A narrow cylindrical air column of length L is closed at one end and open at the other. When excited by a loudspeaker, the column can resonate at discrete frequencies.
(a) Draw the standing‐wave patterns (displacement versus position) for the first three resonant modes of this air column. Clearly label nodes and antinodes.
(b) Explain why only odd harmonics are present in the resonance spectrum of a tube with one end closed and one end open.
(c) Compare the fundamental frequency of this tube with that of another tube of the same length that is open at both ends. Provide a clear justification for any difference.
(d) A thin movable piston is inserted at the open end, effectively shortening the length of the air column by 20%. Predict how the resonant frequencies shift and describe qualitatively how the audible pitch perceived by an observer changes. Support your explanation with appropriate physical reasoning.