Question

An open pipe of length 0.58 m vibrates in the third harmonic with a frequency of 939Hz. What is the speed of sound through the air in the pipe?
344 m/s

363 m/s

369 m/s

357 m/s

350 m/s

Answer 1

363 m/s is the speed of sound through the air in the pipe.

Answer: Option B

Explanation:

The formula used to calculate the wavelength given as below,

      $Wavelength (\lambda)=\frac{\text { wave velocity }(v)}{\text { frequency }(f)}$

      $v=\lambda \times f$   --------> eq. 1

In power system, harmonics define by positive integers of the fundamental frequency. So the third order harmonic is a multiple of the third fundamental frequency. Each harmonic creates an additional node and an opposite node, as well as an additional half wave within the string.

If the number of waves in the circuit is known, the comparison between standing wavelength and circuit length can be calculated algebraically. The general expression for this given as,

         $L=\frac{n \lambda}{2}$

For first harmonic, n =1

         $L=\frac{\lambda}{2}$

For second harmonic, n =2

         $L=\frac{2 \lambda}{2}=\lambda$

For third harmonic, n =3

         $L=\frac{3 \lambda}{2}$

         $\lambda=\frac{2 L}{3}$   -------> eq. 2

Here given f = 939 Hz, L = 0.58 m...And, substitute eq 2 in eq 1 and values, we get

   $v=\frac{2 \times 0.58 \times 989}{3}=\frac{1089.24}{3}=363.08 \mathrm{m} / \mathrm{s}$