Problem 43 - Algebra and Trigonometry

Question

\(29-43\) . These exercises deal with logarithmic scales. Inverse Square Law for Sound A law of physics states that the intensity of sound is inversely proportional to the square of the distance \(d\) from the source: \(I=k / d^{2} .\) (a) Use this model and the equation $$ B=10 \log \frac{I}{I_{0}} $$ (described in this section) to show that the decibel levels \(B_{1}\) and \(B_{2}\) at distances \(d_{1}\) and \(d_{2}\) from a sound source are related by the equation $$ B_{2}=B_{1}+20 \log \frac{d_{1}}{d_{2}} $$ (b) The intensity level at a rock concert is 120 \(\mathrm{dB}\) at a distance 2 \(\mathrm{m}\) from the speakers. Find the intensity level at a distance of \(10 \mathrm{m} .\)

Step-by-Step Solution

Answer

At 10m, the intensity level is approximately 106.02 dB.

Step 1: Understanding the Inverse Square Law

The inverse square law for sound states that the intensity \( I \) of sound is inversely proportional to the square of the distance \( d \) from the source, given by the equation \( I = \frac{k}{d^2} \), where \( k \) is a constant that represents the power of the sound source.

Step 2: Understanding Decibel Level

Decibel level, \( B \), is defined using the formula \( B = 10 \log \left( \frac{I}{I_0} \right) \), where \( I \) is the intensity of the sound and \( I_0 \) is a reference intensity, usually the threshold of hearing.

Step 3: Relate Decibel Levels at Two Distances

Given that \( I_1 = \frac{k}{d_1^2} \) and \( I_2 = \frac{k}{d_2^2} \), substituting into the decibel formula gives:\[B_1 = 10 \log \left( \frac{k / d_1^2}{I_0} \right), \quad B_2 = 10 \log \left( \frac{k / d_2^2}{I_0} \right)\]Subtract \( B_1 \) from \( B_2 \):\[B_2 - B_1 = 10 \log \left( \frac{k / d_2^2}{I_0} \right) - 10 \log \left( \frac{k / d_1^2}{I_0} \right)\]Simplify to get:\[B_2 - B_1 = 10 \log \left( \frac{d_1^2}{d_2^2} \right)\]This further simplifies to:\[B_2 = B_1 + 20 \log \left( \frac{d_1}{d_2} \right)\]This is the relationship you are asked to show in part (a).

Step 4: Calculate Intensity Level at 10m from a 2m Reference Point

Given at 2m, \( B_1 = 120 \text{ dB} \). You need to find the decibel level at \( d_2 = 10 \text{ m} \) using:\[B_2 = B_1 + 20 \log \left( \frac{d_1}{d_2} \right)\]Substitute \( d_1 = 2 \text{ m} \) and \( d_2 = 10 \text{ m} \) into the equation:\[B_2 = 120 + 20 \log \left( \frac{2}{10} \right)\]\[B_2 = 120 + 20 \log (0.2)\]\[B_2 = 120 + 20 \times (-0.69897) \approx 120 - 13.9794 \approx 106.02 \text{ dB}\]

Step 5: Conclusion

The intensity level at a distance of 10 meters from the speakers is approximately 106.02 dB.

Key Concepts

Inverse Square Law

The Inverse Square Law is a fundamental concept in physics that explains how the intensity of certain phenomena, like sound, light, or gravity, diminishes with distance. For sound, this law implies that as you move further from an audio source, the sound intensity decreases rapidly. Specifically, sound intensity is inversely proportional to the square of the distance from the source. This can be mathematically expressed as \( I = \frac{k}{d^2} \), where \( I \) represents the intensity of the sound, \( d \) is the distance from the source, and \( k \) is a constant related to the power of the source.

• The further you move from the sound source, the weaker the intensity becomes.
• For example, doubling the distance from the source reduces the intensity to one-fourth.

This principle is vital in many fields, including acoustics and astronomy, as it helps us understand how sound and light spread in different environments.

Decibel Levels

Decibel Levels are a logarithmic scale used to measure sound intensity. Rather than using absolute units, decibels (dB) express sound levels relative to a reference intensity, often the threshold of hearing. A common formula used to calculate decibel levels is \( B = 10 \log \left( \frac{I}{I_0} \right) \), where \( B \) is the decibel level, \( I \) is the sound intensity, and \( I_0 \) is the reference intensity.

• The logarithmic nature of the decibel scale means it can handle the wide range of human hearing, from the quietest whispers to the loudest noises humans can tolerate.
• A change of 10 dB represents a tenfold change in intensity.

This scale is essential for accurately conveying sound levels in practical, everyday situations.

Sound Intensity

Sound Intensity is a measure of the sound power per unit area. It is typically measured in watts per square meter. The calculation of sound intensity becomes particularly important in scenarios where understanding the energy emitted by a sound source is crucial. In practice, intensity levels can impact design considerations for speakers, soundproofing, and environmental noise management.

• Sound intensity is not the same as loudness, which is a perceptual phenomenon.
• Intensity levels tell us how powerful a sound is at a certain distance from the source.

As demonstrated by the Inverse Square Law, sound intensity significantly decreases as the distance from the source increases, which directly impacts how we perceive sound in different settings.

Logarithmic Functions

Logarithmic Functions are mathematical tools used to describe processes that change multiplicatively, rather than additively, and are central to understanding sound measurements in decibels. In the context of sound, these functions help capture the human ear's ability to perceive sound intensity changes more accurately. The relationship between sound intensity and perceived loudness is logarithmic because our ears detect differences in ratios (multiplicative changes) rather than absolute differences.

• Logarithms help us condense large numerical ranges into more manageable scales, such as the decibel scale.
• This is crucial for expressing very high or very low values in a more comprehensible way.

By applying logarithmic functions, we derive meaningful insights into how sounds are measured, compared, and understood.