If you were to build a musical instrument that uses a pipe with one closed end, what would be the most important thing to remember about the harmonics it can produce? It can only produce even-numbered harmonics. It can only produce odd-numbered harmonics. It can produce both even and odd-numbered harmonics. It can't produce any harmonics.

The correct answer is:<br /><br />**It can only produce odd-numbered harmonics.**<br /><br />### Explanation:<br />A pipe with one closed end behaves as a **closed-end resonator**, and the physics of sound waves in such a system dictates that it can only produce **odd-numbered harmonics**.<br /><br />1. **Fundamental Frequency (First Harmonic):**<br /> - In a closed-end pipe, the closed end must be a **node** (point of no displacement), while the open end must be an **antinode** (point of maximum displacement).<br /> - The length of the pipe corresponds to one-quarter of the wavelength of the fundamental frequency.<br /><br />2. **Harmonics:**<br /> - Only odd multiples of the fundamental frequency are possible because the wave pattern must maintain a node at the closed end and an antinode at the open end.<br /> - For example:<br /> - 1st harmonic: \( \lambda = 4L \) (fundamental frequency)<br /> - 3rd harmonic: \( \lambda = \frac{4L}{3} \)<br /> - 5th harmonic: \( \lambda = \frac{4L}{5} \), and so on.<br /><br />3. **Even-numbered harmonics:**<br /> - Even-numbered harmonics cannot form because they would require a second node or antinode where the physical constraints of the pipe do not allow it.<br /><br />Thus, when designing a musical instrument with a pipe that has one closed end, you must remember that it will only produce **odd-numbered harmonics**.