Resonance tube - class-XII

   An empty vessel with a base is like a closed organ pipe , and fundamental frequency of a closed organ pipe is given by ,

In metallic rod longitudinal waves travels in vibrations.
Option "B" is correct. 

A resonance tube is a classic laboratory apparatus used to determine the speed of sound in air by measuring the lengths of air columns that resonate with a tuning fork of known frequency.

Booming sound indicates that at that length, $l _1$, the air column is in resonance with the given frequency.
and that length is,
$l _1= \lambda /4=10$
or, $\lambda = 40cm$
The next resonance length will be :
$l _2=3\lambda/4=30 cm$

$\quad \displaystyle\frac{5\lambda}{2} = 82.5\space cm$

$\quad \lambda = 33\space cm\quad and \quad v = f\lambda = 330\space ms^{-1}$ 

In a resonance tube, the resonance lengths are l1 = lambda/4, l2 = 3lambda/4, and l3 = 5lambda/4. The difference between consecutive resonances is lambda/2. Thus, l3 = l2 + (l2 - l1) = 2*l2 - l1.

Speed, wavelength and amplitude change as it is traveling through different material on the other side frequency must remain constant to conserve energy (which is dependent solely on frequency).

Velocity of sound in air $V _{air}=300 ms^{-1}$, and $\lambda _{air}=16 cm=0.16 m$.

In a closed air column, the frequency of mth mode of vibration is given by:

Kundt's tube is used to determine the velocity of sound in solids, liquids as well as in gases. In Kundt's tube the longitudinal waves are produced in air column and in rod. The nodes are detected by powder particles and the velocity of sound is determined. Also, the sound velocity in liquids is determined using Kundt's tube. The resonance tube, organ pipes are used to determine velocity of sound in air only. And sonometer determines velocity of sound in solids only.  

$\begin{array}{l} Let, , { n _{ 1 } }, , harmonic, , is, , corresponding, , to, , 50.7, cm, , and, , { n _{ 2 } }, , harmonic, , is, , corresponding, , 83.9, cm. \ { { Sin } }ce, , both, , one, , con\sec  utive, , harmonics \ \therefore their, , difference=\frac { \lambda  }{ 2 }  \ \therefore \frac { \lambda  }{ 2 } =\left( { 83.9-50.7 } \right) cm \ \frac { \lambda  }{ 2 } =33.2, cm \ \lambda =66.4, cm \ \therefore \frac { \lambda  }{ 4 } =16.6, cm \ length, , corresponding, , to, , fundamental, , e, , must, , be, , close, to, , \frac { \lambda  }{ 4 } , and, , 50.7, cm, \ must, , be, , and, , odd, , multiple, , of, , this, , length, , 16.6\times 3=49.8, cm. \ Therefore,, 50.7, , is, , { 3^{ rd } }, , harmonic \ e+50.7=\frac { { 3\lambda  } }{ 4 }  \ e=49.8-50.7=-0.9, cm \ speed, , of, , sound,v=f\lambda  \ \therefore v=500\times 66.4, cm/\sec  =332, m/s \end{array}$

Fundamental frequency (first harmonic frequency) of an open organ pipe is given by ,

For closed pipe $l _1=\dfrac{v}{4n}$
$v=2n(l _2-l _1)$
$n=\dfrac{v}{2(l _2-l _1)}$$=\dfrac{330}{2\times (0.49-0.16)}=500 Hz$

The length of the air column inside the resonance tube is determined by the amount of water kept in the tube. 
If a water filled tube is inclined, the length of resonance column would be different at different lines parallel to the tube, and thus a number of resonating lengths would be created.

Answer is B.

The end correction is given as follows.
End corrections $\displaystyle =\frac { { l } _{ 2 }-3{ l } _{ 1 } }{ 2 }$
$\displaystyle =\frac { 49cm-3\times 16cm }{ 2 } =0.5cm$.
Hence, the end corrections will be 0.5 cm.

    In an experiment to determine the speed of sound using a resonance column , prongs of the tuning fork are kept in a vertical plane so that both the prongs can send sound waves inside the tube .

Frequency of first resonance in closed pipe    $\nu = \dfrac{v}{4L _1}$      ....(1)
where $L _1 = 20 \ cm$
Frequency of next resonance in closed pipe  $\nu = \dfrac{3v}{4L _2}$      ....(2)
Equating (1) and (2), we get
$\dfrac{v}{4L _1} = \dfrac{3v}{4L _2}$
$\implies$  $L _2 = 3L _1  = 3\times 20 = 60 \ cm$

Frequency in a resonant tube=$f=p\dfrac{v}{2l}$