Question

If ${\lambda }_1,{\lambda }_2,{\lambda }_3$ are the wavelengths of the waves giving resonance in the fundamental, first and second overtone modes respectively in an open organ pipe, then the ratio of the wavelengths ${\lambda }_1:{\lambda }_2:{\lambda }_3$ is:

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

Answer 1

The correct option is A $1:2:3$

When air is blown into the open organ pipe, the air column vibrates anti nodes are formed at the ends and nodes is formed in between

If $l\to$ length of pipe

$\therefore l=\frac{{\lambda }_1}{2}⟹{\lambda }_1=2l$

$\therefore v=n_1{\lambda }_1=n_{1}2l$

$\therefore n_1=\frac{v}{2l}\to \left(1\right)$

For second node,

$l={\lambda }_2$ or $v=n_2{\lambda }_2=n_{2}l$

$\therefore n_2=\frac{v}{l}=2n_1\to \left(1\right)$

Similarly, $n_3=\frac{v}{{\lambda }_3}=\frac{3v}{2l}=3n_1$

$n_1:n_2:n_3=1:2:3$