Question

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

Answer 1

In the case of an air column vibrating in a pipe closed at one end, there are longitudinal stationary waves of sound formed in it.

At the closed end, the air molecules are not free to vibrate and hence over there a displacement node can only be formed.

Vibration is the least.

Conversely, at the open end the molecules are free to vibrate. Hence over there, always a displacement antinode is formed.

Vibration is the maximum.

We know that the distance between a node and an adjacent antinode is always $\lambda /4.$

In the first mode, the length of the air column $L={\lambda }_1/4.$

Frequency $f=V/\lambda$

Hence frequency becomes $f_1=\frac{V}{4L}$ as shown in figure (a)

In the 2nd mode, the length of the air column $L=3{\lambda }_2/4.$

Hence frequency becomes $f_2=\frac{3V}{4L}$ as shown in figure (b). $\therefore f_2=3f_1$

In the 3rd mode, the length of the air column $L=5{\lambda }_3/4.$

Hence frequency becomes $f_3=\frac{5V}{4L}$ as shown in figure (c). $\therefore f_3=5f_1$

Similarly $f_4=7f_1$

This shows that only odd multiples of the first fundamental frequency are present in such a case.

• Harmonics means the multiples of the fundamental frequency (may or may not) present in a case of a stationary wave.
  
• Overtones means the frequencies higher than the fundamental one really present.
  

You can thus see that only odd harmonics are present as overtones in the case of an air column vibrating in a pipe closed at one end.

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${\lambda }_1=\frac{81}{173}$

${\lambda }_2=\frac{81}{170}$

$f_1-f_2=10$ smaller $\lambda$ means bigger $f$. Hence not $f_2-f_1$

$V=?$

We know that $f=\frac{V}{\lambda }$

$f_1-f_2=10$

$\therefore \frac{V}{{\lambda }_1}-\frac{V}{{\lambda }_2}=10$

$\therefore V\times ({\lambda }_1^{-1}-{\lambda }_2^{-1})=10$

$V\times \left(\frac{173-170}{81}\right)=10$

$V=\frac{81\times 10}{3}=270$

velocity of sound in air =270m/s