Question

The diagram above in fig. shows three ways in which the string of length $l$ in an instrument can vibrate. Which has the frequency four times that of the first ?

• A. iii
• B. ii
• C. i
• D. i and iii

Answer 1

The correct option is A iii

The string on a musical instrument is (almost) fixed at both ends, so any vibration of the string must have nodes at each end. For instance the string with length L could have a standing wave with wavelength twice as long as the string (wavelength $\lambda =2L$) as shown in the first sketch in the next series. This gives a node at either end and an antinode in the middle.
Let's work out the relationships among the frequencies of these modes. For a wave, the frequency is the ratio of the speed of the wave to the length of the wave: $f=\frac{v}{\lambda }$. Compared to the string length L, you can see that these waves have lengths 2L, L, 2L/3, L/2. We could write this as 2L/n, where n is the number of the harmonic.
The fundamental or first mode has frequency, $f_1=$ $\frac{v}{{\lambda }_1}$ $=\frac{v}{2L}$,
The second harmonic has frequency, $f_2=$ $\frac{v}{{\lambda }_2}$ $=\frac{2v}{2L}=2f_1$,
The third harmonic has frequency, $f_3=$ $\frac{v}{{\lambda }_3}$ $=\frac{3v}{2L}=3f_1$,
The fourth harmonic has frequency, $f_4=$ $\frac{v}{{\lambda }_4}$ $=\frac{4v}{2L}=4f_1$, and, to generalise,
The nth harmonic has frequency, $f_n=$ $\frac{v}{{\lambda }_n}$ $=\frac{nv}{2L}=n{f}_1$.
All waves in a string travel with the same speed, so these waves with different wavelengths have different frequencies as given in the diagrams. The mode with the lowest frequency $\left(f_1\right)$ is called the fundamental or the principle note. The $n^{th}$ mode has frequency n times that of the fundamental. All of the modes (and the sounds they produce) are called the harmonics of the string. The frequencies f, 2f, 3f, 4f etc are called the harmonic series.
Hence, the answer is diagram (iii).