Sound Waves Legacy Problem #30 Guided Solution
Problem*
Haley and Ariel are working on their Musical Instruments Project. They are creating a Pop Bottle Orchestra using a collection of pop bottles filled with water to varying heights. By blowing over the top of the pop bottles, the air column inside acts as a closed-end air column with the water surface at the bottom forcing a vibrational node within the standing wave pattern. Each bottle is 34.2-cm tall. Assuming a speed of sound of 345 m/s and a first harmonic wave pattern, determine how much water must be in the bottle in order for it to produce a 416 Hz sound wave.
Audio Guided Solution
A very common physics project for students is to create a pop bottle orchestra using a collection of pop bottles. By partially filling the pop bottles with water, the student can blow over the top of the bottle, causing the air that remains within the bottle to begin to resonate with the frequency. The length of the air column that is left in that bottle is going to inversely affect the frequency or pitch which that bottle will play when blown over the top. Here we have a couple of students attempting to figure out how much water they need to put inside of a bottle in order to leave sufficient amount of air in order to produce a frequency of 416 Hz. We're told that the speed of sound is 345 meters per second, and we essentially need to use the strategy of finding out how long the air column is and subtracting that amount from the height of the bottle. That will give us the height of the water that is in that bottle. After all, if you think about it, if you partially fill a 34.2 cm tall bottle with water, then the sum of the height of the water plus the height of the air is going to equal the height of the bottle. So, to solve the problem, we need to begin by taking the speed, 345 meters per second, and the frequency, 416 Hz, and using it in the wave equation in order to calculate the wavelength, the lambda. So, V equals F lambda, and rearranged, we can say lambda equals V divided by F. Plugging in 345 and 416 for V and F, I can solve for lambda, and I get 0.8293 meters. Now, that's not the length of the air column. That's the length of the wave that's resonating within the air column, and if I understand the relationship between length and wavelength for the first harmonic, then I can find out how long the air column is, that is, how much air remains within that bottle. So, I have to recognize that this is going to act as a closed-in air column, with the water closing off the air column at the very bottom. And so, at the open end, where you blow across the top of the bottle, that's the open end, and what I would know for the first harmonic is within the length of the bottle, there's one-fourth of the wavelength. So now that I've calculated the wavelength as 0.8293, I can divide it by 4, and get the length of the air column. Comes out to be 0.2073 meters. Now, that's meters, and the height of the bottle is given in centimeters. So I'm going to take my 0.2073 meters, and I'm going to change it to centimeters, 20.73 centimeters. That's the height of the air column. All that's in that bottle is air and water, so 20.7 centimeters of the 34.2 centimeter tall bottle is taken up by air. The remaining 13.5 centimeters must be taken up by water. So simply by subtracting the 20.7 centimeters from the 34.2 centimeter tall bottle, I get my answer, 13.5.
Solution
13.5 cm
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.