When a wave's intensity doubles, what is the relative change in decibels?

When considering how changes in intensity relate to decibels, it’s essential to understand the logarithmic nature of the decibel scale. The formula used to calculate the change in decibels (dB) related to a change in intensity is:

[ \Delta dB = 10 \log_{10} \left( \frac{I_2}{I_1} \right)

]

In this equation, (I_2) represents the new intensity and (I_1) represents the original intensity. If the intensity doubles, (I_2) becomes (2I_1). Plugging this into the formula gives:

[ \Delta dB = 10 \log_{10} \left( \frac{2I_1}{I_1} \right) = 10 \log_{10} (2) ]

The value of (\log_{10}(2)) is approximately 0.301. Thus, when calculated:

[ \Delta dB \approx 10 \times 0.301 = 3.01 \text{ dB} ]

Rounding this result gives a relative change of about +3 dB when intensity