The wave equation
$$u_{xx}(x,t)=\frac {1}{c^2}u_{tt}(x,t) $$
requires two initial conditions because the equation is second order:
IC1: $$u(x,0)= f(x)$$
IC2: $$u_{t}(x,0)= g(x)$$
But when it is factored:
$$u_{tt} - c^2 u_{xx} = \bigg( \frac{\partial }{\partial t} - c \frac{\partial }{\partial x} \bigg) \bigg( \frac{\partial }{\partial t} + c \frac{\partial }{\partial x} \bigg) u(x,t) = 0$$
it only requires one initial condition when each of the factors is set equal to zero:
$$\bigg( \frac{\partial }{\partial t} - c \frac{\partial }{\partial x} \bigg) u(x,t)= 0 $$ $$\bigg( \frac{\partial }{\partial t} + c \frac{\partial }{\partial x} \bigg)u(x,t)=0 $$
IC1: $$u(x,0)= f(x)$$
because the 'factors' are first order.
So I understand mathematically why in the first case two ICs are needed whereas in the second case only one IC is needed. But if the factored and unfactored wave equations are equivalent--containing all of the same information-- , I don't understand intuitively or physically the difference in the number of required ICs.
My question is: Intuitively and physically, why does the wave equation need two ICs when the 'factored' wave equation needs only one?
See also:
Intuition into why the wave equation needs the second initial condition (e.g. velocity)
Intuitively, why are only two initial conditions needed for the wave equation? Why not 3 or 4?
https://math.stackexchange.com/q/2706776/147776
https://physics.stackexchange.com/a/403761/45664
EDIT 6/2/18 SEE @jcandy ANSWER BELOW FOR CORRECTIONS TO THIS QUESTION AND FOR THE ANSWER
The equations with the factors should have been written
$$\bigg( \frac{\partial }{\partial t} - c \frac{\partial }{\partial x} \bigg) u(x,t)= v $$ $$\bigg( \frac{\partial }{\partial t} + c \frac{\partial }{\partial x} \bigg)v(x,t)=0 $$
with an IC given for each equation.