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Harmonics are fundamental concepts in physics connected to the vibrations of physical objects. When a system like a pipe or string vibrates, it tends to produce a series of frequencies. These frequencies are known as the harmonics. The lowest frequency is the fundamental frequency, and the higher ones are called overtones. In the context of organ pipes, these vibrations are what create different sounds or pitches.

An important aspect of harmonics is their pattern. In a closed pipe, only odd harmonics are present. This means that if the fundamental frequency is the first harmonic, the "first overtone" is actually the third harmonic, producing a frequency three times the fundamental. In contrast, open pipes support both even and odd harmonics. The overtone series can have all whole number multiples of the fundamental frequency. This distinction is key in understanding how pipes produce sound and why they sound different from one another.

The speed of sound is a crucial factor in determining the frequencies of sound waves in different media. This speed is the rate at which sound waves travel through a medium. In air, the speed of sound is approximately 343 meters per second at room temperature. This speed can vary with temperature and pressure, which can affect how sound waves are perceived.

When calculating the frequencies of harmonics, the speed of sound ( v ) is a vital variable. In formulas for harmonics in pipes, such as for closed or open pipes, the speed of sound plays a role in computing the frequency of sound waves produced. Understanding this allows us to calculate how long or short a pipe needs to be to produce a specific pitch when played.

Closed and open pipes function differently and produce unique sets of harmonics. A closed pipe is closed at one end and open at the other, whereas an open pipe is open at both ends. These structural differences lead to different patterns of standing waves and thus different harmonic series.

In closed pipes, the boundary condition allows for only odd harmonics. Therefore, if you were to measure the sound frequencies, you wouldn't find the second harmonic, only the first, third, and so on. Open pipes, however, can generate both even and odd harmonics. This means they can produce a wider range of tones.

In practical terms, these differences explain why certain musical instruments produce specific sounds and how instrument makers design them to create desired pitches and tones. Understanding the physics behind closed and open pipes is critical for anyone studying acoustics or working with sound.