1)

 The equation of simple harmonic progressive wave is given by y=a sin 2 $\pi$  (bt-cx). The maximum particle velocity will be twice the  wave velocity  if 


A) c= $\pi$ a

B) $c=\frac{1}{2\pi a}$

C) $c=\frac{1}{\pi a}$

D) c= 2 $\pi$ a

Answer:

Option C

Explanation:

 Given wave equation  , y= a sin 2 $\pi$  (bt-cx) 

 Compairing the above equation with the general equation of the progressive  wave which is given as 

$y=A_{0} \sin 2 \pi  (ft-\frac{x}{\lambda})$

 We get,  frequency , f=b, wavelength, $\lambda=\frac{1}{c}$ and amplitude of the wave , A0=a

 As, we know , that the maximum velocity of the particle,

   $V_{max}=A_{0}\omega= a\times 2\pi b$  ......(i)

 Wave velocity,  $V_{wave}=f\lambda$

 $\Rightarrow$    $ V_{wave}=\frac{b}{c}$ .......(ii)

 it is given that, 

$ V_{max}=2V_{wave}$

 So, by substituting the values from 

 Eqs. (i) and (ii)  in the above realation , we get

$a2\pi b=2\frac{b}{c}$

$\therefore$    $c=\frac{1}{a\pi}$