Ten cars in a circle at a boom box competition produce a 120-decibel sound intensity level at the center of the circle. What is the average sound intensity level produced there by each stereo, assuming interference effects can be neglected?
Let’s start by recording some of the important information given in the statement. We’re told that the sound intensity at the center of the circle is 120 decibels; we’ll call that 𝐼 sub c. And we’re asked to solve for the average sound intensity level produced by each stereo; we’ll call that 𝐼 sub avg.
Let’s begin by drawing a diagram of this circle The ten cars, each represented by a gold dot, all have stereos producing sound waves.
These stereos and waves are pointed towards the center of the circle. Here, the waves combine and create an intensity level of 120 decibels that we’ve called a 𝐼 sub c, for the intensity at the center.
If we assume that each stereo produces a sound intensity level of 𝐼 sub avg, then that means we can write that 10 times I sub avg is equal to 𝐼 sub c. When we say that interference effects can be neglected, we mean constructive and destructive interference that happens when waves overlap.
Outside of those effects, there still is a cumulative addition effect of combining these ten sound waves from the ten stereos. So 10 times the average sound intensity produced by each stereo is equal to 𝐼 sub c, which is equal to 120 decibels.
Now we might be tempted to solve for 𝐼 sub avg simply by dividing both sides of the equation by 10 and assuming that that would give us an answer of 12 point zero decibels. It’s at this point we need to remember important facts about the decibel scale.
The decibel scale, unlike some other number scales, is logarithmic. That means that when the power of the sound source is multiplied by 10, then the decibel value of the sound intensity created by that soars only has 10 added to it rather than multiplied. For example, if you had a source that created sound intensity of 30 decibels and then you multiply the power of that source by 10, the sound intensity that resulted would not be 300 decibels but 40 decibels.
As it relates to our situation, this means that when we divide both sides of the equation we’ve developed by 10, in order to solve for the average sound intensity created by a single stereo, we don’t find a resulting sound intensity level of 12 point zero decibels but rather one-tenth of 120 decibels is 110 decibels.
This is the average sound intensity level produced by each stereo in each car in the circle, as a result of working with the decibel logarithmic scale.