The fundamental frequency of a pipe that is open at both ends is 524 Hz. If one end is now closed, find the wavelength

Verified step by step guidance

1. Understand the concept of fundamental frequency: For a pipe open at both ends, the fundamental frequency is the lowest frequency at which the pipe resonates. The formula for the fundamental frequency \( f \) of a pipe open at both ends is \( f = \frac{v}{2L} \), where \( v \) is the speed of sound in air and \( L \) is the length of the pipe.
2. When one end of the pipe is closed, the pipe becomes a closed pipe, and the fundamental frequency changes. The fundamental frequency for a pipe closed at one end is given by \( f' = \frac{v}{4L} \).
3. Since the fundamental frequency of the pipe when open at both ends is 524 Hz, use the relationship \( f = \frac{v}{2L} \) to express the length \( L \) in terms of the speed of sound \( v \) and the frequency \( f \).
4. Substitute the expression for \( L \) from the open pipe into the formula for the closed pipe's fundamental frequency \( f' = \frac{v}{4L} \) to find the new fundamental frequency when one end is closed.
5. To find the wavelength \( \lambda \) when one end is closed, use the relationship \( \lambda = \frac{v}{f'} \), where \( f' \) is the new fundamental frequency calculated in the previous step.

Key Concepts

Fundamental Frequency

The fundamental frequency is the lowest frequency at which a system vibrates. For a pipe open at both ends, it corresponds to the first harmonic, where the length of the pipe is half the wavelength of the sound wave. This frequency is crucial for determining the wavelength when the pipe's configuration changes.

Open and Closed Pipe Harmonics

An open pipe supports harmonics where both ends are antinodes, while a closed pipe has one end as a node and the other as an antinode. This change affects the harmonic series and the wavelength, as a closed pipe's fundamental frequency is half that of an open pipe, altering the wave pattern inside the pipe.

Wavelength Calculation

Wavelength is the distance between successive crests of a wave. In a closed pipe, the fundamental frequency results in a quarter-wavelength fitting into the pipe. To find the wavelength when one end is closed, use the relationship between frequency, speed of sound, and wavelength: λ = v/f, where v is the speed of sound and f is the frequency.