a. Find a solution of the form: us(x,t) = X (x) sin that satisfies the partial differential equation (1) and the boundary conditions (2) and (3). (Clue: Don't worry about the initial conditions yet) b. Where is the rope stationary (i.e. us (x, t) = 0)? c. For what values of is your solution invalid?

Suppose you shake the end of a rope of dimensionless length 1 at a certain frequency w. The opposite end of the rope is fixed to a wall. We aim to find the special frequencies w at which certain points along the rope remain fixed in mid air. We negelect gravity and friction and model the waves on the rope using the 1D wave equation: Uy = Ugg, D 0. (3) We assume the rope has zero initial position and velocity wlzml) = 0, O