Problem 27 - Pearson Physics

Question

What is the wavelength of the second harmonic in a \(2.5\)-m-long pipe that is open at both ends?

Step-by-Step Solution

Answer

The wavelength of the second harmonic is 2.5 meters.

Step 1: Identify the harmonic

In a pipe open at both ends, the harmonics are formed by standing waves. The second harmonic corresponds to a standing wave with two full wavelengths fitting inside the pipe.

Step 2: Determine the relationship between wavelength and pipe length

For a pipe open at both ends, the length of the pipe is equal to a whole number of half-wavelengths. In the second harmonic, the pipe length is equal to two full wavelengths. This means the length of the pipe (L) is equal to the wavelength (\(\lambda\)).

Step 3: Calculate the wavelength of the second harmonic

Given that the length of the pipe, L, is 2.5 meters, and the entire length is equal to the wavelength for the second harmonic, the wavelength \(\lambda\) of the second harmonic is 2.5 meters.

Key Concepts

Second Harmonic

Standing waves in a pipe open at both ends create unique patterns known as harmonics. The second harmonic specifically involves the wave pattern where the entire length of the pipe contains exactly two complete wavelengths. This is sometimes also called the "first overtone." For a standing wave to form, nodes and antinodes establish themselves in specific positions along the pipe. In the second harmonic, the pipe exhibits two nodes at both ends and one node in the middle, with antinodes situated between them.
Understanding harmonics can aid in musical instrument tuning and sound engineering, as different harmonics produce different tones. For instance, the second harmonic sounds at twice the frequency of the fundamental (first) harmonic. Hence, it is crucial in music to create richer and more diverse soundscapes.

Wavelength Calculation

In physics, understanding how to calculate the wavelength of harmonics, especially in pipes, is essential. The wavelength (\(\lambda\)) can be determined by a simple relationship with the pipe length (\(L\)). For a second harmonic in an open pipe:

• Length of the pipe equals two full wavelengths.
• This means for a second harmonic, \(L = 2\lambda\).
• Therefore, the wavelength is directly equal to the pipe length.

Thus, identifying these relationships helps solve the wavelength problems efficiently. In the context of our example, the calculation shows that each harmonic pattern can vary, altering the frequency and pitch of the sound observed.

Open Pipe Harmonics

Pipes open at both ends allow for unique properties in sound manipulation through standing waves. Open pipes support harmonics where the antinodes are at both ends, leading to whole-number multiples of half wavelengths fitting inside the pipe.
These conditions create a series of harmonics: first harmonic (fundamental frequency), second harmonic, third harmonic, and so on. Each harmonic corresponds to a specific frequency that is an integer multiple of the fundamental frequency. Because each end of the pipe must be an antinode, this imposes an interesting restriction, clearly differentiating open pipe harmonics from their closed end counterparts.
This characteristic tuning allows for versatile applications in various musical instruments, such as flutes and organ pipes, attributing to their distinct sounds. Understanding open pipe harmonics is valuable in acoustics, as it helps in designing instruments and controlling sound ranges. By analyzing how these harmonics occur, engineers and musicians can predict the behavior of sound waves in their instruments.