Fixed Rope and Wave Solution

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 \(v=1\)?

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