Problem 65 - University Physics with Modern Physics

Question

An organ pipe has two successive harmonics with frequencies 1372 and 1764 \(\mathrm{Hz}\) (a) Is this an open or a stopped pipe? Explain. (b) What two harmonics are these? (c) What is the length of the pipe?

Step-by-Step Solution

Answer

(a) Open pipe. (b) 3rd and 4th harmonics. (c) Pipe length is approximately 0.438 m.

Step 1: Identify the Fundamental Frequency

The two successive harmonic frequencies are given as 1372 Hz and 1764 Hz. The difference between these two frequencies is 1764 - 1372 = 392 Hz. This difference represents the fundamental frequency because, in a harmonic series, consecutive harmonics differ by the fundamental frequency.

Step 2: Determine Pipe Type

In an open pipe, harmonics occur at all integer multiples of the fundamental frequency, while in a stopped pipe, harmonics occur at odd integer multiples. The first harmonic for both types is the fundamental frequency. Here, 1372 Hz and 1764 Hz can be written as 1372 = 3 * 392 and 1764 = 4 * 392, respectively. This indicates that the odd and even multiples are consecutive, meaning these harmonics belong to an open pipe.

Step 3: Identify the Harmonics

Since it is an open pipe and the fundamental frequency is 392 Hz, the frequencies of the harmonics can be expressed as n * 392 Hz where n is an integer. Therefore, 1372 Hz = 3 * 392 Hz (3rd harmonic), and 1764 Hz = 4 * 392 Hz (4th harmonic). Thus, the two harmonics are the 3rd and 4th harmonics of an open pipe.

Step 4: Calculate the Length of the Pipe

For an open pipe, the wavelength of the fundamental frequency (first harmonic) is given by \(\lambda = \frac{v}{f}\), where \(v\) is the speed of sound (approximately 343 m/s at room temperature), and \(f\) is the fundamental frequency (392 Hz). Thus, \(\lambda = \frac{343}{392} \approx 0.8755\,m\). The length of the pipe is half the wavelength of the fundamental frequency, or \(L = \frac{\lambda}{2} \approx \frac{0.8755}{2} \approx 0.438\,m\).

Key Concepts

Open vs Stopped Pipe

Understanding open and stopped pipes is crucial for figuring out the nature of harmonics in musical instruments and sound systems.

Open pipes, like a flute, have both ends open. This allows air to vibrate freely within the tube. The harmonics produced in open pipes occur at all integer multiples of the fundamental frequency. So if the fundamental frequency is, say, 100 Hz, the harmonics would be at 200 Hz, 300 Hz, 400 Hz, and so on.

• Open pipes: Harmonics at all integer multiples.
• Example: Fundamental frequency of 100 Hz, harmonics at 200 Hz, 300 Hz, etc.

Stopped pipes, on the other hand, have one closed end, like a clarinet. This affects the vibration patterns such that only odd multiples of the fundamental frequency are present. With a fundamental frequency of 100 Hz, the harmonics would be at 300 Hz, 500 Hz, etc.

• Stopped pipes: Harmonics at odd integer multiples.
• Example: Fundamental frequency of 100 Hz, harmonics at 300 Hz, 500 Hz, etc.

In our exercise, the presence of consecutive harmonics at 1372 Hz and 1764 Hz suggests an open pipe. This is because they are integer multiples of the fundamental frequency and do not follow the pattern of stopped pipes.

Fundamental Frequency

The fundamental frequency is the lowest frequency produced by any vibrating object, and it's the first harmonic. In our exercise, finding the fundamental frequency was essential to solving the problem.

The two given harmonics were 1372 Hz and 1764 Hz.

• Consecutive harmonics have a difference equal to the fundamental frequency.

By subtracting these two, 1764 Hz - 1372 Hz, we found the fundamental frequency to be 392 Hz.

This frequency sets the base value for all harmonics in the series. Knowing that the fundamental is the first harmonic in open pipes, the harmonics can be represented as multiples of this frequency.

• Fundamental frequency: 392 Hz.
• Controls all harmonic frequencies of the pipe.

This illustrates why identifying the fundamental frequency helps in predicting the specific harmonics formed within the pipe.

Pipe Length Calculation

Calculating the length of the pipe ties together our understanding of frequency, waves, and harmonics. This section explains the process using the fundamental frequency, which we calculated as 392 Hz.
For an open pipe, the fundamental frequency corresponds to the first harmonic.

• The formula involves sound speed: ewline \( \lambda = \frac{v}{f} \) where \( \lambda \) is the wavelength, \( v = 343 \, \text{m/s} \, \text{at room temperature}, \) and \( f \) is the frequency.

Substituting values, we get:

• \( \lambda = \frac{343}{392} \approx 0.8755 \, \text{m} \)

In open pipes, the length of the pipe is half the wavelength of the fundamental frequency.
This gives us:

• Pipe Length \( L = \frac{\lambda}{2} = \frac{0.8755}{2} \approx 0.438 \, \text{m} \)

Understanding this calculation helps illustrate the relationship between the physical dimensions of a pipe and the sound waves it produces.