% Speed in degrees per hour for various Earth-Moon-Sun astronomical attributes, as given in Tides, Surges and Mean Sea-Level, D.T. Pugh.
clear EMS;
% T + s - h                         +15            w0: Nominal day, ignoring the variation followed via the Equation of Time.
EMS.T = +360/(1.0350)/24;          %+14.492054485  w1: is the advance of the moon's longitude, referenced to the Earth's zero longitude, one full rotation in 1.0350 mean solar days.
EMS.s = +360/(27.3217)/24;         % +0.5490141536 w2: Moon around the earth in 27.3217 mean solar days.
EMS.h = +360/(365.2422)/24;        % +0.0410686388 w3: Earth orbits the sun in a tropical year of 365.24219879 days, not the 365.2425 in 365 + y/4 - y/100 + y/400. Nor with - y/4000.
EMS.p = +360/(365.25* 8.85)/24;    % +0.0046404    w4: Precession of the moon's perigee, once in 8.85 Julian years: apsides.
EMS.N = -360/(365.25*18.61)/24;    % -0.00220676   w5: Precession of the plane of the moon's orbit, once in 18.61 Julian years: negative, so recession.
EMS.pp= +360/(365.25*20942)/24;    % +0.000001961  w6: Precession of the perihelion, once in 20942 Julian years.
% T + s = 15.041068639°/h is the rotation of the earth with respect to the fixed stars, as both are in the same sense.
%                   Reference                      Angular Speed           Degrees/hour  Period in Days.  Astronomical Values.
% Sidereal day      Distant star                   ws = w0 + w3 = w1 + w2  15.041                0.9973
% Mean solar day    Solar transit of meridian      w0 = w1 + w2 - w3       15                    1
% Mean lunar day    Lunar transit of meridian      w1                      14.4921               1.0350
% Month Draconic    Lunar ascending node           w2 + w5                   .5468              27.4320
% Month Sidereal    Distant star                   w2                        .5490              27.3217    27d07h43m11.6s  27.32166204
% Month Anomalistic Lunar Perigee (apsides)        w2 - w4                   .5444              27.5546
% Month Synodic     Lunar phase                    w2 - w3 = w0 - w1         .5079              29.5307    29d12h44m02.8s  29.53058796
% Year Tropical     Solar ascending node           w3                        .0410686          365.2422   365d05h48m45s   365.24218967  at 2000AD. 365.24219879 at 1900AD.
% Year Sidereal     Distant star                                             .0410670          365.2564   365d06h09m09s   365.256363051 at 2000AD.
% Year Anomalistic  Solar perigee (apsides)        w3 - w6                   .0410667          365.2596   365d06h13m52s   365.259635864 at 2000AD.
% Year nominal      Calendar                                                                   365 or 366
% Year Julian                                                                                  365.25
% Year Gregorian                                                                               365.2425
% Obtaining definite values is tricky: years of 365, 365.25, 365.2425 or what days? These parameters also change with time.

clear Tide;
%                                          w1 w2 w3 w4 w5 w6
Tide.Name{1} = 'M2';   Tide.Doodson{ 1} = [+2  0  0  0  0  0]; Tide.Title{ 1} = 'Principal lunar, semidiurnal';
Tide.Name{2} = 'S2';   Tide.Doodson{ 2} = [+2 +2 -2  0  0  0]; Tide.Title{ 2} = 'Principal solar, semidiurnal';
Tide.Name{3} = 'N2';   Tide.Doodson{ 3} = [+2 -1  0 +1  0  0]; Tide.Title{ 3} = 'Principal lunar elliptic, semidiurnal';
Tide.Name{4} = 'L2';   Tide.Doodson{ 4} = [+2 +1  0 -1  0  0]; Tide.Title{ 4} = 'Lunar semi-diurnal: with N2 for varying speed around the ellipse';
Tide.Name{5} = 'K2';   Tide.Doodson{ 5} = [+2 +2 -1  0  0  0]; Tide.Title{ 5} = 'Sun-Moon angle, semidiurnal';
Tide.Name{6} = 'K1';   Tide.Doodson{ 6} = [+1 +1  0  0  0  0]; Tide.Title{ 6} = 'Sun-Moon angle, diurnal';
Tide.Name{7} = 'O1';   Tide.Doodson{ 7} = [+1 -1  0  0  0  0]; Tide.Title{ 7} = 'Principal lunar declinational';
Tide.Name{8} = 'Sa';   Tide.Doodson{ 8} = [ 0  0 +1  0  0  0]; Tide.Title{ 8} = 'Solar, annual';
Tide.Name{9} = 'nu2';  Tide.Doodson{ 9} = [+2 -1 +2 -1  0  0]; Tide.Title{ 9} = 'Lunar evectional constituent: pear-shapedness due to the sun';
Tide.Name{10} = 'Mm';  Tide.Doodson{10} = [ 0 +1  0 -1  0  0]; Tide.Title{10} = 'Lunar evectional constituent: pear-shapedness due to the sun';
Tide.Name{11} = 'P1';  Tide.Doodson{11} = [+1 +1 -2  0  0  0]; Tide.Title{11} = 'Principal solar declination';
Tide.Constituents = 11;
% Because w0 + w3 = w1 + w2, the basis set {w0,...,w6} is not independent. Usage of w0 (or of EMS.T) can be eliminated.
% For further pleasure w2 - w6 correspond to other's usage of w1 - w5.

% Collect the basic angular speeds into an array as per A. T. Doodson's organisation. The classic Greek letter omega is represented as w.
clear w;
% w(0) = EMS.T + EMS.s - EMS.h;	% This should be w(0), but MATLAB doesn't allow this!
w(1) = EMS.T;	
w(2) = EMS.s;
w(3) = EMS.h;
w(4) = EMS.p;
w(5) = EMS.N;
w(6) = EMS.pp;

    % Prepare the basis frequencies, of sums and differences. Doodson's published coefficients typically have 5 added
% so that no negative signs will disrupt the layout: the scheme here does not have the offset.
disp('Name °/hour  Hours   Days');
for i = 1:Tide.Constituents
  Tide.Speed(i) = sum(Tide.Doodson{i}.*w);    % Sum terms such as DoodsonNumber(j)*w(j) for j = 1:6.
  disp([int2str(i),' ',Tide.Name{i},' ',num2str(Tide.Speed(i)),' ',num2str(360/Tide.Speed(i)),' ',num2str(15/Tide.Speed(i)),' ',Tide.Title{i}]);
end

clear Place;
% The amplitude H and phase for each constituent are determined from the tidal record by least-squares
% fitting to the observations of the amplitudes of the astronomical terms with expected frequencies and phases.
% The number of constituents needed for accurate prediction varies from place to place.
% In making up the tide tables for Long Island Sound, the National Oceanic and Atmospheric Administration
% uses 23 constituents. The eleven whose amplitude is greater than .1 foot are:
Place(1).Name = 'Bridgeport, CT';	% Counting time in hours from midnight starting Sunday 1 September 1991.
%                  M2      S2     N2    L2    K2     K1     O1    Sa   nu2    Mm     P1...
Place(1).A = [  3.185   0.538  0.696 0.277 0.144  0.295  0.212 0.192 0.159 0.108  0.102];	% Tidal heights (feet)
Place(1).P = [-127.24 -343.66 263.60 -4.72 -2.55 142.02 505.93 301.5 45.70 86.82 340.11];	% Phase (degrees). 
% The values for these coefficients are taken from http://www.math.sunysb.edu/~tony/tides/harmonic.html
% which originally came from a table published by the US. National Oceanic and Atmospheric Administration.

% Calculate a tidal height curve, in terms of hours since the start time.
PlaceCount = 1;
Colour=cellstr(char('b','r','g','c','m','y','k'));	% A collection.
clear y;
step = 0.125; LastHour = 720; % 8760 hours in a year.
n = LastHour/step + 1;
y(1:n,1:PlaceCount) = 0;
t = (0:step:LastHour);
for it = 1:PlaceCount
  i = 0;
  for h = 0:step:LastHour
    i = i + 1;
    y(i,it) = sum(Place(it).A.*cosd(Tide.Speed*h + Place(it).P)); %Sum terms A(j)*cos(speed(j)*h + p(j)) for j = 1:Tide.Constituents.
  end      % Should use cos(ix) = 2*cos([i - 1]*x)*cos(x) - cos([i - 2]*x), but, for clarity...
end

figure('Position', [100, 100, 850, 400]); clf; hold on;
title('Tidal Height, Bridgeport, CT');
xlabel('Days since 0h Sunday 1st September 1991'); ylabel('Feet');
xlim([0 50]);

for it = 1:PlaceCount
  plot(t,y(1:n,it),Colour{it});
end
%legend(Place(1:PlaceCount).Name,'Location','NorthWest');