Animations

% animate(seconds) re-runs the script every frame with t = time.
animate(6)

x = linspace(0, 4*pi, 500);
w = 2*pi/3;                       % angular frequency
right = sin(x - w*t);             % moves right
left  = 0.5*sin(2*x + w*t);       % moves left, half the amplitude

plot(x, right, 'LineWidth', 1.5), hold on
plot(x, left, '--')
plot(x, right + left, 'LineWidth', 2.5), hold off
ylim([-2 2])
title(sprintf('Superposition of two waves   t = %.2f s', t))
legend('right-moving', 'left-moving', 'sum')

animate(12)

[X, Y] = meshgrid(linspace(-10, 10, 56));
R = sqrt(X.^2 + Y.^2) + eps;
Z = sin(R - 2*t) ./ R;

surf(X, Y, Z)
shading flat
colormap plasma
view(-37.5 + 30*t, 28)
zlim([-0.3 1])
title('sin(r − 2t) / r')

animate(10)
x = linspace(-pi, pi, 800);
n = 1 + floor(1.5 * t);            % more terms as time goes on

y = zeros(size(x));
for k = 1:n
  y = y + 4/pi * sin((2*k - 1) * x) / (2*k - 1);
end

plot(x, sign(x), 'k:'), hold on
plot(x, y, 'LineWidth', 2), hold off
ylim([-1.6 1.6])
title(sprintf('Square wave from %d sine terms', n))
subtitle('Note the Gibbs overshoot near the jumps')

animate(8)
s = linspace(0, 2*pi, 600);
delta = 2*pi * t / 8;
plot(sin(3*s + delta), sin(2*s), 'LineWidth', 2)
axis equal
xlim([-1.2 1.2]), ylim([-1.2 1.2])
title(sprintf('x = sin(3s + δ), y = sin(2s)    δ = %.2f', delta))

animate(10)
s = linspace(0, 8*pi, 900);
plot3(cos(s), sin(s), s/(8*pi), 'LineWidth', 2)
hold on
scatter3(cos(2*t), sin(2*t), 2*t/(8*pi) * 4, 60, 'r', 'filled')
hold off
view(36*t, 20)
axis equal
title('A bead on a helix')

animate(9)
x = linspace(-3*pi, 3*pi, 600);
n = 1 + floor(t);                    % number of terms grows with time
y = zeros(size(x));
for k = 0:n-1
  y = y + (-1)^k * x.^(2*k + 1) / factorial(2*k + 1);
end
plot(x, sin(x), 'k:', 'LineWidth', 1.5), hold on
plot(x, y, 'LineWidth', 2), hold off
ylim([-2.5 2.5])
title(sprintf('sin x and its Taylor polynomial of degree %d', 2*n - 1))

animate(12)
f = @(tt, u) [10*(u(2) - u(1)); u(1)*(28 - u(3)) - u(2); u(1)*u(2) - 8/3*u(3)];
[tt, u] = ode45(f, [0 25], [1; 1; 1]);
n = max(2, round(numel(tt) * t / 12));
plot3(u(1:n, 1), u(1:n, 2), u(1:n, 3), 'LineWidth', 0.8)
hold on
scatter3(u(n, 1), u(n, 2), u(n, 3), 50, 'r', 'filled')
hold off
xlim([-20 20]), ylim([-28 28]), zlim([0 50])
view(-30 + 10*t, 15)
title('Lorenz attractor')