Problem 26 - College Physics

Question

\(\bullet\) 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

Length of the pipe is approximately 0.2885 m, wavelength with one end closed is 1.154 m, and frequency is 297 Hz.

Step 1: Determine Speed of Sound

We'll assume the speed of sound in air is approximately 343 m/s. This is a standard reference value, and you'll often find it recommended for calculations of this type unless otherwise specified.

Step 2: Calculate Fundamental Wavelength for Open Pipe

A pipe open at both ends has nodes at both ends for its fundamental frequency, which corresponds to half a wavelength within the pipe. From the frequency formula, we have:\[ \text{Wavelength}, \ \lambda = \frac{v}{f} \]where \(v\) is the speed of sound, and \(f\) is the frequency. Substituting the given values:\[ \lambda = \frac{343 \, \text{m/s}}{594 \, \text{Hz}} \approx 0.577 \, \text{m} \].

Step 3: Determine Length of Pipe

Since the pipe supports half a wavelength when open at both ends, the length \(L\) of the pipe is:\[ L = \frac{\lambda}{2} = \frac{0.577}{2} \approx 0.2885 \, \text{m} \].

Step 4: Calculate Fundamental Wavelength for Closed Pipe

When one end of the pipe is closed, the pipe supports a quarter wavelength for its fundamental frequency; one end is a node, and the other is an antinode. The full wavelength in this situation is four times the length of the pipe:\[ \lambda_{closed} = 4 \times L = 4 \times 0.2885 = 1.154 \, \text{m} \].

Step 5: Determine New Frequency for Closed Pipe

Using the new wavelength \(\lambda_{closed}\), we calculate the frequency when one end is closed:\[ f_{closed} = \frac{v}{\lambda_{closed}} = \frac{343 \, \text{m/s}}{1.154 \, \text{m}} \approx 297 \, \text{Hz} \].

Key Concepts

Fundamental Frequency

The fundamental frequency of a wave in a pipe is the lowest frequency at which the pipe resonates. This resonance occurs when the length of the pipe is in harmony with the frequency of the wave traveling through it.
This concept is vital in understanding how musical instruments like flutes or organs produce notes. For a pipe open at both ends, the fundamental frequency represents a condition where there is a node (point of no displacement) at each end of the pipe, creating half a wavelength inside the pipe.
To summarize the relationship, the fundamental frequency \( f \) is influenced by the speed of sound \( v \) and the wavelength \( \lambda \). Understanding these relationships can aid in predicting how changes in frequency affect the sound produced.

Speed of Sound

The speed of sound in air is typically around 343 meters per second (m/s). This value, however, can change slightly based on environmental conditions such as temperature and humidity.
The speed of sound is crucial for calculating other sound properties like wavelength and frequency. It appears in the formula \( \lambda = \frac{v}{f} \), where \( \lambda \) is the wavelength and \( f \) is the frequency.

• A higher speed of sound yields a longer wavelength for the same frequency.
• Conversely, a lower speed of sound results in a shorter wavelength.

In calculations and exercises, assuming the speed of sound as 343 m/s simplifies the process, allowing for consistent results when measuring sound waves in air.

Wavelength Calculation

Wavelength is the spatial period of a wave, essentially the distance over which the wave's shape repeats. For sound waves in a pipe open at both ends, \( \lambda \) is linked directly to both the speed of sound and the frequency using the formula:\[ \lambda = \frac{v}{f} \]Where \( v \) signifies speed of sound (typically known or given), and \( f \) indicates the sound's frequency. For example, if the frequency of a sound wave in a pipe is given as 594 Hz and the speed of sound is 343 m/s, then the wavelength \( \lambda \) is approximately 0.577 meters.
It's essential to grasp how to manipulate these formula elements, as they serve as the base for further calculations involving sound properties in physics.

Open and Closed Pipes

Open and closed pipes behave differently with regard to resonance.
A pipe that is open at both ends supports standing waves with nodes at the ends and resonates at frequencies where approximately half a wavelength fits inside the pipe.
Conversely, a pipe closed at one end resonates with a node at the closed end and an antinode at the open end, supporting quarter-wavelength resonances. This occurs because the air column in the pipe vibrates in a different pattern, affecting the sound frequency.

• In an open pipe, the fundamental frequency is given by half a wavelength.
• In a closed pipe, the fundamental frequency results from a quarter of a wavelength.

Understanding these differences is key in applications involving sound resonance, including the design and tuning of wind instruments.