Sound Waves Legacy Problem #8 Guided Solution
Problem*
Determine the decibel rating of the following sound sources and their estimated sound intensities.
- Science office at 5 PM on a weeknight: I = ~1 x 10-9 W/m2
- South's student library after school: I = ~1 x 10-6 W/m2
- Period 7 at the beginning of class: I = ~1 x 10-4 W/m2
- Titan Dome on a Friday night during basketball season: I = 8.1 x 10-3 W/m2
- Fall Out Boy concert - front row: I = 7.4 x 10-2 W/m2
Audio Guided Solution
The decibel rating of a sound gives a measure of the intensity of that sound wave at any given location from the source. Calculating the decibel rating demands that you use the equation that can best be found if you click the link back to the overview page to this set of problems. The equation goes something like this. dB, for decibels, is equal to 10 times the log of the quantity, intensity, divided by 1 times 10 to the negative 12. There are three steps here that you need to take in order to calculate the decibel rating of any sound. First, you need to take the intensity and divide it by what's called the intensity of the threshold of hearing, 1 times 10 to the negative 12. So if I were to do that for part A, I need to take 1 times 10 to the negative 9 and divide it by 1 times 10 to the negative 12. When I do that, I get 1,000 or 10 to the third, meaning that the science office at 5 p.m. on a weeknight is 1,000 times more intense than the so-called threshold of hearing or the sound which a person with good hearing can just barely hear. So now what I have is 1,000 as a ratio of the intensity of the science office to that of the threshold of hearing. Now what I need to do is my second step is I need to push the log button on my calculator. I need to take the log of 1,000. When I take the log of 1,000, my calculator tells me the answer is 3. That's the number of bells. That's the bell rating of this sound. The deci of decibels is a Greek prefix which means one-tenth bell. So if you have three bells and you want to know how many one-tenth bells you have, you need to multiply by 10. That's the third part of the equation. Multiply the 3 times the 10 and you get 30 decibels. So we would say that 30 dB is the decibel rating of the science office at 5 p.m. on a weeknight. Now we could do South Student Library after school. Its intensity is 1 times 10 to the negative 6 watts per meter squared. So we first need to do as our first step the ratio of that number to 1 times 10 to the negative 12. So we divide 1 times 10 to the negative 6 by 1 times 10 to the negative 12. What we get is 1 times 10 to the 6, meaning that the library is 10 to the 6 times 1 million times more intense than the threshold of hearing. Now what I need to do with that 10 to the 6 is take the log of it. Taking the log of 10 to the 6 gives me 6. Now if I multiply by 10 I get 60 decibels. I can repeat the similar process for 1 times 10 to the negative 4 in part C. And when I do that I end up getting 10 to the 8. I take the log of that and I get 8. And then I multiply by 10 and I get 80 decibels. Now in part D I take 8.1 times 10 to the negative 3 and you have to divide it by 1 times 10 to the negative 12. And when you do that you get 8.1 times 10 to the 9. Now if you take the log of that you end up getting 9.9085. And if you multiply that by 10 you get 99.085 decibels. We can round that to two significant digits. You can repeat the process for the Fall Out Boy concert and you end up with around 109 decibels.
Solution
- 30 dB
- 60 dB
- 80 dB
- 99 dB
- 109 dB
Habbits of an Effective Problem Solver
- Read the problem carefully and develop a mental picture of the physical situation. If necessary, sketch a simple diagram of the physical situation to help you visualize it.
- Identify the known and unknown quantities and record in an organized manner, often times they can be recorded on the diagram itself. Equate given values to the symbols used to represent the corresponding quantity (e.g., \(\descriptive{v}{v,velocity} = 345\unit{\meter\per\second}\), \(\descriptive{λ}{λ,wavelength} = 1.28 \unit{m}\), \(\descriptive{f}{f,frequency} = \colorbox{gray}{Unknown}\)).
- Use physics formulas and conceptual reasoning to plot a strategy for solving for the unknown quantity.
- Identify the appropriate formula(s) to use.
- Perform substitutions and algebraic manipulations in order to solve for the unknown quantity.
Read About It!
Get more information on the topic of Sound Waves at The Physics Classroom Tutorial.