What are the sound intensity levels for sound waves of intensity 3.0 x 10^-6 W/m²?

검증된 단계별 안내

Step 1: Recall the formula for calculating the sound intensity level (in decibels, dB): \( L = 10 \cdot \log_{10} \left( \frac{I}{I_0} \right) \), where \( I \) is the intensity of the sound wave, and \( I_0 \) is the reference intensity, typically \( 1.0 \times 10^{-12} \; \text{W/m}^2 \).

Step 2: Substitute the given intensity \( I = 3.0 \times 10^{-6} \; \text{W/m}^2 \) and the reference intensity \( I_0 = 1.0 \times 10^{-12} \; \text{W/m}^2 \) into the formula: \( L = 10 \cdot \log_{10} \left( \frac{3.0 \times 10^{-6}}{1.0 \times 10^{-12}} \right) \).

Step 3: Simplify the fraction inside the logarithm: \( \frac{3.0 \times 10^{-6}}{1.0 \times 10^{-12}} = 3.0 \times 10^{6} \).

Step 4: Take the logarithm of \( 3.0 \times 10^{6} \): \( \log_{10}(3.0 \times 10^{6}) = \log_{10}(3.0) + \log_{10}(10^{6}) \). Use the property \( \log_{10}(10^{n}) = n \) to simplify \( \log_{10}(10^{6}) = 6 \).

Step 5: Combine the results: \( L = 10 \cdot (\log_{10}(3.0) + 6) \). Use a calculator or logarithmic table to find \( \log_{10}(3.0) \), then multiply by 10 to get the final sound intensity level in decibels.

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주요 개념

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Sound Intensity

Sound intensity is defined as the power per unit area carried by a sound wave. It is measured in watts per square meter (W/m²) and represents how much energy the sound wave transmits through a given area. Higher intensity levels correspond to louder sounds, while lower levels indicate quieter sounds.

Decibel Scale

The decibel (dB) scale is a logarithmic scale used to measure sound intensity levels. It quantifies sound intensity relative to a reference intensity, typically 10⁻¹² W/m², using the formula L = 10 log₁₀(I/I₀), where L is the sound level in decibels, I is the intensity of the sound, and I₀ is the reference intensity. This scale allows for easier comparison of sound levels, as it compresses a wide range of intensities into a manageable numerical format.

Logarithmic Relationships

Logarithmic relationships are fundamental in physics, particularly in the context of sound intensity and the decibel scale. Since the human ear perceives sound intensity logarithmically, a small increase in decibel level corresponds to a significant increase in actual intensity. Understanding this relationship is crucial for interpreting sound levels and their effects on human perception.