Note: This page has been translated by MathWorks. Click here to see

To view all translated materials including this page, select Country from the country navigator on the bottom of this page.

To view all translated materials including this page, select Country from the country navigator on the bottom of this page.

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

function.

The standard second-order wave equation is

$$\frac{{\partial}^{2}u}{\partial {t}^{2}}-\nabla \cdot \nabla u=0.$$

To express this in toolbox form, note that the `solvepde`

function solves problems of the form

$$m\frac{{\partial}^{2}u}{\partial {t}^{2}}-\nabla \cdot (c\nabla u)+au=f.$$

So the standard wave equation has coefficients $$m=1$$, $$c=1$$, $$a=0$$, and $$f=0$$.

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

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);

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);

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

Set the following initial conditions:

$$u(x,0)=\mathrm{arctan}\left(\mathrm{cos}\left(\frac{\pi x}{2}\right)\right)$$.

$${\frac{\partial u}{\partial t}|}_{t=0}=3\mathrm{sin}(\pi x)\mathrm{exp}\left(\mathrm{sin}\left(\frac{\pi y}{2}\right)\right)$$.

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);

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

Specify the solution times as 31 equally-spaced points in time from 0 to 5.

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

Set the `SolverOptions.ReportStatistics`

of `model`

to `'on'`

.

```
model.SolverOptions.ReportStatistics ='on';
result = solvepde(model,tlist);
```

457 successful steps 38 failed attempts 992 function evaluations 1 partial derivatives 112 LU decompositions 991 solutions of linear systems

u = result.NodalSolution;

Create an animation to visualize the solution for all time steps. 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 $$z$$-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.