%
%  plot k(x)
%

a = 0.5;
%a = 1.2;

x = 0:pi/50:2*pi;
k = x + a*(1-cos(x));

figure(1);               clf
plot(x,k)
title('\bf one period of k(x)','fontweight','bold')
xlabel('\bf x');             ylabel('\bf k(x)')
axis([0 2 0 2]*pi);   axis('square')

disp('  pause:  hit any key to continue');  pause

%
%  level curves of:  k(x) - t = X_0 
%

x  = -pi:pi/50:pi;
t  = 0:pi/50:2*pi;

% make a 2D array of coordinates:  x & t
[xg, tg] = meshgrid(x,t);

% make 2D arrays: X(x,t) & u(x,t)
XX = xg + a*(1-cos(xg)) - tg;
uu = sin(XX);

% vector of values X0 = X(x0,0)
x0 = -3*pi:pi/6:3*pi;
X0 = x0 + a*(1-cos(x0));

% plot level curves of XX with values X0

figure(1);               clf
contour(x,t,XX,X0,'k');
title('level curves of:  X(x,t) = k(x) - t','fontweight','bold')
xlabel('\bf x');             ylabel('\bf t')
axis([-1 1 0 2]*pi);   axis('square')