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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:


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 rectangle 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 rectangle with an arrow 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 partial derivative 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.

    Dialog box for specifying the solver parameters for hyperbolic equations

  11. Solve the PDE by selecting Solve > Solve PDE or clicking the partial derivative with the green triangle 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.

    3-D solution plot in color for time = 5