Fixed Rope and Wave Solution

Classical Mechanics Level 3

A rope of length 1 is fixed to a wall at $x=0$ and shaken at the other end so that

$x(1,t) = \sin \omega t.$

Which of the following is a possible displacement of the rope as a function of $x$ and $t$ consistent with these boundary conditions, assuming the waves of the rope propagate with velocity $v=1$ ?

\frac{\sin \omega x}{\sin \omega} \sin \omega t

\frac{\sin \omega x}{\cos \omega} \sin \omega t

\cos \omega x \sin \omega t

\frac{\sin \omega x}{\sin \omega} \cos \omega t