Oscillation mode question

[ductape]

In the problem, A string's tension is adjusted so that the speed of sound waves on the string equals the speed of sound in the air. The fundamental mode of oscillation is set up on the string, and in a pipe with one end open and one end closed with a length of half of the string resonance is created. What oscillation mode does that sound set up, fundamental, 1st, 2nd, or 3rd overtone?

I don't quite get the meaning of this question, I could use some clarification. The string is resonating at its fundamental frequency, so doesn't that mean that it will be the fundamental oscillation mode? Or do these two frequencies add to give the 1st overtone?

 

[berkeman]

I think the subtlety here may have to do with the "fundamental" frequency of the pipe, versus the "fundamental" frequency of the string. What is the shape of the fundamental oscillation y(x) on the string with its two ends fixed? What is the shape of a fundamental sound pressure oscillation P(x) with one end of the tube open and the other end closed? How are they different.

And assuming that the string excites a sound that has a wavelength that you described in your y(x) answer, how does that sound wavelength relate to the fundamental sound pressure distribution P(x) that you described above?

 

[m2003]

It is important to note that the fundamental frequency of a string and the fundamental mode of oscillation may not always be the same. In this case, if the tension of the string is adjusted to match the speed of sound in the air, the fundamental frequency of the string will also match the fundamental frequency of the pipe with one end open and one end closed. However, the mode of oscillation may differ.

The fundamental mode of oscillation on a string is when the entire length of the string vibrates with a single antinode (point of maximum displacement) in the center and two nodes (points of no displacement) at the ends. In the case of the pipe with one end open and one end closed, the fundamental mode of oscillation is when the air column inside the pipe vibrates with an antinode at the open end and a node at the closed end.

Therefore, the oscillation mode created in the pipe would be the fundamental mode of oscillation, not the fundamental frequency of the string. This is because the fundamental frequency of the string is not the same as the fundamental mode of oscillation in the pipe. The two frequencies may add to give the 1st overtone, but that is not the mode of oscillation being created in the pipe. It is important to distinguish between frequency and mode of oscillation in this scenario.