What Oscillation Mode Is Created in a Pipe with Mismatched Resonance Conditions?

[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?