a wind instrument that produces only odd-numbered standing wave modes has what configuration of its ends?

A wind instrument that produces only odd-numbered standing wave modes must have one end open and one end closed (an open–closed tube).

Why open–closed only gives odd modes

• In an open–closed tube, the closed end is a node (no air movement) and the open end is an antinode (maximum air movement).

• Only patterns that fit a node at one end and an antinode at the other are allowed, which correspond to 14λ,34λ,54λ,…\frac{1}{4}\lambda,\frac{3}{4}\lambda,\frac{5}{4}\lambda,\dots 41​λ,43​λ,45​λ,…, i.e., harmonics with mode numbers n=1,3,5,…n=1,3,5,\dots n=1,3,5,… (odd only).

Why not open–open or closed–closed?

• Open–open tubes have antinodes at both ends, allowing standing waves with n=1,2,3,…n=1,2,3,\dots n=1,2,3,…, so both even and odd harmonics appear.

• Closed–closed tubes have nodes at both ends, which also support all integer modes n=1,2,3,…n=1,2,3,\dots n=1,2,3,…, giving both even and odd harmonics.

Answer: open–closed configuration (one end open, one end closed).

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