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Ch 16: Traveling Waves
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 16: Traveling WavesProblema 38a
Capitolo 16, Problema 38a

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

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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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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.
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Sound Intensity Level and the Decibel Scale

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.
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Sound Intensity Level and the Decibel Scale

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.
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