OpenStax College Physics for AP® Courses, Chapter 17, Problem 62 (Problems & Exercises)

This is College Physics Answers with Shaun Dychko. This question asks us what sound intensity level in decibels is needed for these different frequencies: 60 hertz, 3000 hertz and 8000 hertz in order to have the same perceived loudness as a 40 decibel sound at 1000 hertz. So at 1000 hertz, 40 decibel sound falls in this green line here which is labeling the 40 phons. Now this unit of phon is sort of a human unit; it's measuring the perception of loudness that a human would have and this vertical axis is measuring what a device would measure in... it's almost watts per square meter but not quite because you know, they have gone 10 times logarithm of the intensity divided by reference intensity defined this number here on this vertical axis but it's related to this measurable quantity of watts per square meter whereas this line here—this green line and this orange line— they are measuring what a human would report as the loudness and that's affected by, you know, resonance in the ear canal and other things that make certain frequencies seem louder. Okay! So at 60 hertz, we need to find that on this horizontal axis... this is the frequency axis... 60 hertz is about here say and we'll follow that straight upwards and see where it crosses the green line and then we'll go across to the left here and see whatever that corresponds to on the vertical axis. So at about 60 hertz, this 40 phons line is at about 67 on this intensity level scale here. That's what I'm guessing... it's sort of an eyeball thing and halfway between 60 and 80 would be 70 and this isn't quite halfway but it is a little bit more than a quarter so I'm calling it 67 decibels. And then for 3000 hertz, this is a special frequency that's in the conversational range and the hearing system is set up to perceive these frequencies even more loudly for a given intensity level which is to say that the green line dips down at these frequencies where the conversations happen. So we have made a line straight up from 3000 hertz and it's intersecting this 40 phons line here and let's extend that point to the left and see what its vertical coordinate is and that looks, you know, about 37 say... halfway between 20 and 40 would be 30 and let's suppose that's 37. And then for 8000 hertz—same process— 8000 is here, take that up to this green line and then extend that to the left and we see that that's about it's almost between 40 and 60 so it's almost 50... I don't know, I would say 51 for the decibel level there and there we go! So we can see that you need different intensity levels to have the same perceived volume for these different frequencies.