The initial conditions.

  Since the wave equation is second order in time, two conditions are needed in the time direction. They are both initial conditions; the first is a given initial deflection f₁(x) of the string,

 

u(x,0) = f₁(x),

while the second is a given vertical velocity f₂(x) of the string,  

$$\d ut(x,0) = f_2(x).$$ (13)