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This simulation was made using 40 pendula, each having frequency between 25 and 35 cycles per minute. The top right graph shows the phase of each pendula and the top-down shows the phase difference with the slowest pendulum.

The evolving patterns displayed by the pendula are really mesmerizing, but they don't hide any deep physical mechanism. They occur just by "coincidence".

Yes, "coincidence" because, although they are not physically meaningful, the patterns are related to the least common multiple of the frequencies as the graphics on the right point!

MATLAB Program:

function h = pendulum_waves(t_max, f_min, f_max, n_pendula)
% h = pendulum_waves(t_max, f_min, f_max, f_min)
% Animates n_pendula pendulawith frequence between f_min and f_max.
%
% h         : a structure containing the graphical objects handles.
% t_max     : simulation duration (s).
% f_min     : minimum frequency (Hz).
% f_max     : maximum frequency (Hz).
% n_pendula : number of pendula.
%
% Typical values:
% t_max = 4.5*60;
% f_min = 25/60;
% f_max = 35/60;
% n_pendula = 40;

% Initializang video object:
FILENAME = 'pendulum_wave';
frame_rate = 60;
vid = VideoWriter(FILENAME,'Uncompressed AVI');
vid.FrameRate = frame_rate;
open(vid);

% Constants:
W = 2*pi*linspace(f_min,f_max,n_pendula)'/frame_rate;
L = 9.81./W.^2;
THETA_0 = asin(-max(L)/4./L);                                              % arbitrary initial angle

% Initial conditions:
PHASE = zeros(n_pendula,1);
X = L.*sin(THETA_0);
Y = -L.*cos(THETA_0);

% Initializing graphical objects:
h.f = figure('Color', 'k', ...
'Position', [0 0 1280 720], ...
'InvertHardcopy', 'off', ...
'Visible', 'off'); % turn on to watch in "real time"

h.a(1) = axes(h.f, 'Position', [0.1 0.0 0.5 1.0], ...
'Color', 'k', ...
'XColor', 'k', ...
'YColor', 'k', ...
'DataAspectRatioMode', 'manual', ...
'DataAspectRatio', [1 1 1], ...
'XLim', 1.1*[X(1), -X(1)], ...
'YLim', [1.1*Y(1), 0.9*Y(end)]);
h.a(2) = polaraxes(h.f, 'OuterPosition', [0.5 0.5 0.5 0.5], ...
'Color', 'k', ...
'ThetaColor', [0.6111 0.7205 0.7361], ...
'RColor', [0.6111 0.7205 0.7361], ...
'ThetaTickLabel', {}, ...
'RTick', 0:0.2:1, ...
'RTickLabel', {}, ...
'ThetaZeroLocation', 'top', ...
'RLim', [0 1]);
h.a(3) = polaraxes(h.f, 'OuterPosition', [0.5 0.0 0.5 0.5], ...
'Color', 'k', ...
'ThetaColor', [0.6111 0.7205 0.7361], ...
'RColor', [0.6111 0.7205 0.7361], ...
'ThetaTickLabel', {}, ...
'RTick', flipud(L), ...
'RTickLabel', {}, ...
'ThetaZeroLocation', 'top', ...
'RLim', [0 max(L)]);

h.p = gobjects(length(L),4);
for i = 1:length(L)
h.p(i,1) = plot(h.a(1), [0; X(i)], [0; Y(i)], '-',...
'Color', [0.4028 0.4340 0.5278]);
h.p(i,2) = plot(h.a(1), X(i), Y(i), 'o', ...
'MarkerSize', 12, ...
'Color', hsv2rgb([i/(1.5*length(L)) 1 1]), ...
'MarkerFaceColor', hsv2rgb([i/(1.5*length(L)) 1 1]));
h.p(i,3) = plot(h.a(2), PHASE(i), 1, 'o', ...
'Color', hsv2rgb([i/(1.5*length(L)) 1 1]), ...
'MarkerFaceColor', hsv2rgb([i/(1.5*length(L)) 1 1]));
h.p(i,4) = plot(h.a(3), mod(PHASE(i),PHASE(1)), L(i), 'o', ...
'Color', hsv2rgb([i/(1.5*length(L)) 1 1]), ...
'MarkerFaceColor', hsv2rgb([i/(1.5*length(L)) 1 1]));
end

% Initializang video object:
FILENAME = 'pendulum_wave';
frame_rate = 60;
vid = VideoWriter(FILENAME,'Uncompressed AVI');
vid.FrameRate = frame_rate;
open(vid);

% Simulation and video capture:
for t = 0:t_max*frame_rate
PHASE = W*t;
THETA = THETA_0.*cos(PHASE);
X =  L.*sin(THETA);
Y = -L.*cos(THETA);
for i = 1:n_pendula
set(h.p(i,1),'XData', [0; X(i)], ...
'YData', [0; Y(i)]);
set(h.p(i,2),'XData', X(i), ...
'YData', Y(i));
'RData', 1);
'RData', L(i));
end
drawnow;
FRAME = getframe(h.f);
writeVideo(vid,FRAME);
end
close(vid);
close all;
end