Problem 43 - Precalculus Mathematics for Calculus

Question

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

(a) Derived formula: \(B_2 = B_1 + 20 \log \frac{d_1}{d_2}\). (b) 106 dB at 10 m.

Step 1: Derive Intensity Ratios

To relate the decibel levels at different distances, start by considering the formula for intensity: \( I = \frac{k}{d^2} \). Let \( I_1 = \frac{k}{d_1^2} \) represent the intensity at distance \( d_1 \), and \( I_2 = \frac{k}{d_2^2} \) for distance \( d_2 \). Then the ratio of intensities \( \frac{I_1}{I_2} = \frac{d_2^2}{d_1^2} \).

Step 2: Apply Decibel Equation

Using the decibel equation \( B = 10 \log \frac{I}{I_0} \) for two distances with the same reference intensity \( I_0 \), we have: \( B_1 = 10 \log \frac{I_1}{I_0} \) and \( B_2 = 10 \log \frac{I_2}{I_0} \). The difference between these two expressions gives us: \( B_2 - B_1 = 10 \log \frac{I_2}{I_1} \). Substituting the intensity ratio from Step 1, we get: \( B_2 - B_1 = 10 \log \left(\frac{d_1}{d_2}\right)^2 = 20 \log \frac{d_1}{d_2} \).

Step 3: Conclusion for Part (a)

We have shown the desired relationship \( B_2 = B_1 + 20 \log \frac{d_1}{d_2} \) by deriving the expression based on the inverse square law and the logarithmic decibel equation.

Step 4: Apply Formula for Part (b)

Given \( B_1 = 120 \text{ dB} \) at \( d_1 = 2 \text{ m} \) and \( d_2 = 10 \text{ m} \), use the derived formula: \( B_2 = B_1 + 20 \log \frac{d_1}{d_2} \). Compute: \( B_2 = 120 + 20 \log \frac{2}{10} = 120 + 20 \log \frac{1}{5} \).

Step 5: Calculate the Logarithm

Calculate \( \log \frac{1}{5} \) which is \( \log (0.2) \approx -0.699 \). Thus, \( 20 \log \frac{1}{5} = 20 \times (-0.699) = -13.98 \).

Step 6: Final Decibel Level

Therefore, \( B_2 \approx 120 - 13.98 \approx 106.02 \text{ dB} \). The intensity level at a distance of 10 m is approximately 106 dB.

Key Concepts

Inverse Square Law

The Inverse Square Law is a fundamental principle in physics that describes how certain physical quantities, like sound intensity, diminish with increasing distance from their source. In the case of sound, the law states that intensity is inversely proportional to the square of the distance from the sound source. This means if the distance from the source doubles, the intensity is reduced to one-fourth of its original value.

• Mathematically, it is expressed as \( I = \frac{k}{d^2} \), where \( I \) is the intensity, \( d \) is the distance, and \( k \) is a constant based on the initial intensity level.
• This relationship holds true for sound because it spreads spherically in a three-dimensional space, leading to an exponential decrease in intensity as it travels further away from its source.

Understanding this concept is crucial when analyzing sound levels at varying distances, as it allows you to calculate how much sound insulation or amplification may be needed to achieve desired audio levels in a specific area.

decibel levels

Decibel levels are a measure used to describe sound intensity in a logarithmic scale. This scale is particularly useful because the human ear perceives sound intensity logarithmically, meaning we perceive a sound's intensity proportional to its logarithm rather than its linear value.

• The decibel equation \( B = 10 \log \frac{I}{I_0} \) helps convert intensity levels to decibel levels, where \( B \) is the decibel level, \( I \) is the intensity, and \( I_0 \) is a reference intensity, usually the threshold of hearing.
• This logarithmic scale compresses the wide range of human hearing into a more manageable scale, making it easier to understand and compare sound levels.
• For example, if a sound is 10 times more intense, it appears 10 dB higher, and if it’s 100 times more intense, it appears 20 dB higher. This is because of the logarithmic nature.

When calculating decibel levels at different distances, like in rock concerts or industrial settings, having a basic grasp of decibel calculations ensures sound levels remain safe and within legal and comfort standards.

logarithmic equations

Logarithmic equations are equations that involve the logarithm of a variable or expression. They are particularly helpful in cases where you need to solve problems that deal with exponential scales, such as sound intensity measures.Logarithms are the inverse operations to exponentiation. In simpler terms, a logarithm answers the question, "To what power must we raise a specific number (base) to get another number?" For a logarithm \( y = \log_b(x) \), the equation \( b^y = x \) holds true.

• In sound measurements, the base of the logarithm is usually 10, reflecting the decibel scale where \( B = 10 \log \frac{I}{I_0} \).
• In calculations involving inverse square laws and decibel levels, logarithmic equations simplify the complexities inherent within these relationships, making it easier to determine how changes in one quantity affect another.

Thus, understanding the mechanics of logarithmic equations allows for efficient problem-solving in acoustics and helps in understanding how different distances affect sound intensity and perception at various locations.