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CODE
These matlab programs generate two different wave simulations. The first one is of a linear equation and the second is of a non-linear equation called the Sine-Gordon equation. At the end of each code is a 3D graph. To view the animated graph simply copy each M file into your matlab directory and run either, runprog or runprog2.

Linear equation 
% Matlab code


% finitediff.m

function U=finitediff(f,g,a,b,c,n,m)

%FINITEDIFF approximates solution to the Wave equation,
% where f and g are string args. and f=u(x,0) and g=ut(x,0), 
% a is length of string, b is end time, c is the constant
% in the Wave equation, n is the number of steps in x direction
% and m is the number of steps in the time direction.  Code from
% Numerical Methods, Mathews-Fink, 1999, pg.524.

h=a/(n-1);
k=b/(m-1);
r=c*k/h;
r2=r^2;
r22=r^2/2;
s1=1-r^2;
s2=2-2*r^2;
U=zeros(n,m);
for i=2:n-1
     U(i,1)=feval(f,h*(i-1));
             U(i,2)=s1*feval(f,h*(i-1))+k*feval(g,h*(i-1))+r22*(feval(f,h*i)+feval(f,h*(i-2)));
end

for j=3:m
	    for i=2:(n-1)
           U(i,j)=s2*U(i,j-1)+r2*(U(i-1,j-1)+U(i+1,j-1))-U(i,j-2);
    	end
end

U=U';



% pro1.m
function y=pro1(x)
y=sin(pi*x)+sin(2*pi*x);



% pro2.m
function yy=pro2(x)
yy=0;



% runprog
clear
n=10;
timend=.5; length=1;
c=2;
U=finitediff('pro1','pro2',length,timend,c,n,n);
x=linspace(0,length,n);
t=linspace(0,timend,n);

figure(1)
surf(x,t,U), xlabel('x-axis'), ylabel('time')
zlabel('displacement'), title('Wave Equation')
shading interp,

M=moviein(n);

for j=1:n
    	figure(2)
	    plot(x,U(j,:)), axis([0 length -length*2 length*2])
	    M(:,j) = getframe;
end


movie(M,4,3)


3D Graph


Non-Linear equation 
% sinegordan.m
function U=sinegordan(f,g,a,b,n,m)

%SINEGORDON approximates solution to the Sine-Gordon equation,
%	where f and g are string args. and f=u(x,0) and g=ut(x,0),
%	a is x-max(x>0), b is end time, n is the number of steps in
%	x directionand m is the number of steps in the time
%	direction.


h=a/(n-1);       %Delta x
k=b/(m-1);       %Delta t
r=k/h;
r2=r^2;
r22=r^2/2;
s1=1-r^2;
s2=2-2*r^2;
U=zeros(n,m);

for i=2:n-1,
   U(i,1)=feval(f,h*(i-1));
   U(i,2)=s1*feval(f,h*(i-1))+k*feval(g,h*(i-1))+r22*...
      (feval(f,h*i)+feval(f,h*(i-2))-sin(feval(f,h*(i-1))));
end




% runprog2.m
clear 
n=300; 
timend=5; length=5; 
c=2; 
U=sinegordan('pro1','pro2',length,timend,n,n); 
x=linspace(0,length,n); 
t=linspace(0,timend,n); 

figure(1) 
surf(x,t,U),axis tight, xlabel('x-axis'), ylabel('time') 
zlabel('displacement'), title('Sine-Gordon Equation') 
shading interp, 

M = moviein(n); 

for j=1:2:n 
      figure(2) 
      plot(x,U(j,:)), axis([0 length -10 10]) 
      M(:,j) = getframe; 
end 


movie(M)