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Wave Equation on Square Domain

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

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 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);
ylim([-1.1 1.1]);
axis equal
title("Geometry With Edge Labels Displayed")

Specify PDE coefficients.


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,"neumann","Edge",([1 3]),"g",0);

Create and view a finite element mesh for the problem.

ylim([-1.1 1.1]);
axis equal
xlabel x
ylabel y

Set the following initial conditions:

  • u(x,0)=arctan(cos(πx2)).

  • ut|t=0=3sin(πx)exp(sin(πy2)).

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

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);
431 successful steps
33 failed attempts
930 function evaluations
1 partial derivatives
107 LU decompositions
929 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.

umax = max(max(u));
umin = min(min(u));
for i = 1:n
    pdeplot(model,"XYData",u(:,i),"ZData",u(:,i), ...
    axis([-1 1 -1 1 umin umax]); 
    xlabel x
    ylabel y
    zlabel u
    M(i) = getframe;

To play the animation, use the movie(M) command.