I want to verify my understanding of D'Alembert equation. Please let me know my statements are correct or not.

D'Alembert equation:

$\displaystyle u(x,t)=\frac{1}{2}[f^*(x-ct)+F^*(x+ct)]+\int_{x-ct}^{x+ct} g^*(s)ds$

$\displaystyle where\;\; g(x) \frac{\partial u}{\partial t}(x,0)$

1) $\displaystyle f^*(x-ct)\;\;,\;\;F^*(x+ct)$ represent wave propagate right and left on the x-axis with velocity of c.

2) g(x) is velocity along y-axis, not in direction of x-axis. g(x) DO NOT affect the propagation velocity c. g(x) only affect the shape of the wave while propagating along x-axis.