D'Alembert's formula

In mathematics, and specifically partial differential equations (PDEs), d´Alembert's formula is the general solution to the one-dimensional wave equation:
 * $$u_{tt}-c^2u_{xx}=0,\, u(x,0)=g(x),\, u_t(x,0)=h(x),$$

for $$-\infty < x<\infty,\,\, t>0$$

It is named after the mathematician Jean le Rond d'Alembert, who derived it in 1747 as a solution to the problem of a vibrating string.

Details
The characteristics of the PDE are $$x \pm ct = \mathrm{const}$$ (where $$\pm$$ sign states the two solutions to quadratic equation), so we can use the change of variables $$\mu = x + ct$$ (for the positive solution) and $$\eta = x-ct$$ (for the negative solution) to transform the PDE to $$u_{\mu\eta} = 0$$. The general solution of this PDE is $$u(\mu,\eta) = F(\mu) + G(\eta)$$ where $$F$$ and $$G$$ are $$C^1$$ functions. Back in $$x, t$$ coordinates,


 * $$u(x,t) = F(x+ct) + G(x-ct)$$
 * $$u$$ is $$C^2$$ if $$F$$ and $$G$$ are $$C^2$$.

This solution $$u$$ can be interpreted as two waves with constant velocity $$c$$ moving in opposite directions along the x-axis.

Now consider this solution with the Cauchy data $$u(x,0)=g(x), u_t(x,0)=h(x)$$.

Using $$u(x,0) = g(x)$$ we get $$F(x) + G(x) = g(x)$$.

Using $$u_t(x,0) = h(x)$$ we get $$cF'(x)-cG'(x) = h(x)$$.

We can integrate the last equation to get $$cF(x)-cG(x)=\int_{-\infty}^x h(\xi) \, d\xi + c_1.$$

Now we can solve this system of equations to get $$F(x) = \frac{-1}{2c}\left(-cg(x)-\left(\int_{-\infty}^x h(\xi) \, d\xi +c_1 \right)\right)$$ $$G(x) = \frac{-1}{2c}\left(-cg(x)+\left(\int_{-\infty}^x h(\xi) d\xi +c_1 \right)\right).$$

Now, using $$u(x,t) = F(x+ct)+G(x-ct)$$

d'Alembert's formula becomes: $$u(x,t) = \frac{1}{2}\left[g(x-ct) + g(x+ct)\right] + \frac{1}{2c} \int_{x-ct}^{x+ct} h(\xi) \, d\xi.$$

Generalization for inhomogeneous canonical hyperbolic differential equations
The general form of an inhomogeneous canonical hyperbolic type differential equation takes the form of: $$u_{tt} - c^2 u_{xx} = f(x,t),\, u(x,0)=g(x),\, u_t(x,0)=h(x),$$ for $$-\infty < x < \infty, \,\, t > 0, f \in C^2(\R^2,\R) $$.

All second order differential equations with constant coefficients can be transformed into their respective canonic forms. This equation is one of these three cases: Elliptic partial differential equation, Parabolic partial differential equation and Hyperbolic partial differential equation.

The only difference between a homogeneous and an inhomogeneous (partial) differential equation is that in the homogeneous form we only allow 0 to stand on the right side ($$f(x,t) = 0$$), while the inhomogeneous one is much more general, as in $$f(x,t)$$ could be any function as long as it's continuous and can be continuously differentiated twice.

The solution of the above equation is given by the formula: $$u(x,t) = \frac{1}{2}\bigl( g(x+ct) + g(x-ct)\bigr) + \frac{1}{2c} \int_{x-ct}^{x+ct} h(s)\, ds + \frac{1}{2c} \int_0^t \int_{x-c(t-\tau)}^{x+c(t-\tau)} f(s,\tau) \, ds \, d\tau .$$

If $$g(x) = 0 $$, the first part disappears, if $$h(x) = 0 $$, the second part disappears, and if $$f(x) = 0 $$, the third part disappears from the solution, since integrating the 0-function between any two bounds always results in 0.