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

Verified step by step guidance

1. Understand that a pipe open at both ends supports standing waves with nodes at both ends. The fundamental frequency (first harmonic) has a wavelength that is twice the length of the pipe.
2. For a pipe open at both ends, the fundamental frequency \( f_1 \) is given by \( f_1 = \frac{v}{2L} \), where \( v \) is the speed of sound in air and \( L \) is the length of the pipe.
3. When one end of the pipe is closed, the pipe supports standing waves with a node at the closed end and an antinode at the open end. The fundamental frequency for this configuration is \( f_1' = \frac{v}{4L} \).
4. Since the fundamental frequency of the open pipe is given as 524 Hz, use the relationship \( f_1 = \frac{v}{2L} \) to express the length \( L \) in terms of the speed of sound \( v \) and the given frequency.
5. To find the wavelength \( \lambda' \) of the fundamental frequency when one end is closed, use the relationship \( \lambda' = 4L \), where \( L \) is the length of the pipe determined from the open pipe configuration.

Key Concepts

Fundamental Frequency

The fundamental frequency is the lowest frequency at which a system oscillates. For a pipe open at both ends, it is determined by the speed of sound in air and the length of the pipe. The fundamental frequency is the first harmonic, and it sets the basis for understanding how the frequency changes when the pipe's conditions are altered.

Open and Closed Pipes

An open pipe has both ends open, allowing air to move freely, creating antinodes at both ends. A closed pipe has one end closed, forming a node at the closed end and an antinode at the open end. This change affects the harmonic series and the wavelengths of the sound waves produced, altering the fundamental frequency and harmonics.

Wavelength and Frequency Relationship

The wavelength of a wave is inversely related to its frequency, given by the equation λ = v/f, where λ is the wavelength, v is the speed of sound, and f is the frequency. When the pipe's configuration changes, such as closing one end, the effective length of the pipe changes, affecting the wavelength and frequency of the sound produced.