Coupled oscillation problem, describing subsequent motion

[lark0901]

Homework Statement

A string with no initial displacement is set into motion by being struck over a length 2s about its center. This center section is given an initial velocity v_0. Describe the subsequent motion.

Homework Equations

Coupled oscillations...

The Attempt at a Solution

I've managed to solve to a particular point, but I can't go any further..
And I'm not sure of whether I am doing right, so ..I need help!

 

[member 553083]

The string will begin to vibrate as a result of the initial velocity given to it. The subsequent motion will be a coupled oscillation, with a node at the center and two antinodes at the ends of the string. The nodes and antinodes will move along the string with a velocity v, which is given by the equation v = v_0 / 2 The displacement of any point on the string at any time t is given by the equation x(t) = v_0 t sin (kx) where k = π/2s is the wave number. At any point in time, the velocity of the string at that point is given by v(t) = v_0 cos (kx)