Question

What is meant by harmonics? Show that only odd harmonics are present as overtones in case of an air column vibrating in a pipe closed at one end. The wavelength of two sound waves in air are $\dfrac{{81}}{{173}}m$ and $ \dfrac{{81}}{{170}}m$. They produce 10 beats per second. Calculate the velocity of sound in air.

Answer 1

Hint: Harmonics are the integer multiple of fundamental frequency or main frequency.
Fundamental frequency of closed organ pipe is given by:
$f = \dfrac{v}{{4l}}$, f is the frequency, v is the velocity of progressive wave and l is the length of the resonator).
Similarly, other odd harmonics are $f = \dfrac{{3v}}{{4l}}$,$f = \dfrac{{5v}}{{4l}}$...........

Complete step by step answer:
Let’s define harmonics first and then we will derive the expressions for odd harmonics of closed organ pipes.
Harmonics: The lowest resonant frequency of an object or body is called fundamental frequency. A harmonic is defined as the positive integer multiple of the fundamental frequency.
Now, we will derive the expression for odd harmonics in a closed organ pipe.
Let l be the length of the resonator:
Then,
$ \Rightarrow l = \dfrac{\lambda }{4}$ ($\lambda $ is the wavelength of the sound and l is the length of the resonator). ....from 1
$ \Rightarrow f = \dfrac{v}{\lambda }$ (f is the frequency, v is the velocity of the progressive wave)
$ \Rightarrow f = \dfrac{v}{{4l}}$ (substituting the value of $\lambda $ from equation 1 ). .......2
Expression in equation 2 is the fundamental harmonics.
Further we can derive for third harmonic:
$ \Rightarrow l = \dfrac{{3\lambda }}{4}$( for third harmonic)
$ \Rightarrow f = \dfrac{v}{\lambda }$(Again the frequency relation)
$ \Rightarrow f = \dfrac{{3v}}{{4l}}$(Substituting the value of $\lambda $)..........3
Equation 3 is the third harmonics
Likewise we will obtain only odd harmonics.
Numerical part:
We are given 10beats per second which means frequency
Magnitude of frequency change $f_1 - f_2 = 10$
We can also write as:
$ \Rightarrow \dfrac{v}{{{\lambda _1}}} - \dfrac{v}{{{\lambda _2}}} = 10$( because $f = \dfrac{v}{\lambda }$ )..............1
$ \Rightarrow v = 10[\dfrac{{{\lambda _1}{\lambda _2}}}{{{\lambda _1} + {\lambda _2}}}]$ ( we have found the value of v from equation 1)
$ \Rightarrow v = 10[\dfrac{{\dfrac{{81}}{{173}} \times \dfrac{{81}}{{170}}}}{{\dfrac{{81}}{{173}} + \dfrac{{81}}{{170}}}}]$( substituting the values of wavelength)
$
   \Rightarrow v = 10[\dfrac{{\dfrac{{6561}}{{29410}}}}{{\dfrac{{13770}}{{14013}}}}] \\
   \Rightarrow v = 10 \times 27.00 \\
   \Rightarrow v = 270m{s^{ - 1}} \\
 $ (doing the division in both numerator and denominator)

Therefore, the velocity of sound in air is 270m/s.

Note:
Open organ pipe (a wind instrument in which sound is produced by setting into vibration an air column in it) which is open at both the ends, does not necessarily contain only odd harmonics. Open organ pipe has integer multiples of fundamental frequency v, 2v, 3v, 4v.