Question

If ${\lambda }_1,{\lambda }_2$ and ${\lambda }_3$ are the wavelengths of the waves which give resonance in the fundamental first and second overtones respectively in a organ pipe closed at one end, the ratio of the wave lengths, ${\lambda }_1:{\lambda }_2:{\lambda }_3:$ is (Neglecting end correction)

• A. $1:2:3$
• B. $1:3:5$
• C. $3:2:1$
• D. $15:5:3$

Answer 1

The correct option is D $15:5:3$

Given ${\lambda }_1,{\lambda }_2\&{\lambda }_3$ for the fundamental frequency , first overtone and second overtone

in a closed organ pipe. we have to find their ratio.

In a closed organ pipe,

The fundamental frequency is the first harmonic, the first overtone is the third harmonic, second overtone is the fifth harmonic and so on

(even harmonics are missing)

For fundamental frequency only one Node and one Anti-Node is formed.

so, ${\lambda }_1=4L$ where $L=$ length of the organ pipe

For first overtone or third harmonic two Nodes and two Anti-Nodes are formed.

$3{\lambda }_2=4L\Rightarrow {\lambda }_2=\frac{4L}{3}$

For second overtone or fifth harmonic three Nodes and three Anti-Nodes are formed.

$5{\lambda }_3=4L\Rightarrow {\lambda }_3=\frac{4L}{5}$

So ${\lambda }_1:{\lambda }_2:{\lambda }_3=4L:\frac{4L}{3}:\frac{4L}{5}$

multiplying by $\frac{15}{4L}$

${\lambda }_1:{\lambda }_2:{\lambda }_3=15:5:3$

So D is the correct answer.