Problem 28 - Pearson Physics

Question

What are the conditions necessary for a standing wave in a pipe that is open at one end?

Step-by-Step Solution

Answer

Standing waves require a node at the closed end, an antinode at the open end, and occur at odd harmonics.

Step 1: Understand the Problem

We need to find out the conditions required for a standing wave in a pipe that is open at one end. A standing wave occurs due to the interference of two waves traveling in opposite directions.

Step 2: Recognize the Open-Closed Pipe Configuration

In a pipe that is open at one end and closed at the other (called a closed-open pipe), there are specific conditions for the formation of standing waves. The closed end allows for a node (point of zero displacement), while the open end forms an antinode (point of maximum displacement).

Step 3: Use the Harmonic Series for a Closed-Open Pipe

Standing waves in a closed-open pipe only occur at certain frequencies, known as the harmonic frequencies. For a closed-open pipe, only odd harmonics (1st, 3rd, 5th, etc.) are possible. The wavelength condition is given by \( L = \frac{(2n-1)\lambda}{4} \), where \( L \) is the length of the pipe, \( \lambda \) is the wavelength, and \( n \) is a positive integer.

Step 4: Identify the Fundamental Frequency

The lowest frequency at which a closed-open pipe can support a standing wave is called the fundamental frequency (or first harmonic). For a pipe open at one end and closed at the other, this fundamental frequency corresponds to \( n = 1 \). The wavelength for this frequency is \( \lambda = 4L \).

Step 5: Check the Boundary Condition

Ensure the boundary conditions are met: a node at the closed end and an antinode at the open end. This ensures the wave reflects correctly to form a standing wave pattern between these points.

Key Concepts

Harmonic Frequencies

Harmonic frequencies are a key concept when studying standing waves in pipes. They represent the specific frequencies at which standing waves are able to form due to the interference of two waves moving in opposite directions. In the context of a closed-open pipe, harmonic frequencies only occur at odd harmonic numbers. This means that you will only see the first harmonic, third harmonic, fifth harmonic, and so on.
The reason for this odd-only harmonic pattern lies in the boundary conditions of a closed-open pipe. The closed end must always be a node (a point where the wave has zero amplitude). On the flip side, the open end must be an antinode, where the wave reaches its maximum amplitude. These conditions create a series of resonant frequencies that satisfy these boundary conditions.
To calculate the harmonic frequencies in a closed-open pipe, the relationship is governed by the equation:

• For odd harmonics, the length of the pipe, \( L \), relates to the wavelength of the standing wave, \( \lambda \), as follows: \( L = \frac{(2n-1)\lambda}{4} \), where \( n \) is any positive integer representing the harmonic number.

Understanding this relationship helps you predict at what harmonics a standing wave can be observed in such a pipe.

Closed-Open Pipe

In acoustics, a closed-open pipe, also known as a pipe that is open at one end and closed at the other, includes some distinct properties that affect how sound waves resonate within them. These pipes form standing waves due to the reflection of waves between the closed and open ends.
The closed end of the pipe sets the condition for a node. This is the point where wave motion is essentially paused or zeroed out. Contrastively, the open end is characterized by an antinode, where the wave's vibration reaches its maximum amplitude. This dynamic between node and antinode is crucial for generating standing waves in such a pipe.
Below are the significant aspects of closed-open pipes:

• The configuration supports only odd harmonics, due to the node-to-antinode structure.
• The wavelength conditions resulting from the node-to-antinode structure ensure certain wavelength multiples resonate, creating standing waves.
• The boundary conditions at the closed and open ends dictate the resonant frequencies and determine which harmonics can form.

This understanding of closed-open pipes is beneficial in designing instruments and understanding the fundamentals of acoustics.

Fundamental Frequency

The fundamental frequency is the lowest frequency at which a system can support standing waves. In a closed-open pipe, this frequency is also known as the first harmonic. It's the frequency where the simplest wave pattern is formed, consisting of only one node at the closed end and one antinode at the open end.
For a closed-open pipe, the fundamental frequency occurs when \( n = 1 \). This creates a wavelength four times the pipe's length, expressed as \( \lambda = 4L \). This frequency sets the tone for all subsequent odd harmonics in the system, acting as a baseline for the harmonic series.
The fundamental frequency is significant for a few reasons:

• It determines the pitch of the note produced in musical contexts.
• It acts as the starting point from which all other harmonically related frequencies (overtones) are derived.
• A comprehension of this frequency allows us to manipulate and predict resonant behavior in acoustic systems like musical instruments.

Understanding the fundamental frequency is essential for anyone dealing with sound and vibration, offering a foundational insight into resonance.