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A stopped pipe, also called a closed pipe, is a hollow tube with one end sealed and the other open. This configuration is common in various wind instruments, like clarinets and test tubes, when interacting with sound waves. The closed end of the pipe creates a boundary where the air can't move, forming a node here. Meanwhile, at the open end, air can move freely, forming an antinode. This setup means that the length of the stopped pipe supports a standing wave consisting of a quarter of a wavelength.

When air is blown across the open end, the vibrations inside the pipe produce a standing wave. This is the simplest form of vibration it can sustain and is known as the fundamental frequency. In a stopped pipe, the fundamental frequency corresponds to a standing wave with a single loop between the node and antinode, taking up a quarter wavelength.

The fundamental frequency is the lowest frequency at which a system naturally vibrates. In this context, it refers to the simplest standing wave pattern in the air column of a stopped pipe. The formula to find the fundamental frequency of a standing wave in a stopped pipe is given by:
\[ f = \frac{v}{4L} \]Here:

• \( f \) is the fundamental frequency.
• \( v \) is the speed of sound in the air, measured in meters per second (m/s).
• \( L \) is the length of the air column inside the stopped pipe, measured in meters.

For instance, in a test tube acting as a stopped pipe, if the air column's length is 14 cm, the fundamental frequency with sound speed at 344 m/s can be mathematically expressed as:

\[ f = \frac{344 \text{ m/s}}{4 \times 0.14 \text{ m}} = 614.29 \text{ Hz} \]
This indicates the air inside resonates primarily at approximately 614.29 hertz, creating the first harmonic.

The speed of sound refers to how fast sound waves travel through a medium. It is influenced by factors such as air temperature and pressure. For many calculations, a standardized speed of sound in air is used, which is typically around 344 meters per second at room temperature.

This value is crucial in determining wave properties such as frequency and wavelength in problems involving sound waves. It forms part of the fundamental frequency equation, linking the physical attributes of the medium (air in this case) and the resultant auditory properties We use this when calculating how sounds behave in different situations, like within a stopped pipe or through air columns of varying lengths.

The air column length is the distance between the closed node and the open antinode in a stopped pipe. This length determines the wavelength of the standing wave formed. In a stopped pipe, the fundamental frequency occurs when the air column is one quarter of the sound's wavelength.

When half of the test tube is filled with water, the air column length effectively reduces by half. This change requires recomputation of the fundamental frequency, as the new air column length (\[ L' \]) affects the standing wave’s wavelength within the pipe.

For example, for a water-filled test tube, the air column becomes 0.07 meters, resulting in a recalculated fundamental frequency of:
\[ f = \frac{344 \text{ m/s}}{4 \times 0.07 \text{ m}} = 1228.57 \text{ Hz} \]This doubling of frequency as the air column length halves demonstrates how sensitive sound waves are to changes in the length of the column in closed-ended configurations.