Problem 29 - Pearson Physics

Question

What conditions produce a standing wave in a pipe that is open at both ends?

Step-by-Step Solution

Answer

Standing waves in open pipes form when the pipe length equals any integer multiple of half the wavelength.

Step 1: Understand the Basics

A standing wave in a pipe is formed by the superposition of two waves traveling in opposite directions. When a pipe is open at both ends, waves can reflect and interfere with each other to create standing waves.

Step 2: Identify Boundary Conditions

For a pipe open at both ends, the open ends form nodes for pressure (points of minimum pressure variation, as air pressure is constant there) and antinodes for displacement (points of maximum movement). This is because the air is free to move at these ends.

Step 3: Determine Wavelengths and Frequency

Standing waves form when the pipe length is an integer multiple of half the wavelength, i.e., \[ L = n \frac{\lambda}{2} \] where \( L \) is the length of the pipe, \( \lambda \) is the wavelength, and \( n \) is any positive integer (1, 2, 3,...).

Step 4: Calculate Harmonics

The fundamental frequency (first harmonic) occurs when \( n = 1 \), so the wavelength of this wave is given by \( \lambda_1 = 2L \). Higher harmonics or overtones occur when \( n \) is greater than 1, such as the second harmonic with overtones at \( n = 2, \lambda_2 = L \), third harmonic at \( n = 3, \lambda_3 = \frac{2L}{3} \), etc.

Step 5: Apply the Frequency Formula

The frequency \( f \) of the standing wave is related to the wavelength and speed \( v \) of sound in air by the formula: \[ f = \frac{v}{\lambda} \] For each harmonic, substitute the respective value of \( \lambda \) to calculate the frequency of standing waves.

Step 6: Conclusion

In conclusion, standing waves are formed in a pipe open at both ends when the length of the pipe is an integer multiple of half the wavelength. This causes nodes in pressure and antinodes in displacement at both open ends.

Key Concepts

Harmonics

Harmonics are the different modes or patterns of vibration that a system, like a stretched string or a column of air in a pipe, can support. In the context of a pipe open at both ends, harmonics refer to the distinct standing wave patterns that are established within the pipe. These are caused by the constructive interference of waves. Each harmonic is characterized by a specific frequency and wavelength.

For a pipe open at both ends, harmonics are classified as

• Fundamental frequency (first harmonic), where the whole length of the pipe is one-half the wavelength of the wave.
• Second harmonic, where the pipe contains one full wavelength.
• Third harmonic, which contains one and a half wavelengths, and so on.

All these frequencies are integer multiples of the fundamental frequency, making them predictable and structured.

Wavelength

Wavelength is the distance between consecutive points of a wave in phase (such as crest to crest or trough to trough). It is a crucial element in understanding standing waves in a pipe because it determines the pattern and frequency of sound produced.

When considering a pipe open at both ends, the wavelength of the standing wave has specific relationships with the pipe's length:

• The fundamental wavelength is double the length of the pipe.
• For the second harmonic, the wavelength equals the length of the pipe.
• With the third harmonic, the wavelength is two-thirds the length of the pipe.

This means that the wavelengths of the standing waves are determined by both the length of the pipe and the harmonic order.

Frequency

Frequency refers to how often the wave oscillates at any given point. It is measured in hertz (Hz), indicating the number of cycles a wave completes per second. For standing waves in a pipe open at both ends, frequency is crucial to characterize the sound it produces.

The frequency of a standing wave is found using the relation\[ f = \frac{v}{\lambda} \]

• \(v\) is the speed of sound in air, typically around 340 m/s.
• \(\lambda\) represents the wavelength associated with each harmonic.

The frequency will change with each harmonic because the wavelength changes, allowing us to hear different pitches. Higher harmonics have higher frequencies as they contain shorter wavelengths.

Pipe Open at Both Ends

A pipe open at both ends has distinct acoustic properties because both ends allow the air to move freely, forming nodes in pressure and antinodes in displacement at these points. This is because these points are free to oscillate without impedance.

This means:

• Pressure remains constant (nodes) at both ends.
• Maximum displacement (antinodes) occurs at both ends.

This arrangement leads to the formation of standing waves with clear harmonic structure, where the wavelength is clearly defined by the integer multiples of half the pipe's length. This is attributed to the specific boundary conditions at the pipe's ends, crucial for music instruments like flutes and organ pipes to produce harmonious sounds. By manipulating the length or the speed of sound, different harmonics can be achieved.