Problem 31 - Pearson Physics

Question

The fundamental frequency of a pipe that is open at both ends is \(200 \mathrm{~Hz}\). If you cut the pipe in half, will the fundamental frequency of each half be greater than, less than, or equal to \(200 \mathrm{~Hz}\) ? Explain.

Step-by-Step Solution

Answer

The frequency is greater; it becomes 400 Hz.

Step 1: Understanding the Problem

We have a pipe that is open at both ends with a fundamental frequency of 200 Hz. The task is to determine how the fundamental frequency changes when the pipe is cut in half.

Step 2: Frequency Formula for Open Pipes

The fundamental frequency \( f_1 \) of a pipe open at both ends is given by the formula \( f_1 = \frac{v}{2L} \), where \( v \) is the speed of sound and \( L \) is the length of the pipe.

Step 3: Effect of Cutting the Pipe

Cutting the pipe in half reduces the length \( L \) to \( \frac{L}{2} \). Since the frequency formula is directly related to the length \( L \), we replace \( L \) in the formula with \( \frac{L}{2} \): \( f'_1 = \frac{v}{2(\frac{L}{2})} = \frac{v}{L} = 2f_1 \).

Step 4: Calculate the New Frequency

Substituting the original fundamental frequency \( f_1 = 200 \) Hz into the equation, the new frequency \( f'_1 = 2 \times 200 = 400 \) Hz.

Key Concepts

Open Pipes

Open pipes, such as those found in musical instruments like flutes and organ pipes, have specific properties that determine the notes they produce. An open pipe is a tube that is open at both ends, allowing air to vibrate freely between them. This openness creates standing waves that are crucial for sound production.

For open pipes, the natural or fundamental frequency is the lowest frequency at which the pipe resonates. This occurs when there is a node at each open end and one antinode in the middle. Because the pipe is open on both ends, it supports harmonics that are odd multiples of the fundamental frequency. This means both the fundamental and overtones can be produced. Hence, the open pipe acts like a resonant cavity for sound.

The length of the pipe directly affects the fundamental frequency: the longer the pipe, the lower the frequency, and vice versa. This is because the length of the pipe determines the wavelength of the sound that can fit perfectly within it, establishing standing wave patterns that the pipe can support.

Frequency Formula

In understanding how open pipes produce sound, we utilize a specific frequency formula. The fundamental frequency of an open pipe, denoted as \( f_1 \), can be calculated using the equation \( f_1 = \frac{v}{2L} \). Here:

• \( f_1 \) is the fundamental frequency.
• \( v \) is the speed of sound in air.
• \( L \) is the length of the pipe.

This formula tells us that frequency is inversely proportional to the length of the pipe. Therefore, if the pipe's length decreases, the frequency increases since \( L \) is in the denominator of the fraction. This is why, when a pipe is cut in half, the new frequency is twice as high.
For example, given a pipe with an initial frequency of 200 Hz, when we halve its length, the fundamental frequency becomes \( \frac{v}{L} \), doubling the original frequency. As a result, the new fundamental frequency becomes 400 Hz after halving the pipe's length.

Speed of Sound

The speed of sound, an essential factor in calculating pipe frequencies, describes how fast sound waves travel through a medium. In general, the speed of sound in air is about 343 meters per second at room temperature (20°C or 68°F). This speed might vary slightly due to environmental factors like temperature, humidity, and air pressure.

When dealing with the formula \( f_1 = \frac{v}{2L} \), \( v \) represents this speed. The speed of sound affects how quickly waves propagate along the length of the pipe, impacting the frequency that is produced. A higher speed of sound would increase frequency for the same pipe length and vice versa.
It's important to note that while the formula itself is theoretical, understanding the speed of sound helps in comprehending how sound behavior changes based on pipe length and other characteristics of the pipe. Moreover, knowing this speed aids in understanding how changes in the medium's conditions might affect the sound wave propagation—and ultimately, what we hear when an open pipe resonates.