Sound Waves Legacy Problem #25 Guided Solution
Problem*
Middle C on a piano keyboard corresponds to a frequency of 262 Hz. Determine the length of an open-end air column whose fundamental matches this frequency. Use a value of 348 m/s as the speed of sound.
Audio Guided Solution
In this problem, we read of an open-end air column that's vibrating in a fundamental frequency of 262 Hz. That's F1. What we wish to calculate is we wish to calculate the length of the air column that would cause it to vibrate with that frequency. Waves travel through the air of the air column with a speed of 348 meters per second. That's V. So what I know is an F and a V, and what I'm looking for is the length of the air column. Now, immediately when you get F and V, you should think wave equation. I can find wavelength, and that wavelength is related to the length of the air column. It's related by its standing wave pattern, and for a first harmonic open-end air column, the standing wave pattern would show a half of a wavelength between the ends. So the length of the air column is one-half the wavelength. The strategy for this problem is to use the F and the V in the wave equation to calculate wavelength and then use the standing wave pattern to find the length of the air column. So F equal 262 Hz, V equal 348 meters per second, and V equal F times lambda, where lambda is the wavelength. Rearranging, wavelength equal V divided by F, 348 divided by 262 Hz, and that gives me a wavelength of 1.3282 meters. Now I know that within the length of the air column, there's a half of a wavelength, so L equal one-half times 1.3282 meters, that comes out to be 0.6641 meters, which I can round to three significant digits, or 66.4 centimeters.
Solution
0.664 m or 66.4 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.