What is the length of a tube that has a fundamental frequency of 176 hertz and a first overtone of 352 hertz if the speed of sound is 343 metres per second?
We’re told in this statement that the fundamental frequency of this tube is 176 hertz, which we’ll call 𝑓 sub one. We’re also told that the speed of sound is 343 metres per second. What we’ll call 𝑣 the tube that we’re considering in this scenario. If we have a tube with both ends closed like this, then the fundamental wavelength of that tube corresponding to the fundamental frequency 𝑓 sub one would look like this, where 𝐿 is the length of the tube. That’s the length we want to solve for in this exercise.
To do so, we can recall that in general the speed of a wave 𝑣 is equal to the frequency of the wave times its wavelength. In our case, we can write 𝑣, the speed of sound, is equal to 𝑓 sub one, the fundamental frequency, times 𝜆 sub one, the fundamental wavelength, that we’ve drawn. Imagine we were to extend this wave until it is one complete wavelength and in fact this is the fundamental wavelength 𝜆 sub one, looking at the diagram we can see that 𝜆 sub one is equal to two times the length of the tube. So we can replace 𝜆 sub one in our equation with two 𝐿.
We now rearrange this equation to solve for 𝐿 and see that it’s equal to 𝑣 divided by two times 𝑓 sub one, the fundamental frequency. We’re given both 𝑣 and 𝑓 sub one and can plug those values in. When we calculate this fraction, we find that it’s equal to 0.974 metres. That’s how long the tube is with this fundamental frequency at this speed of sound.