Problem 59 - Pearson Physics

Question

A car horn blasts out sound with an intensity level of \(68 \mathrm{~dB}\). How many such car horns would be required to reach an intensity level of \(78 \mathrm{~dB}\) ?

Step-by-Step Solution

Answer

10 car horns are required.

Step 1: Understand Decibel Scale

The decibel scale is logarithmic, not linear, which means that each step in decibels represents a multiplicative change in intensity. Specifically, an increase of 10 dB means the sound intensity has increased by a factor of 10.

Step 2: Decibel Intensity Formula

The formula for sound intensity level in decibels (dB) is given by:\[L = 10 \cdot \, log_{10}\left( \frac{I}{I_0} \right)\]where \(L\) is the sound level in decibels, \(I\) is the intensity, and \(I_0\) is the reference intensity.

Step 3: Calculate Intensity Ratio

To determine how many times more intense 78 dB is than 68 dB, we calculate the difference:\[78 \mathrm{~dB} - 68 \mathrm{~dB} = 10 \mathrm{~dB}\]Since a 10 dB increase equals a 10-fold increase in intensity, the intensity at 78 dB is 10 times the intensity at 68 dB.

Step 4: Determine Number of Horns Required

Since each horn produces an intensity level of 68 dB, to achieve a total intensity level of 78 dB, we need 10 horns. This is because each one increases the intensity by a factor of 1, and using 10 will increase it by a factor of 10.

Key Concepts

Sound Intensity

Sound intensity is a measure of the power carried by sound waves per unit area in a direction perpendicular to that area. It is usually measured in watts per square meter (W/m²). Higher sound intensity generally means a louder sound.

In the context of our exercise, when we talk about the intensity of a car horn, we are referring to the amount of energy it emits into the air as sound waves. The more intense a sound, the more energy it has, and this is why blasts from multiple car horns can produce a notable increase in the total sound intensity.

• Sound intensity relates directly to how we perceive the loudness of a sound.
• It is important to note that intensity is different from pitch; pitch refers to the frequency of sound, while intensity refers to its loudness.

Understanding sound intensity is key to solving problems related to how many sources (like car horns) are needed to reach a certain loudness level measured in decibels.

Logarithmic Scale

A logarithmic scale is one in which equal increments correspond to multiplicative changes, rather than additive changes. Logarithmic scales are used in many scientific fields because they can represent large ranges of values in a more compact way.

In acoustics, the decibel scale, which measures sound intensity levels, is logarithmic. This means that each step on the decibel scale represents a multiplication of the intensity, not just a simple addition.

• An increase of 10 dB means the intensity increases by a factor of 10.
• This approach is helpful in audio as human hearing is sensitive to proportional rather than absolute changes in sound intensity.

This makes it easier to express and compare the wide range of sound intensities we encounter. For example, increasing the sound intensity level by 10 dB (as in our exercise from 68 dB to 78 dB) corresponds to multiplying the factor of intensity by 10. This is why understanding the logarithmic nature of the decibel scale is fundamental in audio physics and engineering.

Decibel Formula

The decibel (dB) formula is a useful tool for understanding and calculating sound intensity levels. The formula is:\[L = 10 \cdot \log_{10}\left( \frac{I}{I_0} \right)\]where \(L\) is the sound level in decibels, \(I\) is the sound intensity, and \(I_0\) is a reference intensity, often set at the threshold of hearing, \(10^{-12} \text{ W/m}^2\).

This formula allows us to convert between raw sound intensities and their more usable decibel levels. When solving problems like determining how many car horns are necessary to reach a certain decibel level, the formula helps us calculate how an increase in physical sound intensity translates into the decibel scale.

• It is important because it lets us relate sound power to a perceived sound level.
• The logarithmic nature of the formula helps in scaling human-audible sound levels to fit within a compact decibel range.

Grasping the decibel formula is crucial for working with scenarios involving variable sound intensities, such as those commonly encountered in audio analysis and design.