Question

For a particular resonance tube, following are four of the six harmonics below $1000Hz$
$300,{\text{ }}600,\;750,\;and{\text{ }}900Hz$
The two missing harmonics are
(A) $75,{\text{ }}150$
(B) $150,{\text{ 4}}50$
(C) $400,{\text{ 80}}0$
(D) $250,{\text{ 40}}0$

Answer 1

Hint
To solve this question, we need to use the formula of the frequency of the closed organ pipe. Then by finding the H.C.F. of the values of the frequencies given in the question, we can find the missing frequencies.
$\Rightarrow f = \dfrac{{nv}}{{2L}}$, where $f$ is the ${n^{th}}$ harmonic frequency for a closed organ pipe of length $L$ and $n$ is a positive integer.

Complete step by step answer
We know that the ${n^{th}}$ harmonic frequency in a closed organ pipe is given by the relation
$\Rightarrow f = \dfrac{{nv}}{{2L}}$, $n = 1,2,3......$ (1)
By putting different values of $n$, we get the different frequencies.
So, each harmonic is a multiple of the fundamental frequency ${f_0} = \dfrac{v}{{2L}}$
Therefore, from (1), we have
$\Rightarrow f = n{f_0}$ (2)
The frequencies given in the question are of
$300,{\text{ }}600,\;750,\;and{\text{ }}900Hz$
To find the missing frequencies, first we need to find the fundamental frequency ${f_0}$, which the given frequencies are the multiples of.
To find the fundamental frequency, we need to find the H.C.F. of the frequencies given in the question.
So, H.C.F. of $300,{\text{ }}600,\;750,\;and{\text{ }}900 = 150$
Hence, the fundamental frequency, ${f_0} = 150Hz$
From (2) we get
$\Rightarrow f = 150n$
Putting the values of $n$ from $1$ to $6$, we get the six harmonics as follows
$150,{\text{ }}300,{\text{ 450, }}600,\;750,\;and{\text{ }}900Hz$
So, the missing frequencies are $150,{\text{ 450Hz}}$
Hence, the correct answer is option (B); $150,{\text{ 450}}$

Note
If we don’t know the exact formula for the ${n^{th}}$ harmonic frequency, then also we can solve these types of questions. This is because the harmonic frequencies are always a multiple of the corresponding fundamental frequency. So, we just need to find out the value of the H.C.F. of the frequencies given the problem, which will be the fundamental frequency. By putting the different values of $n$, we will get the missing frequencies.