this formula for different values of \(I\).

Solve for I=10^-14

Substitute \(I = 10^{-14}\) and \(I_0 = 10^{-16}\) into the formula: \(\beta(I)=10 \log _{10}\left(\frac{10^{-14}}{10^{-16}}\right)\). Simplifying the fraction and calculating the logarithm gives \(\beta = 20 \) decibels.

Solve for I=10^-9

Substitute \(I = 10^{-9}\) and \(I_0 = 10^{-16}\) into the formula: \(\beta(I)=10 \log _{10}\left(\frac{10^{-9}}{10^{-16}}\right)\). Simplifying the fraction and calculating the logarithm gives \(\beta = 70 \) decibels.

Solve for I=10^-6.5

Substitute \(I = 10^{-6.5}\) and \(I_0 = 10^{-16}\) into the formula: \(\beta(I)=10 \log _{10}\left(\frac{10^{-6.5}}{10^{-16}}\right)\). Simplifying the fraction and calculating the logarithm gives \(\beta = 95 \) decibels.

Solve for I=10^-4

Substitute \(I = 10^{-4}\) and \(I_0 = 10^{-16}\) into the formula: \(\beta(I)=10 \log _{10}\left(\frac{10^{-4}}{10^{-16}}\right)\). Simplifying the fraction and calculating the logarithm gives \(\beta = 120 \) decibels.

Key concepts

Decibels

Decibels are a unit used to measure the intensity of sound. They are incredibly useful in comparing different sound levels in a way that our ears actually perceive them. Sound intensity can vary immensely, from the faintest whisper to the roar of a rocket engine. The scale we use, called the decibel scale, is logarithmic, which helps us handle such vast differences in intensity with manageable numbers. To understand this better, when we refer to a whisper (around 20 decibels), a busy street corner (70 decibels), or the loud noise of an air hammer (95 decibels), we're not just using big numbers to show that these sounds are louder, but that each increase is perceived exponentially louder.

• A sound of 30 decibels is ten times more intense than a sound of 20 decibels.
• Therefore, a normal conversation at 60 decibels is significantly louder than a whisper.

By using decibels, we can describe how loud a sound is in a way that reflects real-world perception.

Logarithms

Logarithms are a mathematical tool that helps us manage very large or very small numbers easily. In sound intensity, they allow us to quantify immense differences in sound pressure levels with simple numbers. When calculating decibels ( \( \beta = 10 \log_{10}\left(\frac{I}{I_{0}}\right) \)), the logarithm compresses the scope of sound intensities into a more compact range. This makes it easier to represent and compare sounds that can range over several orders of magnitude in intensity. If you're wondering why we use logarithms, consider how our ears perceive sound. Our hearing senses variations in intensity logarithmically, which means that equal increments on the decibel scale correspond to equal perceptual changes.

• A change from 40 to 50 decibels represents the same relative increase in loudness as from 90 to 100 decibels.
• This logarithmic behavior aligns perfectly with human hearing.

Intensity of Sound

The intensity of sound refers to the amount of energy that the sound wave carries per unit area. It's measured in watts per square centimeter (or meters, etc.), providing a concrete measure of how strong a sound wave is at its source. The formula \( \beta(I) = 10 \log_{10}\left(\frac{I}{I_{0}}\right) \) uses the intensity ratio, comparing the given intensity \( I \) to a reference \( I_0 \), defined as the threshold of hearing, at \( 10^{-16} \) watts per square centimeter.This reference point represents the softest possible sound the average human ear can detect, making it a baseline for measuring all other sounds. When you double the intensity \( I \), the decibels don’t double. Instead, they increase by a certain amount, which corresponds to our understanding of sound as humans.

• For example, the whisper mentioned above has an intensity of \( 10^{-14} \) watts/cm².
• This is 100 times the threshold of hearing, and hence translates to 20 decibels.

Threshold of Pain

The threshold of pain is the point where sound becomes painfully loud to the human ear. It's generally at around 120 decibels, though some people might experience discomfort before or beyond this level. In terms of sound intensity, this is a whopping \( 10^{-4} \) watts/cm², contrasting with \( 10^{-16} \) watts/cm² for the threshold of hearing. Such a huge difference in intensity showcases why decibels are necessary—to compress that wide range into manageable numbers.

• The intensity of 120 decibels is over a trillion times greater than that of sounds at the threshold of hearing.
• Personal tolerance for pain at this level varies greatly among individuals.

Reaching or surpassing this threshold can cause physical damage to the ears, underlining the importance of understanding sound levels and protecting our hearing.