# University of Wisconsin Green Bay

A particular musical instrument is found to have only odd harmonics. If the instrument plays a low A (220 Hz), what are the frequencies of the first two overtones? What resonates to provide the sound that is amplified in this instrument?

• In this problem, you are directly told to consider overtones of vibration. Overtones are standing wave patterns that result from the superposition of identical waves in the same place at the same time. Therefore, this is a 1-dimensional standing wave problem. Regardless of the details of a question, always begin with the key physics of the situation.

• Even though you are asked for the overtones, always begin with a picture of the fundamental oscillation. The fundamental oscillation is the simplest wave pattern that meets the boundary conditions. In this case, you are not given direct information about the boundary conditions. Instead, you are told that only odd harmonics are present. If you have experience with other problems, you will recognize this to mean that the instrument is a closed pipe. The boundary conditions at the two ends are then different--the closed end is a position node (or pressure anti-node) and the open end is a position anti-node (or a pressure node.) The simplest pattern that accomplishes this is one fourth of a wavelength

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Once you have the fundamental oscillation, you can build the overtones in order just by making them increasingly complicated. Each addition of a node takes you to a higher order overtone. Note that as soon as you recognized that only odd harmonics means you have a closed pipe, you have answered the second question.

• Standing wave pictures give a picture of, and a way to measure, the the wavelength of the oscillation of interest. In this case, you are asked to consider the frequencies associated with the wavelengths. Wavelength and frequency are always related by the wave equation:

v = f λ

• We now know the frequencies of the first two overtones, and we previously determined the instrument is an open pipe. No further solution is necessary

• In this problem, the word "overtones" should flag that you want to examine standing waves, even if you aren't yet comfortable recognizing the relationship between standing waves, resonance, and musical instruments. All you need in order to understand 1-dimensional standing waves is a picture of the waves and the wave equation. In this case, you were not told whether the standing waves were in a string, an open pipe, or a closed pipe. You had instead the information that only odd harmonics were present, which constrains the boundary conditions and leads you to the realization that the instrument is a closed pipe.

That said, make sure to relate your answers to your musical knowledge. You know that overtones are higher frequencies than the fundamental and that harmonics are integer multiples of the fundamental.