Problem 75 - Pearson Physics

Question

An organ pipe that is closed at one end is \(3.5 \mathrm{~m}\) long. What is its fundamental frequency?

Step-by-Step Solution

Answer

The fundamental frequency is approximately 24.5 Hz.

Step 1: Identify the Type of Pipe

The problem mentions that the organ pipe is closed at one end. This makes it a closed pipe, and closed pipes have specific harmonic properties.

Step 2: Determine the Speed of Sound

To find the fundamental frequency, we need the speed of sound in air, which is typically around \(343 \, \text{m/s}\) at room temperature.

Step 3: Use the Fundamental Frequency Formula for a Closed Pipe

The formula for the fundamental frequency of a closed pipe is: \(f = \frac{v}{4L}\), where \(f\) is the frequency, \(v\) is the speed of sound, and \(L\) is the length of the pipe.

Step 4: Insert Known Values into the Formula

Substitute the known values into the formula: \(f = \frac{343}{4 \times 3.5}\).

Step 5: Calculate the Fundamental Frequency

Perform the calculation: \(f = \frac{343}{14}\), which simplifies to \(f \approx 24.5 \, \text{Hz}\).

Key Concepts

Closed Pipe

A closed pipe is an interesting and fundamental concept in acoustics, especially when discussing musical instruments like organs. Such a pipe is closed at one end and open at the other. This configuration leads to unique acoustic phenomena compared to open pipes.

In a closed pipe, the closed end acts as a node, meaning no air movement occurs there, while the open end is an antinode, where the maximum air vibration happens. This setup causes only odd harmonics - frequencies that are odd multiples of the fundamental frequency - to be produced.

The fundamental frequency, also known as the first harmonic, has a wavelength that is four times the length of the pipe. Consequently, it can be expressed by the formula: \( f = \frac{v}{4L} \), where \( f \) is the fundamental frequency, \( v \) is the speed of sound, and \( L \) is the length of the pipe. Understanding closed pipes helps in grasping the basics of sound production and propagation, not just for musical applications but also for other acoustic devices.

Speed of Sound

The speed of sound is an essential concept in physics and acoustics that influences many aspects of sound propagation, including the fundamental frequency of closed pipes. Sound travels as a wave, and its speed depends on the medium through which it moves. In air, the speed of sound typically measures about \( 343 \, \text{m/s} \) at room temperature (around \( 20^\circ\)C).

Factors can alter this speed, such as:

• Temperature - The warmer the air, the faster sound travels.
• Humidity - Higher humidity levels can increase the speed of sound slightly.
• Pressure - In normal atmospheric conditions, changes in pressure affect sound speed negligibly.

For practical purposes, especially in exercises involving sound waves in pipes, knowing the approximate speed of sound is useful for determining frequencies and understanding wave behavior.
  
Thus, maintaining a conventional speed of \( 343 \, \text{m/s} \) helps standardize calculations when working with acoustic problems.

Harmonic Properties

Harmonic properties are a key part of understanding how instruments like closed pipes produce sound. When dealing with harmonics, we consider the various modes of vibration that can occur within a medium.

For a closed pipe, harmonic properties limit vibrations to odd harmonics: the first, third, fifth, and so on. This is due to the nature of wave reflection at the closed end, creating a pattern distinct from that of an open pipe.

The basic frequency is known as the fundamental frequency.

• First harmonic: \(f_1 = \frac{v}{4L} \)
• Third harmonic: \(f_3 = 3 \times \frac{v}{4L} \)
• Fifth harmonic: \(f_5 = 5 \times \frac{v}{4L} \)

As only odd harmonics are formed, each odd multiple adds vibrational complexity, creating the unique sound tones associated with such pipes.

Understanding harmonic properties in closed pipes not only provides insights into musical acoustics but also enhances our comprehension of sound wave behavior in various configurations.