📂Partial Differential Equations

Wave Equation

Definition¹

The following partial differential equation is called the wave equation.wave equation.

$$ u_{tt} - \Delta u =0 $$

This equation assumes the propagation speed of the wave as a constant $1$. If the propagation speed of the wave is denoted as $c$, then the wave equation becomes,

$$ u_{tt} - c^{2}\Delta u =0 $$

In the case of being nonhomogeneousnonhomogeneous,

$$ u_{tt} - \Delta u = f $$

• $U \subset \mathbb{R}^{n}$ is an open set
• $u : \overline{U}\times [0, \infty) \to \mathbb{R}$
• $t>0$
• $x = (x_{1}, x_{2}, \dots, x_{n}) \in U$
• $u=u(x, t)=u(x_{1}, \dots, x_{n}, t)$
• $\Delta$ is a Laplacian
• $f:U \times [0, \infty) \to \mathbb{R}$

Description

If the second order derivative with respect to time is replaced with terms related to position, it becomes the Helmholtz equation.