Question

The number of possible modes of natural oscillations of air column in a pipe closed at one end of length $170\text{cm}$ below frequency $1450\text{Hz}$ are
[velocity of sound $=340{\text{ms}}^{-1}$]

• A. $14$
• B. $13$
• C. $12$
• D. $15$

Answer 1

The correct option is A $14$

Fundamental frequency for closed organ pipe,
$f_0=\frac{v}{4L}$
$\Rightarrow f_0=\frac{340}{4\times 1.70}=50\text{Hz}$
Natural frequencies can exist as odd multiple of $f_0$.
So, natural frequencies are,
$f_0,3f_0,5f_0......$
$\Rightarrow 50\text{Hz},150\text{Hz},250\text{Hz},.......$
For this arithmetic progression, let us suppose $n^{th}$ term is $1450\text{Hz}$.
So, $t_n=a+(n-1)d$
$\Rightarrow 1450=50+(n-1)100$
$\Rightarrow n=15$
So, total 14 modes are possible for frequency below $1450\text{Hz}$.