Wave Equation on Square Domain: PDE Modeler App

This example shows how to solve a wave equation for transverse vibrations of a membrane on a square. The membrane is fixed at the left and right sides, and is free at the upper and lower sides. This example uses the PDE Modeler app. For a programmatic workflow, see Wave Equation on Square Domain.

A wave equation is a hyperbolic PDE:

$$\frac{{\partial }^{2}u}{\partial t^2}-\Delta u=0$$

To solve this problem in the PDE Modeler app, follow these steps:

1. Open the PDE Modeler app by using the pdeModeler command.

2. Display grid lines by selecting Options > Grid.

3. Align new shapes to the grid lines by selecting Options > Snap.

4. Draw a square with the corners at (-1,-1), (-1,1), (1,1), and (1,-1). To do this, first click the button. Then click one of the corners using the right mouse button and drag to draw a square. The right mouse button constrains the shape you draw to be a square rather than a rectangle.

   You also can use the pderect function:

   pderect([-1 1 -1 1])

5. Check that the application mode is set to Generic Scalar.

6. Specify the boundary conditions. To do this, switch to boundary mode by clicking the button or selecting Boundary > Boundary Mode. Select the left and right boundaries. Then select Boundary > Specify Boundary Conditions and specify the Dirichlet boundary condition u = 0. This boundary condition is the default one (h = 1, r = 0), so you do not need to change it.

   For the bottom and top boundaries, set the Neumann boundary condition ∂u/∂n = 0. To do this, set g = 0, q = 0.

7. Specify the coefficients by selecting PDE > PDE Specification or clicking the button on the toolbar. Select the Hyperbolic type of PDE, and specify c = 1, a = 0, f = 0, and d = 1.

8. Initialize the mesh by selecting Mesh > Initialize Mesh. Refine the mesh by selecting Mesh > Refine Mesh.

9. Set the solution times. To do this, select Solve > Parameters. Create linearly spaced time vector from 0 to 5 seconds by setting the solution time to linspace(0,5,31).

10. In the same dialog box, specify initial conditions for the wave equation. For a well-behaved solution, the initial values must match the boundary conditions. If the initial time is t = 0, then the following initial values that satisfy the boundary conditions: atan(cos(pi/2*x)) for u(0) and 3*sin(pi*x).*exp(sin(pi/2*y)) for ∂u/∂t,

    The inverse tangent function and exponential function introduce more modes into the solution.

    

11. Solve the PDE by selecting Solve > Solve PDE or clicking the button on the toolbar. The app solves the heat equation at times from 0 to 5 seconds and displays the result at the end of the time span.

12. Visualize the solution as a 3-D static and animated plots. To do this:

    1. Select Plot > Parameters.

    2. In the resulting dialog box, select the Color and Height (3-D plot) options.

    3. To visualize the dynamic behavior of the wave, select Animation in the same dialog box. If the animation progress is too slow, select the Plot in x-y grid option. An x-y grid can speed up the animation process significantly.