# Wave Equation on a Circle

## Model Data

**Type:**
quickstart
classic pde

**Physics Modes:**
custom equation

**Keywords:**
wave equation
equation editing

This tutorial explains how to set up and solve a generalized wave equation model. The wave equation is a hyperbolic partial differential equation (PDE) of the form

$$ \frac{\partial^2 u}{\partial t^2} = c\Delta u + f $$

where *c* is a constant defining the
propagation speed of the waves, and *f* is a source term. This equation can not
be solved as it is due to the second order time derivative. However, the problem
can be transformed by reformulating the wave equation as two coupled parabolic
PDEs, that is

$$
\left\{\begin{array}{l}
\ \frac{\partial u}{\partial t} = v \\\

\frac{\partial v}{\partial t} = c\Delta u + f
\end{array}\right.
$$

This dual coupled problem can easily be implemented in FEATool with the custom
equation feature. This example solves the wave equation on a unit circle, with
zero boundary conditions, constant *c = 1*, source term *f = 0*, and initial
condition *u(t=0,x,y) = 1 - ( x ^{2} + y^{2} )*.

This model is available as an automated tutorial by selecting **Model
Examples and Tutorials…** > **Quickstart** > **Wave Equation on a
Circle** from the **File** menu. Or alternatively, follow the linked
step-by-step instructions.