D'Alembert's Formula

$$\begin{array}{rl}u(t,x)& =\frac{1}{2}\left[u_0(x+ct)+u_0(x-ct)+\frac{1}{c}\{U_1(x+ct)-U_1(x-ct)\}\right]\\ & =\frac{1}{2}\left[u_0(x+ct)+u_0(x-ct)+\frac{1}{c}{\int }_{x-ct}^{x+ct}{u}_1(z)dz\right]\end{array}$$

• $u(x,t)$ is the displacement of the wave at position $x$ and time $t$.
• $u_0(x-ct)$ and $u_0(x+ct)$ are the initial displacement functions of the wave traveling to the left and right, respectively.
• $c$ is the wave speed.
• $u_1(z)$ is the initial velocity function of the wave.