Accelerating the pace of engineering and science

# Partial Differential Equation Toolbox

## Wave Equation on a Square Domain

This example shows how to solve the wave equation using the hyperbolic function in the Partial Differential Equation Toolbox™.

### Problem Definition

The standard second-order wave equation is

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

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

c = 1; a = 0; f = 0; d = 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 

### Boundary Conditions

Set zero Dirichlet boundary conditions on left (edge 4) and right (edge 2) and zero Neumann boundary conditions on the top (edge 1) and bottom (edge 3). The zero Neumann boundary condition is the default, so it is not necessary to include this setting.

applyBoundaryCondition(model,'Edge',[2,4],'u',0); applyBoundaryCondition(model,'Edge',([1 3]),'g',0); % Not strictly necessary 

### 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 = 'atan(cos(pi/2*x))'; ut0 = '3*sin(pi*x).*exp(sin(pi/2*y))'; 

### 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

u = hyperbolic(u0,ut0,tlist,model,c,a,f,d); 
428 successful steps 62 failed attempts 982 function evaluations 1 partial derivatives 142 LU decompositions 981 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 movie(M,1);