Problem 95 - Physics

Question

In a discussion person \(\mathrm{A}\) is talking \(1.5 \mathrm{~dB}\) louder than person \(\mathrm{B},\) and person \(\mathrm{C}\) is talking \(2.7 \mathrm{~dB}\) louder than person \(\mathrm{A}\). What is the ratio of the sound intensity of person \(\mathrm{C}\) to the sound intensity of person \(\mathrm{B} ?\)

Step-by-Step Solution

Answer

The ratio of sound intensity is approximately 2.63.

Step 1: Understanding the Decibel Formula

The decibel (dB) scale is a logarithmic scale used to compare sound intensities. The formula for sound intensity level in decibels is given by \( L = 10 \log_{10}(I/I_0) \), where \( L \) is the sound level in decibels, \( I \) is the intensity of the sound, and \( I_0 \) is a reference intensity.

Step 2: Establishing Relationships Between Intensities

We know that person A is talking 1.5 dB louder than person B, and person C is talking 2.7 dB louder than person A. The total difference in decibels between person C and person B is 1.5 + 2.7 = 4.2 dB.

Step 3: Using the Intensity Ratio Formula

To find the intensity ratio, use the formula for decibels: \( \,\Delta L = 10 \log_{10}(I_C/I_B) \). Here \( \,\Delta L = 4.2 \). Thus, \( \log_{10}(I_C/I_B) = 0.42 \).

Step 4: Solving for the Intensity Ratio

To find \( I_C/I_B \), rearrange the equation from the previous step: \( I_C/I_B = 10^{0.42} \).

Step 5: Calculating the Exact Ratio

Compute the value of \( 10^{0.42} \) using a calculator to get the exact intensity ratio. This gives \( I_C/I_B \approx 2.63 \).

Key Concepts

Sound Intensity

Sound intensity refers to the amount of energy carried by sound waves per unit area in a direction perpendicular to that area. It represents the strength or power of a sound wave and is measured in watts per square meter (W/m²). In simpler terms, it helps us understand how "loud" a sound is at a particular point.

When we talk about sound intensity, we are focusing on the physical attribute of sound that can be quantified. Intensity determines how much energy is transmitted through the air to your ears.

Keep in mind:

• High sound intensity can lead to a louder perceived sound.
• Intensity diminishes with distance from the sound source.
• It is a crucial factor in fields like acoustics and audio engineering.

Understanding sound intensity is essential in the context of discussing decibels, which use it as a basis for comparison.

Logarithmic Scale

The logarithmic scale is a way of displaying numerical data over a very wide range of values in a condensed manner. In acoustics, the decibel (dB) scale is logarithmic, meaning each step on the scale represents a tenfold increase or decrease in actual sound intensity.

This scale is particularly useful for measuring sound because the human ear perceives sound intensity logarithmically rather than linearly. For example, an increase of 10 dB roughly corresponds to a tenfold increase in the sound energy and is perceived as "twice as loud".

Some highlights of logarithmic scales:

• They allow us to represent very large or very small ranges of data in a scaled-down manner.
• A small increase in decibels can correspond to a large increase in intensity.
• They help in many scientific calculations where exponential growth or decrease is present.

This is why we use the log scale in sound measurements, helping to quantify "louder" or "quieter" more tangibly.

Sound Level

Sound level, as measured in decibels (dB), represents the loudness or intensity of sound as perceived by the human ear. The sound level is calculated using the formula:
\[ L = 10 \log_{10}(I/I_0) \] where:

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

This calculation allows us to compare the loudness of different sounds.
For example, if one person is speaking at 1.5 dB louder than another, it doesn't seem like much on the linear scale, but it indicates a more noticeable difference in perceived sound intensity. Similarly, a sound level 3 dB higher will have double the intensity of another.

A practical aspect of sound level involves:

• Understanding environmental noise conditions.
• Setting audio equipment accurately.
• Processing sound signals in various technological applications.

Sound levels in decibels provide a consistent, standardized way of assessing and calculating sound intensity, crucial for both everyday life and scientific exploration.