propagating wave created by point source adding random phase considering retarded potential

How can demo a propagating wave created by a point source considering retarded potential and phase change by time?

I successful create the wave, with the retarded potential, but if i change the phase, the whole wave on the plane will change. How can I make the phase change only propagate from the point source and doesnt change the wave that had already propagate out?

This is so far I go

v=1;
lambda = 1;
k = 2*pi/lambda;
w = 2*pi*v/lambda;
x=linspace(-10,10,500);
y=x;
[X,Y]=meshgrid(x,y);
t_lens=100;   
t = linspace(0,2*pi,t_lens);   % time
an=0+(2*pi)*rand(1,t_lens);   % create random phase
E20=0;
for a=1:t_lens
    theta=an(1,a);
    r=sqrt(X.^2+Y.^2);
    u0=1./(1+exp(-2.*(10e4).*(t(a)-r./v)));  % step function u(t-r/v)
    
    E=u0.*exp(1i*(k.*r-w.*t(a)+theta));  % wave propagate from origin
    rE=real(E);
    pcolor(X,Y,rE); axis equal;shading flat;
    caxis([-5,5]);
    drawnow;
end

Thanks

[chia ching lin]

Can any one give me some advice?