Final answer:
The amplitude of a sound wave increases by a factor of 10,000 if the sound intensity level goes up by 40.0 dB.
Step-by-step explanation:
The amplitude of a sound wave is measured in terms of its maximum gauge pressure. The sound intensity level is a logarithmic scale used to measure the perceived loudness of a sound. The formula to calculate the change in sound amplitude is:
Amplitude factor = 10 ^(change in intensity level/ 10)
In this case, if the sound intensity level goes up by 40 dB, the amplitude factor can be calculated as:
Amplitude factor = 10 ^(40/ 10) = 10^4 = 10000
The question at hand is asking about the relationship between sound intensity level in decibels (dB) and the amplitude of a sound wave, specifically how much the amplitude of a sound wave increases if the sound intensity level rises by 40 dB.
Sound intensity level in decibels is a logarithmic scale, meaning that every increase of 10 dB represents a tenfold increase in sound intensity. However, amplitude, which is related to the maximum gauge pressure of the sound wave, increases by a factor of the square root of the intensity.
Therefore, if the sound intensity level increases by 40 dB, the sound intensity itself increases by a factor of 104 (as 40 dB represents four 10-dB increases). Hence, the amplitude of the wave increases by a factor of 102, which is 100 times.
Therefore, the amplitude of the sound wave increases by a factor of 10,000.