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Organ pipes come in various designs, but the specific type we're discussing here is one that is open at one end and closed at the other. This configuration is known as a closed-end pipe.

An organ pipe of this type supports standing waves, which are crucial for creating sound. When air is blown into the pipe, these standing waves form, leading to resonant frequencies.

In a closed-end pipe, the closed end acts as a node, where there is no air movement, and the open end acts as an antinode, where the air moves the most.

This characteristic determines the harmonic series a pipe can produce. Unlike open pipes, closed-end pipes only support odd harmonics, which can affect the types of sound they produce significantly. Understanding these principles is essential when considering musical instruments like organs, where the quality and type of sound are critical.

The pipe length is critical; in this exercise, we see a change in one pipe's length influences the sound and causes a beat frequency when both pipes are played together.

The fundamental frequency is the lowest frequency at which a system like an organ pipe resonates. It is the most critical frequency and forms the basis for all other harmonics and overtones produced.

For a pipe that is closed at one end, the fundamental frequency, denoted by \( f \), is calculated using the formula \( f = \frac{v}{4L} \) where \( v \) is the speed of sound in air, and \( L \) is the length of the pipe.

This formula reflects the specific standing wave pattern permissible in a closed-end pipe, where only one-quarter wavelength fits into the pipe's length.

In the given problem, different lengths cause variations in fundamental frequency, leading to the production of beats when sounds from two pipes interact. Understanding this principle explains why the smallest changes in length or environment can drastically affect the sound quality of a pipe organ or any resonant system.

The speed of sound in air is a vital parameter in acoustics and plays a crucial role in the resonant frequencies of musical instruments like organ pipes.

Typically, the speed of sound in air at room temperature (20°C) is approximately 343 meters per second. This value can fluctuate depending on temperature, humidity, and atmospheric pressure, but for simplicity and most basic calculations, we use this standard figure.

Knowing the speed of sound allows us to link the physical length of a resonant column with the sound frequencies it produces.

• For open-end organ pipes, the speed determines how wavelengths are structured inside the pipe.
• For closed-end pipes, it helps in determining the fundamental frequency using the formula \( f = \frac{v}{4L} \).

Understanding the speed of sound helps us explore how two pipes with slightly different lengths can create a beat frequency due to variations in fundamental frequencies, such as in our exercise.