Problem 35 - Pearson Physics

Question

What is the wavelength of the third harmonic in a \(2.7\)-m-long pipe that is closed at one end?

Step-by-Step Solution

Answer

The wavelength of the third harmonic is 3.6 meters.

Step 1: Understand the Properties of Harmonics in a Closed Pipe

In a closed pipe, the harmonics occur at odd multiples of the fundamental frequency. This means the third harmonic can be expressed as the third odd multiple (n = 3) in such a pipe.

Step 2: Apply the Formula for Wavelength in a Closed Pipe

For a pipe closed at one end, the wavelength of the nth harmonic is given by: \( \lambda_n = \frac{4L}{n} \) where \( L \) is the length of the pipe, and \( n \) is the harmonic number.

Step 3: Calculate the Wavelength of the Third Harmonic

Substitute the length \( L = 2.7 \) meters and \( n = 3 \) into the formula: \[ \lambda_3 = \frac{4 \times 2.7}{3} \]Calculate the result to find the wavelength of the third harmonic.

Step 4: Complete the Calculation and Simplify

Continuing from Step 3, calculate:\[ \lambda_3 = \frac{10.8}{3} = 3.6 \text{ meters} \] This is the wavelength for the third harmonic.

Key Concepts

Wavelength Calculation

Wavelength is a fundamental concept in understanding wave phenomena. It represents the distance between two consecutive points that are in phase on a wave. In simpler terms, think of it as the length of a complete wave cycle. When dealing with pipes, the calculation of wavelength becomes crucial in determining the characteristics of the sound waves produced.

• In a closed pipe, specific harmonics are produced at various lengths, relating the pipe length to the frequency of the sound it emits.
• For closed pipes, harmonics are at odd multiples of the fundamental frequency.
• For the wavelength of the nth harmonic, we use the formula: \( \lambda_n = \frac{4L}{n} \).

This equation tells us how to find the specific wavelength based on the pipe's length and which harmonic we are observing.

Pipe Acoustics

Pipe acoustics refers to how sound behaves inside a tube-like structure. When a pipe is closed at one end, it has a unique impact on the sound waves it can produce. This closed-end reflects sound waves, creating a standing wave pattern.

• In closed pipes, like organ pipes or certain wind instruments, sound waves reflect off the closed end.
• This reflection creates nodes and antinodes within the pipe—the node being a point of minimum amplitude, and the antinode a point of maximum amplitude.
• This standing wave pattern is essential in producing harmonics, which rely on the wave's interference within the pipe.

Understanding pipe acoustics is key to mastering how instruments work and how sound can be controlled using confined spaces.

Third Harmonic

The third harmonic in closed pipes is an intriguing phenomenon where the frequency is three times that of the fundamental frequency. This harmonic corresponds to the third odd multiple and creates a unique standing wave pattern.

• In a closed pipe, the harmonics are odd-numbered: first, third, fifth, and so on, making it different from open pipes.
• The third harmonic specifically doubles the interaction within the pipe compared to simpler harmonics, like the first harmonic.
• For a 2.7-meter-long pipe closed at one end, using the formula \( \lambda_3 = \frac{4 \times 2.7}{3} \), we find that the wavelength is 3.6 meters.

This indicates how standing waves manifest more complexly as harmonic numbers increase, allowing for deeper exploration of sound properties.