wave equation

The wave equation is a partial differential equation whichdescribes certain kinds of waves. It arises in various physicalsituations, such as vibrating , waves, andelectromagnetic waves.

The wave equation in one 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 beobtained by a change of coordinates: $(x,t)⟶(\xi ,\eta )$,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 beobtained 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

$$u(x,t)=\frac{1}{2}[f(x-ct)+f(x+ct)]+\frac{1}{2c}{\int }_{x-ct}^{x+ct}g(s)ds.$$

In general, the wave equation in $n$ is

$$\frac{{\partial }^{2}u}{\partial t^2}=c^2{\nabla }^{2}u.$$

where $u$ is a function of the location variables$x_1,x_2,\mathrm{\dots },x_n$, and time $t$. Here, ${\nabla }^2$ is the Laplacianwith respect to the location variables, which in Cartesian coordinates is given by ${\nabla }^2=\frac{{\partial }^2}{\partial x_1^2}+\frac{{\partial }^2}{\partial x_2^2}+\mathrm{\cdots }+\frac{{\partial }^2}{\partial x_n^2}$.