Question

The level of sound \(\beta\) (in decibels) with an intensity of \(I\) is $$\beta(I)=10 \log _{10} \frac{I}{I_{0}}$$ 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 sound levels for the different intensities are: (a) Whisper: 20 decibels, (b) Busy street corner: 70 decibels, (c) Air hammer: 95 decibels, (d) Threshold of pain: 120 decibels.

Step-by-step solution

Understand the Given Formula

Evaluate the given sound intensity formula. The formula for the sound level in decibels is \(\beta(I) = 10 \log _{10} \frac{I}{I_{0}}\), where \(I\) is the sound intensity to which you want to convert and \(I_{0} = 10^{-16}\) is the reference sound intensity, which corresponds to the faintest sound that can be heard.

Calculate Decibels for Whisper Intensity

Substitute \(I = 10^{-14}\) into the sound level formula to find out the decibel level for a whisper. So, \(\beta(10^{-14}) = 10 \log _{10} \frac{10^{-14}}{10^{-16}} = 10 \log _{10} 10^{2} = 10 * 2 = 20\) decibels.

Calculate Decibels for Busy Street Corner

Substitute \(I = 10^{-9}\) into the sound level formula to find out the decibel level for a busy street corner. So, \(\beta(10^{-9}) = 10 \log _{10} \frac{10^{-9}}{10^{-16}} = 10 \log _{10} 10^{7} = 10 * 7 = 70\) decibels.

Calculate Decibels for Air Hammer

Substitute \(I = 10^{-6.5}\) into the sound level formula to find out the decibel level for an air hammer. So, \(\beta(10^{-6.5}) = 10 \log _{10} \frac{10^{-6.5}}{10^{-16}} = 10 \log _{10} 10^{9.5} = 10 * 9.5 = 95\) decibels.

Calculate Decibels for Threshold of Pain

Substitute \(I = 10^{-4}\) into the sound level formula to find out the decibel level for the threshold of pain. So, \(\beta(10^{-4}) = 10 \log _{10} \frac{10^{-4}}{10^{-16}} = 10 \log _{10} 10^{12} = 10 * 12 = 120\) decibels.

Key concepts

Decibels

When it comes to understanding sound, the decibel is a crucial unit. This unit, symbolized as dB, measures sound level or sound intensity in a way that reflects the logarithmic nature of how we perceive sound. Essentially, the decibel scale is a logarithmic scale used to gauge the intensity of sounds.

Why logarithmic? Well, the human ear can detect a wide range of sound powers, from the slightest whisper to the deafening roar of a jet engine. Because this range is so vast, a linear scale would not be practical for representing sound levels. Instead, the logarithmic scale allows us to compress this range into a manageable set of numbers.

For instance, the sound level in decibels is calculated using the formula \(\beta(I) = 10 \log_{10} \frac{I}{I_{0}}\), where \(I\) is the intensity of the sound in question, and \(I_{0}\) is a reference intensity, typically the lowest intensity discernible by the human ear - around \(10^{-16}\) watt per square centimeter.

Here are some key takeaways about decibels:

• 0 dB marks the threshold of human hearing (reference sound).
• Every increase of 10 dB represents a tenfold increase in sound intensity.
• Normal conversation typically occurs around 60 dB, while a rock concert might reach 110 dB.
• Extended exposure to levels above 85 dB can result in hearing damage.

Understanding decibels is not just important for audio engineers but also for protecting our hearing in everyday environments.

Sound Intensity

Sound intensity is a measure of the power of sound waves as they travel through a medium, expressed in terms of energy per unit area. This concept is essential when calculating sound levels in decibels, which we encounter in various environments, from quiet rooms to noisy city streets.

Measured in watts per square centimeter (W/cm²), sound intensity is not directly proportional to the loudness we perceive. Instead, our perception of loudness increases on a logarithmic scale; hence the use of the logarithmic decibel scale for sound levels.

To better understand this, let's think of sound as ripples in a pond. Intensity is akin to the energy transferred by these ripples; a small stone generates gentle ripples with low intensity, while a large rock creates larger, more forceful ripples, corresponding to greater intensity.

In the exercise given, various sound intensities are considered, ranging from a whisper to the threshold of pain. Calculations demonstrate how these intensities correspond to different decibel levels. For example, a whisper is relatively quiet but is significantly louder than the faintest sound the average human ear can detect.

Key points about sound intensity include:

• It is directly related to the sound's energy.
• Intensity levels help us understand our environment's soundscapes.
• Knowing intensity measures can inform decisions on hearing protection.

Grasping this concept helps students understand the physics of sound and how it influences our daily experiences.

Logarithmic Functions

Logarithmic functions play a pivotal role in translating the physics of sound into something we can understand and measure, like the decibel scale. Fundamentally, a logarithm is the inverse operation to exponentiation, answering the question: to what exponent must we raise a base number to obtain another number?

For example, when we say \(\log_{10}{100} = 2\), we mean that 10 raised to the power of 2 gives us 100. This concept is essential when dealing with quantities that vary over a large range, from very small to very large values – such as sound intensity.

In our sound level calculation, the logarithmic function helps us by compressing the scale of sound intensities into a range that's manageable for our perception. Through the sound level formula \(\beta(I) = 10 \log_{10} \frac{I}{I_{0}}\), logarithms enable us to express a sound's intensity relative to a fixed reference point, making different sounds easy to compare.

Important aspects of logarithmic functions include:

• They are essential in many fields like music, engineering, and scientific measurements.
• A logarithmic scale is used whenever there's a need to deal with geometric progression or exponential growth.
• Understanding logarithms is crucial for interpreting various scientific data, including sound levels.

By understanding logarithmic functions, students can gain insight into complex phenomena in a more intuitive and accessible way.