Problem 24 - University Physics with Modern Physics

Question

The fundamental frequency of a pipe that is open at both ends is 594 Hz (a) How long is this pipe? If one end is now closed, find (b) the wavelength and (c) the frequency of the new fundamental.

Step-by-Step Solution

Answer

(a) 0.289 m; (b) 1.156 m; (c) 296.9 Hz

Step 1: Understanding the Problem: Fundamental Frequency of Open Pipe

For a pipe open at both ends, the fundamental frequency corresponds to the first harmonic where the length of the pipe (L) is half the wavelength of the wave. The formula is given by \( f = \frac{v}{2L} \), where \( f \) is the frequency and \( v \) is the speed of sound (approximately 343 m/s at room temperature).

Step 2: Calculate the Length of the Open Pipe

Rearrange the formula to solve for \( L \): \( L = \frac{v}{2f} \). Given \( f = 594 \) Hz, substitute \( v = 343 \) m/s to find \( L = \frac{343}{2 \times 594} \approx 0.289 \) meters.

Step 3: Wavelength with One End Closed

When one end is closed, the pipe supports quarter-wavelength resonances. The fundamental frequency corresponds to the first harmonic, where \( L = \frac{\lambda}{4} \). So the wavelength (\( \lambda \)) is \( \lambda = 4L \). Using \( L = 0.289 \) meters, we find \( \lambda = 4 \times 0.289 \approx 1.156 \) meters.

Step 4: Frequency of the New Fundamental

With a closed end, the fundamental frequency is given by \( f = \frac{v}{\lambda} \). Substitute \( \lambda = 1.156 \) meters and \( v = 343 \) m/s to find the new frequency: \( f = \frac{343}{1.156} \approx 296.9 \) Hz.

Key Concepts

Open Pipe

An open pipe is a type of resonating air column where both ends are open to the surrounding air. In terms of sound, this setup supports standing waves where the ends of the pipe are always antinodes. Antinodes are points of maximum amplitude, making the sound loud and clear at these points. This characteristic is important as it determines the wavelengths of the sound waves that can resonate within the pipe.
For an open pipe, the fundamental frequency is the lowest frequency at which the pipe resonates. The length of the pipe corresponds to half the wavelength (\( \frac{1}{2} \lambda \)). Therefore, in mathematical terms, the frequency of the wave is given by the formula:

• \( f = \frac{v}{2L} \)
• Where \( f \) is the frequency
• \( v \) is the speed of sound (approximately 343 m/s)
• \( L \) is the length of the pipe.

This relationship helps you calculate important characteristics of sound in musical instruments or acoustics.

Closed Pipe

A closed pipe is similar to an open pipe, but only one end is open while the other is closed. This structure changes the way sound waves travel, producing a different set of harmonics compared to an open pipe. In a closed pipe, the closed end is a node (point of no motion), while the open end remains an antinode (point of maximum motion).
Due to this setup, a closed pipe supports quarter-wavelength resonances. The fundamental frequency in a closed pipe corresponds to a wave whose length is four times the pipe's length. This means the pipe length (\( L \)) is one-fourth of the wavelength (\( \frac{\lambda}{4} \)). As a result, the formula for the wavelength becomes:

• \( \lambda = 4L \)

Compared to open pipes, closed pipes have a lower fundamental frequency because the air column vibrates at longer wavelengths.

Wavelength Calculation

The wavelength of the sound wave is an essential factor in determining the behavior of sound in pipes. For open and closed pipes, it's crucial to calculate this property because it influences the frequency and harmonics produced. The wavelength represents the distance between two consecutive points in phase, like peaks or troughs, in a wave.In practical situations, knowing the length of a pipe helps in calculating the wavelength. For an open pipe, where the length is half the wavelength, and a closed pipe, where the length is a quarter of the wavelength, these relationships are key:

• Open Pipe: \( \lambda = 2L \)
• Closed Pipe: \( \lambda = 4L \)

By understanding these formulas, you can accurately determine the wavelength of sound in different contexts, which aids in designing musical instruments and understanding acoustic phenomena.

Frequency Calculation

Frequency is a critical aspect of sound, defining how high or low a note is perceived. It's the number of wave cycles that pass a point per second, measured in Hertz (Hz). Understanding frequency calculation is essential, especially in the context of sound traveling through pipes.The fundamental frequency of a pipe is dependent on its length and whether it is open or closed. Using the relationship between speed, frequency, and wavelength (\( f = \frac{v}{\lambda} \)), you can determine the frequency of sound for both configurations:

• For Open Pipe: \( f = \frac{v}{2L} \)
• For Closed Pipe: \( f = \frac{v}{\lambda} \) with \( \lambda = 4L \)

By applying these formulas, you can easily compute the fundamental frequency and explore how altering the pipe’s length or configuration affects the sound it produces. This knowledge is particularly useful in the fields of music and acoustics engineering.