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When discussing harmonics in pipes, it's essential to understand that these harmonics are specific frequencies at which the air inside the pipe resonates. Such frequencies correspond to the standing waves that can form inside the pipe due to its fixed boundaries. For example, an open-open pipe, which is open at both ends, supports harmonics that are multiples of the fundamental frequency, hence the term 'harmonic series'. The frequency of the n-th harmonic, denoted as harmonic number n, can be calculated using the formula for an open-open pipe: \[\begin{equation}f_n = \frac{n v}{2L}\end{equation}\]where n is the harmonic number (1, 2, 3, ...), v the speed of sound, and L the length of the pipe. In practice, when an organist plays a note, they’re not just hearing the fundamental pitch; they’re also hearing a mixture of various harmonics, which gives each pipe its unique sound.

The fundamental frequency is the lowest frequency produced by any oscillating system, such as an organ pipe. It's often referred to as the first harmonic. In pipe organs, the fundamental frequency defines the pitch of the note that the pipe is said to play. For a pipe that’s open at both ends, the fundamental frequency is determined by the length of the pipe and the speed of sound within it. Conversely, an open-closed pipe, which has one end closed and one end open, will have a fundamental frequency (first harmonic) that occurs at a lower frequency compared to an open-open pipe of the same length. Their fundamental frequencies have a unique relationship, where the open-closed pipe's fundamental frequency is found at the same pitch as the third harmonic of an open-open pipe. The Mathematical representation is given by:\[\begin{equation}f_1 = \frac{v}{4L}\end{equation}\]where f_1 is the fundamental frequency and L is the length of the open-closed pipe.

The concept of standing waves in pipes is a fascinating phenomenon, crucial for understanding the sound produced by organ pipes. A standing wave is a pattern of vibration that simulates a wave standing still. This behavior occurs as a result of interference between two waves traveling in opposite directions. In organ pipes, these waves are formed by the reflection of sound waves at the ends of the pipe. Open-open pipes have antinodes at both ends, allowing them to support standing waves with wavelengths that are multiples of twice the pipe's length. For open-closed pipes, the closed end is a node, and the open end is an antinode. This setup yields standing waves with wavelengths that are four times the pipe's length for the fundamental frequency. Therefore, understanding how these standing waves form and resonate within organ pipes is fundamental to grasping why particular lengths of pipes resonate at specific frequencies.Standing waves are inherently related to the harmonics and fundamental frequencies discussed in the previous sections, as they are the physical manifestation of these acoustic principles within the confines of organ pipes.