Marées, par PYL pyl@ccr.jussieu.fr  (www.lmm.jussieu.fr/~lagree) janvier 2003

constantes harmoniques pour Brest, Coefficient, Saint Malo

Brest = {Z0 -> 445 , ASA -> 58, GSA -> 230, 
          AQ1 -> 20, GQ1 -> 294, A01 -> 67, G01 -> 342, 
          AK1 -> 65, GK1 -> 89, 
AN2 -> 418, 
          GN2 -> 123, AM2 -> 2160, 
          GM2 -> 142, 
AS2 -> 755, 
          GS2 -> 182, AMN4 -> 19, 
          GMN4 -> 117, 
AM4 -> 54, 
          GM4 -> 164, AMS4 -> 34, GMS4 -> 239};

coef = {Z0 -> 0, ASA -> 0, GSA -> 0, AQ1 -> 0, 
          GQ1 -> 0, A01 -> 0, G01 -> 0, AK1 -> 0, 
          GK1 -> 0, 
AN2 -> 130, GN2 -> 123, 
          AM2 -> 672, 
          GM2 -> 142, 
AS2 -> 235, 
          GS2 -> 182, AMN4 -> 0, 
          GMN4 -> 0, 
AM4 -> 0, GM4 -> 0, 
          AMS4 -> 0, GMS4 -> 0};

SM = {Z0 -> 671, ASA -> 54, GSA -> 224, 
          AQ1 -> 24, GQ1 -> 330, A01 -> 88, G01 -> 354, 
          AK1 -> 90, 
          GK1 -> 126, 
AN2 -> 720, 
          GN2 -> 190, AM2 -> 3680, 
          GM2 -> 208, 
AS2 -> 1432, 
          GS2 -> 258, AMN4 -> 101, 
          GMN4 -> 315, 
AM4 -> 273, 
          GM4 -> 337, AMS4 -> 206, GMS4 -> 35};

nombre de jours écoulés depuis le 1/1/1980

T[J_, M_, A_, heurehiv_] := 
      IntegerPart[
          30.6001 
(1 +  M  +  
                12 IntegerPart[
                    1/(1 + M) +  .7])]  + 
 
        IntegerPart[
          365.25 (A  -  
                IntegerPart[1/(1 + M) +  .7])]  + 
 
        J  +  heurehiv/24  - 723258

pour le javascript

CForm[T[JJ, MM, AAAA, 0] /. IntegerPart -> Mathfloor]

T[1, 1, 1980, 0]

8399.709112/Pi*180/36525

13.176396475158947`

13.1764

positions astronomiques
13.17639673: TMGPDM 
13.17639653: Bouiges p 46    ->
13.176396475 sérane p255

s[t_] := (78.16  + 13.17639673 t)

    h[t_] := (279.82  +  0.98564734 t) 

    p[t_] := (349.50  +  .11140408 t) 

    Np[t_] := (208.1  +  0.05295392 t) 

    p1[t_] := (282.6  +  0.000047069 t)

CosDeg[x_] := Cos[x*N[Pi/180]]

CosDeg[90]

phase[tc_] := {15 heure, s[tc], h[tc], p[tc], Np[tc], p1[tc], 
        90}

phas = N[phase[T[1, 1, 2003, heure]]]

hauteur d'eau

\(\(\(hauteur = 
        Z0/100 + 
\((
ASA\ \ CosDeg[\(-GSA\)\ \ \  + \ 
                      phas . {0, \ 0, \ 1, \ 0, \ 0, \ 0, \ 
                          0}] + 
AK1\ CosDeg[\ \(-\ 
                        GK1\) + \ \ phas . {1, \ 0, \ 1, \ 0, \ 0, \ 0, \ 
                          1}]\  + 
A01\ CosDeg[\(-\ 
                        G01\)\  + \ 
                      phas . {1, \(-2\), \ 1, \ 0, \ 0, \ 
                          0, \(-1\)}]\  + 
AQ1\ CosDeg[\(-GQ1\)\  + \ phas . {1, \(-3\), \ 1, \ 1, \ 0, \ 
                          0, \(-1\)}]\  + 
\((AK1/\(-3\))\)\ CosDeg[\(-GK1\)\  + \ 
                      phas . {1, \ 0, \(-1\), \ 0, \ 0, \ 0, \ 
                          1}]\  + 
\((A01/
                      5.3)\)\ CosDeg[\(-G01\)\  + \ 
                      phas . {1, \(-2\), \ 1, \ 0, \(-1\), \ 
                          0, \(-1\)}]\  + 
\((AK1/\ 
                      7.4)\)\ \ CosDeg[\(-\ GK1\)\  + \ 
                      phas . {1, \ 0, \ 1, \ 0, \ 1, \ 0, \ 
                          1}]\  + 
 (*"Semi-diurne"\ *) 
\ 
                AM2\ \ CosDeg[\(-GM2\) + 
                      phas . {2, \(-2\), \ 2, \ 0\ , 0, \ 0, \ 
                          0}]\  + 
AN2\ \ CosDeg[\(-GN2\)\  + phas . {2, \(-3\), \ 2, \ 1, \ 0, \ 0, \ 
                          0}]\  + 
AS2\ \ CosDeg[\(-GS2\)\  + \ phas . {2, \ 0, \ 0, \ 0, \ 0, \ 0, \ 0}]\  + \ 
\((AN2/
                      7.6)\)\ \ CosDeg[\(-GN2\)\  + \ 
                      phas . {2, \(-4\), \ 2, \ 2, \ 0, \ 0, \ 
                          0}] + 
\ \((AN2/
                      6.3)\)\ \ CosDeg[\(-\ GN2\)\  + \ 
                      phas . {2, \(-4\), \ 4, \ 0, \ 0, \ 0, \ 
                          0}] + 
\ \((AN2/5.3)\)\ \ CosDeg[\(-GN2\)\  + \ 
                      phas . {2, \(-3\), \ 4, \(-1\), \ 0, \ 0, \ 
                          0}]\  + 
\ \((AM2/\(-35\))\)\ \ CosDeg[\(-GM2\)\  + \ phas . {2, \(-1\), \ 2, \(-1\), \ 0, \ 0, \ 
                          0}]\  + 
\((AS2/
                      3.7)\)\ CosDeg[\(-GS2\)\  + \ 
                      phas . {2, \ 0, \ 2, \ 0, \ 0, \ 0, \ 0}] + \ 
\((\ 
                    AS2/17)\)\ CosDeg[\(-GS2\)\  + \ 283 + 
                      phas . {2\ , 0, \(-1\), \ 0, \ 0, \ 0, \ 
                          0}] + 
\ \((AM2/\(-27\))\)\ \ CosDeg[\(-GM2\)\  + 
                      phas . {2, \(-2\), \ 2, \ 0, \(-1\), \ 0, \ 
                          0}]\  + 
\ \((AS2\ /12)\)\ \ CosDeg[\(-GS2\) + \ 
                      phas . {2\ , 0, \ 2, \ 0, \ 1, \ 0, \ 
                          0}]\  + 
 (*\ "Quart-diurne"\ *) \ 

                AMN4\ \ CosDeg[\(-\ GMN4\)\  + \ 
                      phas . {4, \(-5\), \ 4, \ 1, \ 0, \ 0, \ 
                          0}]\  + 
AM4\ \ CosDeg[\(-GM4\)\  + \ phas . {4, \(-4\), \ 4, \ 0, \ 0, \ 0, \ 0}]\  + \ \ 

                AMS4\ \ CosDeg[\(-GMS4\)\  + \ 
                      phas . {4, \(-2\), \ 2, \ 0, \ 0, \ 0, \ 
                          0}]
)\)/
            1000;\)\(
\)
    \)\)

Quelle est la période?

360/24

phas . {2, -2,  0,  0 , 0,  0,  0}

degparheure = 
      D[Expand[N[phas . {2, -2,  0,  0 , 0,  0,  0}]], 
        heure]

360/13.17639673

2 *360/degparheure

au bout de 24h54mn42s la phase est la m^me, mais en fait ce n'est pas juste!!!!

360/28.901966939166666 /2

4  180/28.901966939166666

Solve[(Jl)/(Jl - 1) == %/24, Jl]

2 15 /(27.3216615 /26.3216615 )

2 15 /((27.3216615 /(27.3216615  -  
                  1.0) ))

Jsideral terre= 24h 54mn 42s

Jsideralterre = 24*364.2422/365.24252

Jsideral terre= 23h 56mn 04s

Jlunaire =  
      24*1/(365.2422/364.24252 -  
              1/(360/(13.17639673)))

Jlunaire= 24h 50mn 27s

2 360/Jlunaire

24*1/(1 -  1/(360/(13.17639673)))

hauteurs en fonction du nombre d'heures écoulées

Show[Plot[hauteur /. SM, {heure, 0, 24*14}, 
          PlotPoints -> 2000, DisplayFunction -> Identity],
Plot[6 +  
                AM2  CosDeg[-153.65125 + 2 360/Jlunaire*heure] /1000 /. 
              SM,{heure, 0, 24*14},PlotStyle->RGBColor[0,0,1],DisplayFunction -> Identity],
DisplayFunction -> $DisplayFunction]

la phase initiale

(Simplify[-GM2 + 
                phas . {2, -2,  2,  0 , 0,  0,  0}  /. SM] /. 
          heure -> 0) + 360*568

Show[Plot[hauteur /. SM, {heure, -0, 24*2 }, 
          DisplayFunction -> Identity],  
Plot[6 +  
                AM2  CosDeg[-153.6512507800071    +  
                        2 360/Jlunaire*heure] /1000 /. SM,{heure, 0, 24*2},PlotStyle->RGBColor[0,0,1],DisplayFunction -> Identity], 
DisplayFunction -> $DisplayFunction,PlotRange -> {{0, 49}, {2, 12}},GridLines -> {{0, 6, 12, 18, 24, 30, 36, 42, 28}, {2, 3, 
              4, 5, 6, 7, 8, 9, 10, 11, 12}}]

étude du décallage entre le calcul avec tous les modes en noir, uniquement la lune en vert et un seul mode en bleu

glissement

nj = 100;
Show[Plot[hauteur /. SM, {heure, 24*nj, 24*(nj + 5)}, 
          DisplayFunction -> Identity],  
Plot[6 +  
                AM2  CosDeg[-153.6512507800071    +  
                        degparheure*heure] /1000 /. SM,{heure, 24*nj, 24*(nj + 5)},PlotStyle->RGBColor[0,0,1],DisplayFunction -> Identity], 

Plot[6 +  
                AM2  CosDeg[-153.6512507800071    + 
                        28.984303290232987*heure] /1000 /. SM,{heure, 24*nj, 24*(nj + 5)},PlotStyle->RGBColor[0,1,0],DisplayFunction -> Identity], 


DisplayFunction -> $DisplayFunction ]