When the vibrations of a string are damped, with a resistance which is proportionnal to the velocity. the one dimmensional wave equation becomes Pu t Ju t 01 where is the constant of proportionality and is the wave speed. +2k a. If the initial shape of the string sin (at any point ) and the constant of proportionality is half. Write down the initial boundary value problem that describes the damped vertical vibrations u(x, t) of a string of length which is fixed at both ends and initially at rest, and unit wave speed. (10 Marks) b. Determine the vibrations of the string at each point and at time t, by solving the initial problem obtained in part (a.). (15 Marks)