This tutorial explains how to set up and solve a generalized wave equation model on a circle, like for example for a drum or membrane, and then tracks how the initial parabolic deformed state will affect the resulting shape. The wave equation is one of the classic hyperbolic partial differential equations (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 - ( x2 + y2 ).
This model is available as an automated tutorial by selecting Model Examples and Tutorials… > Quickstart > Wave Equation on a Circle from the File menu, and is also available as the MATLAB simulation m-script example ex_waveequation1. Or alternatively, follow the video tutorial or step-by-step instructions linked below.