When a string is divided into three segments of lengths \(l_1,~l_2\text{ and }l_3,\) the fundamental frequencies of these three segments are \(\nu_1,~\nu_2\text{ and }\nu_3\) respectively. The original fundamental frequency \((\nu)\) of the string is:
1. \(\sqrt{\nu}=\sqrt{\nu_1}+\sqrt{\nu_2}+\sqrt{\nu_3}\)
2. \(\nu=\nu_1+\nu_2+\nu_3\)
3. \(\dfrac{1}{\nu}=\dfrac{1}{\nu_1}+\dfrac{1}{\nu_2}+\dfrac{1}{\nu_3}\)
4. \(\dfrac{1}{\sqrt{\nu}}=\dfrac{1}{\sqrt{\nu_1}}+\dfrac{1}{\sqrt{\nu_2}}+\dfrac{1}{\sqrt{\nu_3}}\)

The equation of a simple harmonic wave is given by 

          $\mathrm{y}=3<mspace\; width="1em"></mspace>\sin \frac{\mathrm{\pi }}{2}(50\mathrm{t}-\mathrm{x})$

where x and y are in meters and t is in seconds. The ratio of maximum particle velocity to the wave velocity is

1. $2\mathrm{\pi }$

2. $\frac{3}{2}\mathrm{\pi }$

3. $3\mathrm{\pi }$

4. $\frac{2}{3}\mathrm{\pi }$

Two waves are represented by the equations ${\mathrm{y}}_1=\mathrm{a}<mspace\; width="1em"></mspace>\sin <mspace\; width="1em"></mspace>(\mathrm{\omega t}+\mathrm{kx}+0.57)\mathrm{m}$ and ${\mathrm{y}}_2=\mathrm{a}<mspace\; width="1em"></mspace>\cos <mspace\; width="1em"></mspace>(\mathrm{\omega t}+\mathrm{kx})\mathrm{m}$, where x is in metre and t in second. The phase difference between them is?

1. 1.25 rad

2. 1.57 rad

3. 0.57 rad

4. 1.0 rad

Sound waves travel at 350 m/s through a warm air and at 3500 m/s through brass. The wavelength of a 700 Hz acoustic wave as it enters brass from warm air :

1. increases by factor 20

2. increases by factor 10

3. decreases by factor 20

4. decreases by factor 10

Each of the two strings of length 51.6 cm and 49.1 cm are tensioned separately by 20N force. Mass per unit length of both the strings is same and equal to 1 $g{m}^{-1}$When both the strings vibrate simultaneously the number of beats is :

1. 5                                                   

2. 7

3. 8                                                   

4. 3

Two waves are represented by the equations $y_1=a\sin \omega t$ and $y_2=a\cos \omega t.$ The first wave 

(1) Leads the second by $\pi$

(2) Lags the second by $2\pi$

(3) Leads the second by $\frac{\pi }{2}$

(4) Lags the second by $\frac{\pi }{2}$

The distance between two consecutive crests in a wave train produced in a string is 5 cm. If 2 complete waves pass through any point per second, the velocity of the wave is :

1. 10 cm/sec

2. 2.5 cm/sec

3. 5 cm/sec

4. 15 cm/sec

The ratio of the speed of sound in nitrogen gas to that in helium gas, at 300 K is 

1.  $\sqrt{2/7}$

2.  $\sqrt{1/7}$

3.  $\sqrt{3}/5$

4.  $\sqrt{6}/5$

The speed of a wave in a certain medium is 960 m/s. If 3600 waves pass over a certain point of the medium in 1 minute, the wavelength is :

(1) 2 metres

(2) 4 metres

(3) 8 metres

(4) 16 metres