Question

A string is clamped on both ends. Which of the following wave equations is valid for a stationary wave set up on this string ? (Origin is at one end of string.)

• A. $y=Asinkxsint$
  
• B. $y=Acoskxsint$
  
• C. $y=Acoskxcost$
  
• D. None of the above

Answer 1

The correct option is B $y=Acoskxsint$

Wave equation of the standing wave is given by the superposition of the two waves travelling in opposite direction.

$y=A\sin (\omega t-kx)+A\sin (\omega t+kx)$

$\Rightarrow y=A\left[\sin \right(\omega t-kx)+\sin (\omega t+kx\left)\right]$

$\Rightarrow y=A\left[2\sin \right(\frac{\omega t-kx+\omega t+kx}{2})+\cos (\frac{\omega t+kx-\omega t-kx}{2}\left)\right]$

$\Rightarrow y=A\left[2\sin \right(\omega t)+\cos (-kx\left)\right]$

$\Rightarrow y=A\left[2\sin \right(\omega t)+\cos (kx\left)\right]$

Hence, option $B$ is correct.