Acoustics and Sound Waves

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:

Sound is a mechanical wave requiring a medium, so its speed depends on the medium's properties like temperature and density. Light is an electromagnetic wave that can travel through a vacuum, and its speed is a fundamental constant independent of the medium's temperature.

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 sum of the intervals between adjacent notes of the major diatonic scale is an octave.
A series of notes arranged, such that their fundamental frequencies have definite ratios is called a musical scale.
In $1588$, Zarlino constructed a musical scale by introducing six notes between an octave. These eight notes constitute major diatonic scale. The first note or the note of the lowest frequency is called keynote and ratio of the frequencies of the two notes is called interval between them. It means two octaves higher means four times the given frequency.
$\therefore$ Required frequency $=4\times 128$
$= 512\ Hz$

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

Forced vibrations: External force is acting on the body. 
Free vibration: Constant amplitude and no external force.
Damped vibration: Amplitude is not constant, it keeps on decreasing due to environmental factors of the system like air resistance.  
Therefore, correct option is B.