Sound Waves Legacy Problem #29 Guided Solution

As you read this problem, you quickly recognize that it's actually two problems in one. One of the problems is about calculating the length of an open-end organ pipe, given the fact that you know the frequency of its first harmonic, or fundamental, and the speed of sound within the pipe. The other one has the same frequency and the same speed, but it's the third harmonic frequency in a closed-end organ pipe, and again, you wish to calculate the length of it. So these two separate independent problems can be performed in much the same way. In Part A, I'm going to begin by taking the values of V, 346 meters per second, and F, 392 hertz, and using it in the wave equation in order to solve for lambda. The wave equation goes V equals F times lambda, where lambda is the wavelength. Rearranging, lambda equals V divided by F, and substituting 346 for V and 392 hertz for F, will yield a value for lambda of 0.8827 meters. Now that's the wavelength, not the length. To find the length, I need to find the length-wavelength relationship. That actual relationship depends on what harmonic it is, and for this open-end pipe, it's the first harmonic. I can click the link to go back to the physics classroom tutorial for open-end air columns, and I can find a collection of diagrams which relate the length to the wavelength. There you'll find that the length of the organ pipe is equal to one-half the wavelength of the first harmonic. So I can take one-half of the value 0.8827 meters, and I end up getting a length of 0.4413 meters. I can change that to centimeters, 44.13 centimeters, and I can also round to three significant digits. I need to repeat nearly the exact same process for the length of the closed-end organ pipe. It begins by first finding the wavelength. Since the V and the F are the same as for the open-end organ pipe, I'm going to get the same wavelength here for the closed-end organ pipe, 0.8827 meters. And like the first part of this problem, now I need to find the length-wavelength relationship. If you click the link that you find on this page that heads back to the physics classroom tutorial, you'll see a collection of diagrams there under the closed-end air columns page, which relate the length and the wavelength. Here we have the third harmonic, and for the third harmonic wave pattern, you'll find that the length is equal to three-quarters of a wavelength. So, now that I know the wavelength, I can just take three-quarters of it, and I get the length of the closed-end air column. It comes out to be 0.6620 meters. I can round that to three significant digits, and I can also convert it to centimeters.