(a) By what factor must the sound intensity be increased to raise the sound intensity level by 13.0 dB? (b) Explain why you don't need to know the original sound intensity

Verified step by step guidance

1. Identify the relationship between sound intensity level (in decibels, dB) and sound intensity (I). The formula to use is \( \beta = 10 \log_{10}(\frac{I}{I_0}) \), where \( \beta \) is the sound intensity level in decibels, \( I \) is the sound intensity, and \( I_0 \) is the reference sound intensity (typically \( 10^{-12} \, \text{W/m}^2 \)).
2. Determine the change in sound intensity level, which is given as 13.0 dB. Use the formula for the difference in sound intensity levels when the intensity changes by a factor of \( k \): \( \Delta \beta = 10 \log_{10}(k) \).
3. Set up the equation to solve for \( k \) using the change in sound intensity level: \( 13.0 = 10 \log_{10}(k) \).
4. Solve for \( k \) by isolating it on one side of the equation: \( \log_{10}(k) = 1.3 \) and then raising 10 to the power of both sides to find \( k \).
5. Explain why the original sound intensity is not needed: The calculation depends only on the ratio of the final intensity to the initial intensity, not on their absolute values. The decibel scale is logarithmic and expresses ratios, making the actual value of the initial intensity irrelevant as long as it is consistent.

Key Concepts

Sound Intensity Level (dB)

The sound intensity level, measured in decibels (dB), quantifies the loudness of sound relative to a reference intensity. The formula for sound intensity level is L = 10 log10(I/I0), where L is the level in dB, I is the sound intensity, and I0 is the reference intensity, typically 10^-12 W/m². An increase of 10 dB represents a tenfold increase in intensity, making the dB scale logarithmic.

Intensity Ratio

To determine the factor by which sound intensity must be increased to achieve a specific change in sound intensity level, we use the relationship between dB and intensity. For a change of ΔL dB, the intensity must be increased by a factor of 10^(ΔL/10). Therefore, a 13 dB increase corresponds to an intensity increase factor of 10^(13/10), which simplifies to approximately 19.95.

Independence from Original Intensity

The calculation of the intensity increase factor for a given change in sound intensity level does not depend on the original intensity because the dB scale is relative. The logarithmic nature of the dB scale means that the ratio of intensities is what matters, not their absolute values. Thus, knowing the original intensity is unnecessary for determining how much to increase the intensity to achieve the desired dB change.