A microphone receiving a pure sound tone is connected to an oscilloscope producing a wave on its screen. The sound intensity is originally 2.00 times 10 to the negative two milliwatts per meter squared, but is increased until the amplitude shown by the oscilloscope increases by 30.00 percent. What is the intensity of the sound after the amplitude increases?
In this statement, we’re told the original sound intensity is 2.00 times 10 to the negative two milliwatts per meter squared; we’ll call that 𝐼 sub zero. We’re also told that, over time, the intensity is increased so that the amplitude displayed by the oscilloscope increases by 30.00 percent; we’ll call that Δ𝑃. We want to know the sound intensity after the amplitude increase; we’ll call that 𝐼 sub 𝑓.
As we begin on our solution, we can recall a proportionality relationship between signal intensity and signal amplitude. The intensity of a wave signal, 𝐼, is proportional to the square of the wave amplitude, 𝐴. If we were to draw a sketch of the oscilloscope screen, we would see a display of the wave represented by its amplitude rather than its intensity.
But because of this proportionality relationship, we can connect those two terms. We’re told that after the signal intensity is increased, the amplitude of the wave on the oscilloscope goes up by 30.00 percent, Δ𝑃.
If 𝐼 sub zero, the initial signal intensity, is proportional to 𝐴 sub zero, what we’ll call the initial amplitude of the signal, squared, then the final signal intensity 𝐼 sub 𝑓 equals 𝐴 sub zero plus Δ𝑃 quantity squared.
We can also write this as 1.300 squared times 𝐴 sub zero squared. But recall that 𝐴 sub zero squared is proportional to 𝐼 sub zero, the initial signal intensity.
If we plug in the given value for 𝐼 sub zero, being careful to write it in terms of watts per meter squared units, then when we multiply these values together we find that 𝐼 sub 𝑓, to three significant figures, equals 3.38 times 10 to the negative fifth watts per meter squared. That’s the final signal intensity that would create an amplitude increase of 30.00 percent.