Question

If ${\lambda }_1,{\lambda }_2,{\lambda }_3$ are the wavelength of the waves giving resonance to the fundamental, first and second overtone modes respectively in a string fixed at both ends. The ratio of the wavelengths ${\lambda }_1:{\lambda }_2:{\lambda }_3$ is

• A. $1:2:3$
• B. $1:3:5$
• C. $1:\frac{1}{2}:\frac{1}{3}$
• D. $1:\frac{1}{3}:\frac{1}{5}$

Answer 1

The correct option is C $1:\frac{1}{2}:\frac{1}{3}$

The following are the wavelength : length ratios:

• 1st harmonic - $\lambda =2l$
• 2nd harmonic - $\lambda =l$
• 3rd harmonic - $\lambda =\frac{2}{3}l$

Now taking the ratios we get:
${\lambda }_1:{\lambda }_2:{\lambda }_3=1:\frac{1}{2}:\frac{1}{3}$