What is the expected gain in decibels when comparing a listener who is 180 inches from a loudspeaker to one who is 600 inches away?

To determine the expected gain in decibels when comparing the sound levels at different distances from a loudspeaker, we can use the inverse square law of sound propagation. This principle states that sound intensity decreases with the square of the distance from the source.

The formula for calculating the change in decibels due to a change in distance is as follows:

[

\text{Change in dB} = 10 \times \log_{10}\left(\frac{d_1^2}{d_2^2}\right)

]

where (d_1) is the distance of the first listener (180 inches) and (d_2) is the distance of the second listener (600 inches).

1. Calculate the ratio of distances squared:

[

\frac{d_1^2}{d_2^2} = \frac{180^2}{600^2} = \frac{32400}{360000} = \frac{1}{11.11}

]

2. Plug this ratio into the decibel change formula:

[

\text{Change in dB} = 10 \times \log_{10}\left(\frac{1}{11.11}\