The fundamental frequency of a pipe that is open at both ends is 524 Hz. How long is this pipe? If one end is now closed

검증된 단계별 안내

Understand that a pipe open at both ends supports standing waves with nodes at both ends. The fundamental frequency corresponds to the first harmonic, where the length of the pipe is half the wavelength of the sound wave.

Use the formula for the speed of sound in air, which is approximately 343 m/s at room temperature. The relationship between speed (v), frequency (f), and wavelength (λ) is given by:

v = f λ

For the fundamental frequency, the wavelength is twice the length of the pipe. Therefore, the length of the pipe (L) can be expressed as:

L = \frac{v}{2 f}

Substitute the given frequency (524 Hz) and the speed of sound (343 m/s) into the formula to find the length of the pipe:

L = \frac{343}{2 \times 524}

If one end of the pipe is closed, the pipe now supports a different set of harmonics. The fundamental frequency for a pipe closed at one end is a quarter wavelength. The new length of the pipe can be calculated using the same speed of sound and the new fundamental frequency, which would be half of the original frequency for the open pipe.

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주요 개념

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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 calculating higher harmonics.

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, creating a node at the closed end and an antinode at the open end. This affects the harmonic series and the wavelengths of the standing waves that can form.

Speed of Sound

The speed of sound in air is approximately 343 meters per second at room temperature. It is crucial for calculating the wavelength of sound waves in a pipe. The relationship between speed, frequency, and wavelength is given by the equation: speed = frequency × wavelength, which helps determine the pipe's length.