When a string is divided into three segments of lengths $l_1, l_2 and l_3,$ the fundamental frequencies of these three segments are $v_1, v_2 and v_3$ respectively. The original fundamental frequency (v) of the string is 

(1) $\sqrt{v}=\sqrt{v_1}+\sqrt{v_2}+\sqrt{v_3}$

(2) $v=v_1+v_2+v_3$

(3) $\frac{1}{v}=\frac{1}{v_1}+\frac{{\displaystyle 1}}{{\displaystyle v_2}}+\frac{{\displaystyle 1}}{{\displaystyle v_3}}$

(4) $\frac{1}{\sqrt{v}}=\frac{1}{\sqrt{v_1}}+\frac{{\displaystyle 1}}{{\displaystyle \sqrt{v_2}}}+\frac{{\displaystyle 1}}{{\displaystyle \sqrt{v_3}}}$

Two sources of sound placed close to each other, are emitting progressive waves given by

${\mathrm{y}}_1$=4 sin 600$\mathrm{\pi t}$ and ${\mathrm{y}}_2$=5 sin 608 $\mathrm{\pi t}$

An observer located near these two sources of sound will hear

(a)4 beats per second with intensity ratio 25:16 between waxing and waning

(b) 8 beats per second with intensity ratio 25:16 between waxing and waning

(c) 8 beats per second with intensity ratio 81:1 between waxing and waning

(d) 4 beats per second with intensity ratio 81:1 waxing and waning

The equation of a simple harmonic wave is 

given by 

          $\mathrm{y}=3 \sin \frac{\mathrm{\pi }}{2}(50\mathrm{t}-\mathrm{x})$

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) $\frac{2}{3}\mathrm{\pi }$

A train moving at a speed of 220 ${\mathrm{ms}}^{-1}$ towards a stationary object, emits a sound of frequency 1000 Hz. Some of the sound reaching the object gets reflected back to the train as an echo. The frequency of the echo as detected by the driver of the train is
(speed of sound in air is 330 ${\mathrm{ms}}^{-1}$)
1. 3500Hz
2. 4000Hz
3. 5000Hz
4. 3000Hz

Two waves are represented by the equations ${\mathrm{y}}_1=\mathrm{a} \sin (\mathrm{\omega t}+\mathrm{kx}+0.57)\mathrm{m}$ and ${\mathrm{y}}_2=\mathrm{a} \cos (\mathrm{\omega t}+\mathrm{kx})\mathrm{m}$, 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

Two particles are oscillating along two close parallel straight lines side by side, with the same frequency and amplitudes. They pass each other, moving in opposite directions when their displacement is half of the amplitude. The mean positions of the two particles lie on a straight line perpendicular to the paths of the two particles. The phase difference is :

(1)zero                                 

(2)$\frac{2\mathrm{\pi }}{3}$

(3)$\mathrm{\pi }$                                     

(4)$\frac{\mathrm{\pi }}{6}$

A transverse wave is represented by y=$A \sin \left(\omega t-kx\right).$ For what value of the wavelength is the wave velocity equal to the maximum particle velocity?

(1) $\mathrm{\pi } \mathrm{A}/2$                                           

(2) $\mathrm{\pi } \mathrm{A}$

(3) $2 \mathrm{\pi } \mathrm{A}$                                           

(4) A