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## Wave Equation on Square Domain

This example shows how to solve the wave equation using the `solvepde` function.

### Problem Definition

The standard second-order wave equation is

` `

To express this in toolbox form, note that the `solvepde` function solves problems of the form

` `

So the standard wave equation has coefficients , , , and .

```c = 1; a = 0; f = 0; m = 1; ```

### Geometry

Solve the problem on a square domain. The `squareg` function describes this geometry. Create a `model` object and include the geometry. Plot the geometry and view the edge labels.

```numberOfPDE = 1; model = createpde(numberOfPDE); geometryFromEdges(model,@squareg); pdegplot(model,'EdgeLabels','on'); ylim([-1.1 1.1]); axis equal title 'Geometry With Edge Labels Displayed'; xlabel x ylabel y ``` ### Specify PDE Coefficients

```specifyCoefficients(model,'m',m,'d',0,'c',c,'a',a,'f',f); ```

### Boundary Conditions

Set zero Dirichlet boundary conditions on the left (edge 4) and right (edge 2) and zero Neumann boundary conditions on the top (edge 1) and bottom (edge 3).

```applyBoundaryCondition(model,'dirichlet','Edge',[2,4],'u',0); applyBoundaryCondition(model,'neumann','Edge',([1 3]),'g',0); ```

### Generate Mesh

Create and view a finite element mesh for the problem.

```generateMesh(model); figure pdemesh(model); ylim([-1.1 1.1]); axis equal xlabel x ylabel y ``` ### Create Initial Conditions

The initial conditions:

• .

• .

This choice avoids putting energy into the higher vibration modes and permits a reasonable time step size.

```u0 = @(location) atan(cos(pi/2*location.x)); ut0 = @(location) 3*sin(pi*location.x).*exp(sin(pi/2*location.y)); setInitialConditions(model,u0,ut0); ```

### Define Solution Times

Find the solution at 31 equally-spaced points in time from 0 to 5.

```n = 31; tlist = linspace(0,5,n); ```

### Calculate the Solution

Set the `SolverOptions.ReportStatistics` of `model` to `'on'`.

```model.SolverOptions.ReportStatistics ='on'; result = solvepde(model,tlist); u = result.NodalSolution; ```
```456 successful steps 37 failed attempts 988 function evaluations 1 partial derivatives 112 LU decompositions 987 solutions of linear systems ```

### Animate the Solution

Plot the solution for all times. Keep a fixed vertical scale by first calculating the maximum and minimum values of `u` over all times, and scale all plots to use those -axis limits.

```figure umax = max(max(u)); umin = min(min(u)); for i = 1:n pdeplot(model,'XYData',u(:,i),'ZData',u(:,i),'ZStyle','continuous',... 'Mesh','off','XYGrid','on','ColorBar','off'); axis([-1 1 -1 1 umin umax]); caxis([umin umax]); xlabel x ylabel y zlabel u M(i) = getframe; end ``` To play the animation, use the `movie(M)` command.

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