JEE MAIN - Physics (2002 - No. 45)
A wave $$y=a$$ $$\sin \left( {\omega t - kx} \right)$$ on a string meets with another wave producing a node at $$x=0.$$ Then the equation of the unknown wave is
$$y = a\,\sin \,\left( {\omega t + kx} \right)$$
$$y = - a\,\sin \,\left( {\omega t + kx} \right)$$
$$y = a\,\sin \,\left( {\omega t - kx} \right)$$
$$y = - a\,\sin \,\left( {\omega t - kx} \right)$$
Explication
To form a node there should be superposition of this wave with the reflected wave. The reflected wave should travel in opposite direction with a phase change of $$\pi $$. The equation of the reflected wave will be
$$y = a\sin \left( {\omega t + kx + \pi } \right)$$
$$ \Rightarrow y = - a\sin \left( {\omega t + kx} \right)$$
$$y = a\sin \left( {\omega t + kx + \pi } \right)$$
$$ \Rightarrow y = - a\sin \left( {\omega t + kx} \right)$$