Question

A closed organ pipe has a fundamental frequency of $1.5kHz$. The number of overtones that can be distinctly heard by a person with this organ pipe will be (Assume that the highest frequency a person can hear is $20,000Hz)$

• A. $4$
• B. $7$
• C. $6$
• D. $5$

Answer 1

The correct option is B $7$

Given,
$f_0$ is fundamental frequency $=1.5kHz$

Highest frequency that can be heard,

$f_{\text{max}}=20000Hz$

Fundamental frequency of closed organ pipe is given by

$f_0\frac{\upsilon }{4L}$

where $\upsilon$ is the velocity of sound in it and $L$ is the length of the pipe.Also, overtone frequencies are given by

$f=(2n+1)\frac{v}{4L}\text{or}f=(2n+1)f_0$

This means, $f_{\text{max}}>f$

So,

$f\text{max}>(2n+1)f_0$

$2000>(2n+1)1500$

$\Rightarrow 2n+1<13.33\Rightarrow 2n<13.33-1$

$\Rightarrow 2n<12.33\text{or}n<6.16$

$\Rightarrow \text{or}n=6\text{(integer number)}$

$\therefore \text{Total number of overtones that can be heard is}7.\left(0\text{to}6\right)$.