Problem 32 - College Physics

Question

\(\bullet\) You blow across the open mouth of an empty test tube and produce the fundamental standing wave of the air column inside the test tube. The speed of sound in air is 344 \(\mathrm{m} / \mathrm{s}\) and the test tube acts as a stopped pipe. (a) If the length of the air column in the test tube is \(14.0 \mathrm{cm},\) what is the frequency of this standing wave? (b) What is the frequency of the fundamental standing wave in the air column if the test tube is half filled with water?

Step-by-Step Solution

Answer

(a) 614.29 Hz (b) 1228.57 Hz

Step 1: Understand the Problem

We need to calculate the frequency of the fundamental standing wave in the air column inside a test tube when the tube acts as a stopped pipe. In part (a), the test tube is empty, and in part (b), it is half filled with water.

Step 2: Identify the Formula for a Stopped Pipe

For a stopped pipe, the fundamental frequency is given by the formula \( f = \frac{v}{4L} \), where \( v \) is the speed of sound, and \( L \) is the length of the air column in the pipe.

Step 3: Convert Length from cm to m

The length of the air column is given as 14.0 cm. Convert this into meters: \( 14.0\, \text{cm} = 0.14\, \text{m} \).

Step 4: Calculate the Frequency for the Empty Test Tube (Part a)

Substitute \( v = 344\, \text{m/s} \) and \( L = 0.14\, \text{m} \) into the formula: \[ f = \frac{344\, \text{m/s}}{4 \times 0.14\, \text{m}} = \frac{344}{0.56} = 614.29\, \text{Hz} \].

Step 5: Calculate Half-filled Test Tube Length (Part b)

When the test tube is half-filled with water, the air column length becomes \( 0.14 / 2 = 0.07\, \text{m} \).

Step 6: Calculate the Frequency for the Half-filled Test Tube (Part b)

Substitute \( L = 0.07\, \text{m} \) into the same formula: \[ f = \frac{344\, \text{m/s}}{4 \times 0.07\, \text{m}} = \frac{344}{0.28} = 1228.57\, \text{Hz} \].

Key Concepts

Standing Waves

Standing waves are fascinating occurrences where certain wave frequencies appear to be standing still, rather than traveling through the medium. This happens due to the superposition of two waves traveling in opposite directions with the same frequency. As these waves overlap, they create points of constructive and destructive interference.
**Constructive interference** at nodes results in points where there is no movement, whereas **destructive interference** at antinodes results in points where movement is maximized.
In the context of our exercise, a standing wave is produced by the air column in a test tube when someone blows across its opening. The boundaries at the ends of the column lead to specific frequencies, forming these standing patterns.

Fundamental Frequency

In physics, the fundamental frequency is the lowest frequency at which a system can vibrate. It's also known as the first harmonic. For a stopped pipe, which our test tube represents, the fundamental frequency creates a specific standing wave pattern where the length of the air column includes exactly one quarter of a wavelength.
This frequency is crucial because it establishes the base tone or pitch that the system produces. The formula to calculate the fundamental frequency in a stopped pipe is given by \(f = \frac{v}{4L}\), where \(v\) is the speed of sound in the material, and \(L\) is the length of the column.

Sound Waves

Sound waves are types of mechanical waves that travel through air (or any other medium), carrying energy from one place to another. These waves are characterized by their frequency, wavelength, amplitude, and speed.
The speed of sound is often denoted as \(v\), and varies depending on the medium through which it moves. In our physics exercise, the speed of sound in air is considered as 344 m/s.
Sound waves demonstrate both wave behaviors, such as reflection and interference, which contribute to creating standing waves. By understanding sound waves, we can analyze how they interact within the air column of our test tube.

Stopped Pipe

A stopped pipe, also known as a closed tube, is a type of resonator where one end is closed off. This setup is similar to a test tube in our exercise.
Consequently, only odd harmonics are produced because the closed end acts as a node (point of no vibration), and the open end acts as an antinode (point of maximum vibration).
The fundamental frequency is achieved when the air column length is a quarter of a wavelength. This primary setup makes it essential to correctly measure the air column length, especially if the volume inside the pipe changes, such as half-filling it with water, altering the resonance conditions and producing different tones.

Physics Formulas

Physics is rich with formulas that help us understand and quantify the natural world. In this exercise, the main formula, \(f = \frac{v}{4L}\), plays a key role.
This equation calculates the frequency of the fundamental standing wave within a closed tube, like our test tube example. Here, \(v\) stands for the speed of sound, a constant under uniform conditions, and \(L\) represents the variable length of the air column.
Converting units properly is crucial, as seen when transforming 14.0 cm to 0.14 m to maintain consistency.

• The formula shows us that the frequency inversely depends on the length: shorter tubes result in higher frequencies.
• This relationship helps explain how changing the air column length by half-filling the test tube leads to different fundamental frequencies.