Question

What is the angular frequency of the third harmonic in a pipe of length $1.5\mathrm{m}$ with one closed end? (Note: The speed of the sound is approximately $100\frac{\mathrm{m}}{\mathrm{s}},)$ (A) 170 radians per second (B) $170\pi$ radians per second (C) $340$ radians per second (D) $340\pi$ radians per second

Short answer

Option B ($170\pi \text{radians per second}$)

Step-by-step solution

Identify the harmonic series and length condition

For a pipe with one closed end, the harmonics are given by odd multiples of the fundamental frequency. Therefore, the harmonics are denoted as $f=(2n-1)f_1$ where $n$ is an integer and $f_1$ is the fundamental frequency.

Calculate the fundamental frequency

The frequency of the fundamental (first harmonic) in a pipe closed at one end is given by $f_1=\frac{v}{4L}$where $v$ is the speed of sound and $L$ is the length of the pipe. Substituting the given values:$f_1=\frac{100}{4\times 1.5}\text{hi}=\frac{100}{6}=16.67\text{Hz}$

Determine the frequency of the third harmonic

Using the equation for the harmonics, the third harmonic ($n=3$) is: $f_3=(2(3)-1)f_1$ Substituting for the fundamental frequency: $f_3=5\times 16.67=83.35\text{Hz}$

Convert frequency to angular frequency

The angular frequency $\omega$ is given by $\omega =2\pi f$. Substituting the third harmonic frequency: $\omega =2\pi \times 83.35\approx 170\pi \text{ radians per second}$

Select the correct option

Among the given options, the correct one corresponding to $170\pi \text{radians per second}$ is option B.

Key concepts

harmonic series

In wave physics, a harmonic series refers to a sequence of frequencies that are integer multiples of a fundamental frequency. These frequencies are important in understanding the vibration patterns of various physical systems.
In the context of a pipe with one closed end, harmonics are represented as odd multiples of the fundamental frequency. This unique characteristic arises from the way sound waves reflect and interfere within the pipe.

• The first harmonic (fundamental frequency) is the lowest frequency and is typically denoted as $f_1$.
• The third harmonic, or the second overtone, is represented by $3f_1$, and so forth.

Recognizing how harmonics build upon the fundamental frequency aids in comprehending the vibration and sound production in such systems.

fundamental frequency

The fundamental frequency is the lowest natural frequency at which a system vibrates. For a pipe closed at one end, it can be calculated using the formula:
$$f_1=\frac{v}{4L}$$
In this equation:

• $v$ is the speed of sound, approximately $100\frac{m}{s}$ in air.
• $L$ is the length of the pipe. In this case, it is $1.5\text{m}$.

Substituting these values, we get:
$$f_1=\frac{100}{4\times 1.5}=16.67\text{Hz}$$
This fundamental frequency is the basis for all other harmonics in this system. By understanding $f_1$, we can determine the frequencies of higher harmonics by multiplying $f_1$ by odd integers.

angular frequency

Angular frequency, represented by $\omega$, is a measure of how quickly an object oscillates and is given by $\omega =2\pi f$. It is measured in radians per second.
To find the angular frequency of the third harmonic of the pipe, we first determine the frequency of the third harmonic. For a pipe with one closed end, the third harmonic is:$$f_3=5f_1$$
Substituting the fundamental frequency $f_1=16.67\text{Hz}$, we get:
$$f_3=5\times 16.67=83.35\text{Hz}$$
Using this in our angular frequency formula:
$$\omega =2\pi \times 83.35\approx 170\pi \text{ radians per second}$$
This calculation helps us understand the specific rate at which the third harmonic oscillates in radians per second.