Vibrations and waves

y(x,t) = Acos(Kx-ωt) ---> Vy(x,t) = (∂y(x,t))/(∂t) = ωAsin(Kx-ωt) ---> (∂^2 y(x,t))/∂x^2 = (∂^2 y(x,t))/(v^2 ∂t^2 )

Can someone help me interpret this derivation and understand what this wave equation really means? This class has a co requisite of Calc 3, however we are still covering limits and continuity of multi variable function in calc.

Thanks