The wave equation is a partial differential equation which describes certain kinds of waves. It arises in various physical situations, such as vibrating strings, sound waves, and electromagnetic waves.

The wave equation in one dimension is $$ \frac{\partial^2 u}{\partial t^2}=c^2\frac{\partial^2 u}{\partial x^2}. $$ The general solution of the one-dimensional wave equation can be obtained by a change of coordinates: $(x,t)\longrightarrow(\xi,\eta)$ , where $\xi=x-ct$ and $\eta=x+ct$ . This gives $\frac{\partial^2 u}{\partial\xi\partial\eta}=0$ , which we can integrate to get d'Alembert's solution: $$ u(x,t)=F(x-ct)+G(x+ct) $$ where $F$ and $G$ are twice differentiable functions. $F$ and $G$ represent waves traveling in the positive and negative $x$ directions, respectively, with velocity $c$ . These functions can be obtained if appropriate initial conditions and boundary conditions are given. For example, if $u(x,0)=f(x)$ and $\frac{\partial u}{\partial t}(x,0)=g(x)$ are given, the solution is $$ u(x,t)=\frac{1}{2}[f(x-ct)+f(x+ct)]+\frac{1}{2c}\int_{x-ct}^{x+ct}g(s)\mathrm d s. $$

In general, the wave equation in $n$ dimensions is $$ \frac{\partial^2 u}{\partial t^2}=c^2\nabla^2 u. $$ where $u$ is a function of the location variables $x_1,x_2,\ldots,x_n$ , and time $t$ . Here, $\nabla^2$ is the Laplacian with respect to the location variables, which in Cartesian coordinates is given by $ \nabla^2=\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}+\cdots+\frac{\partial^2}{\partial x_n^2}$ .