Sound Waves Legacy Problem #9 Guided Solution

In this question, we must convert from the decibel rating of a sound to the intensity level in watts per meter squared. Now that conversion can be done using an equation, or it can be done conceptually simply using an understanding of what a decibel rating means. I�m going to begin this by using an equation. You�ll find the equation on the overview page for this set of problems if you click the link that you find underneath this audio help file. The equation says that to calculate the intensity, you need to first find the bell rating by taking the decibels and dividing by ten. Then once you get the bell rating, you need to raise ten to the power of the bell rating. Then once you�ve done that, your third step involves multiplying the result by one times ten to the negative twelve, the threshold of hearing for sound. So if I were to do that for the first problem, fifty decibels, I need to begin by taking the bell rating. The bell rating for fifty decibels would simply be five bells. Then I need to raise ten to the fifth power, and that gives me one with five zeros after it. Then I have to take the result, one with five zeros after it, or a hundred thousand, and I have to multiply it by one times ten to the negative twelve. And when I do, I get one times ten to the negative seventh watts per meter squared. You can repeat the process for ninety decibels, in which case it would be nine bells and ten to the ninth is ten to the ninth, and then multiply it by one times ten to the negative twelfth, and you end up getting one times ten to the negative third watts per meter squared. And again, for a hundred and ten decibels, that�s eleven bells, or ten to the eleventh, and then if you multiply that by one times ten to the negative twelfth, you get one times ten to the negative first watts per meter squared. Now the more conceptual means of doing this is understanding that what the decibel rating tells you is it tells you the number of times greater that the intensity of a sound is than the so-called threshold of hearing, one times ten to the negative twelfth watts per meter squared. Now when we say the number of times greater, we�re going to be using a logarithmic scale. That is, in terms of powers of ten. So if you have fifty decibels, think of it as five bells after all, the deci of decibels means one-tenth of a bell. So fifty-one-tenth bells is the same as five bells, which means that the sound, whatever intensity it has, is ten to the fifth times more intense than one times ten to the negative twelfth. So that�s five powers of ten greater than one times ten to the negative twelfth, and five powers of ten greater than one times ten to the negative twelfth means that you take the negative twelve as the exponent of ten and you add five to it, which makes it one times ten to the negative seven. If you did the same thing for ninety decibels, you�d think of it as nine bells, or ten to the ninth times more intense than the threshold of hearing. So you change the exponent of one times ten to the negative twelfth to one times ten to the negative third. And that�s how you approach it conceptually.