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Notebook[{

Cell[CellGroupData[{
Cell["Linear Water Waves", "Title",
  TextAlignment->Center],

Cell[TextData[{
  "We solve for very small ampitude waves in a layer of \"water\" between \
z=-H and z=0.  All fields are assumed to be proportional to ",
  Cell[BoxData[
      \(TraditionalForm\`e\^\(i(kx - \[Omega]t)\)\)]],
  ".  The velocity potential is \[Phi] and the surface displacement is \
\[Eta].  We know that a separable solution for \[Phi] is \[Phi]=f(z)",
  Cell[BoxData[
      \(TraditionalForm\`e\^\(i(kx - \[Omega]t)\)\)]],
  "where f(z)=A",
  Cell[BoxData[
      \(TraditionalForm\`e\^kz\)]],
  "+B",
  Cell[BoxData[
      \(TraditionalForm\`e\^\(-kz\)\)]],
  ".  We assume \[Eta]=C",
  Cell[BoxData[
      \(TraditionalForm\`e\^\(i(kx - \[Omega]t)\)\)]],
  ".  A, B and C are complex  constants.  We find \[Omega], A and B.  See \
Kundu, page 190."
}], "Text"],

Cell[BoxData[
    \(\[Phi] := f*phase\)], "Input"],

Cell[BoxData[
    \(\[Eta] := C*phase\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(phase = Exp[I \((k*x - \[Omega]*t)\)]\)], "Input"],

Cell[BoxData[
    \(E\^\(I\ \((k\ x - t\ \[Omega])\)\)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(f = A*Exp[k*z] + B*Exp[\(-k\)*z]\)], "Input"],

Cell[BoxData[
    \(B\ E\^\(\(-k\)\ z\) + A\ E\^\(k\ z\)\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "The bottom boundary condition is w=0, or ",
  Cell[BoxData[
      \(TraditionalForm\`\[PartialD]\[Phi]\/\[PartialD]z\)]],
  "=0, at z=-H."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(bottombc = \((D[\[Phi], z] == 0)\) /. z -> \(-H\)\)], "Input"],

Cell[BoxData[
    \(E\^\(I\ \((k\ x - t\ \[Omega])\)\)\ 
        \((A\ E\^\(\(-H\)\ k\)\ k - B\ E\^\(H\ k\)\ k)\) == 0\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "The top dynamic boundary condition is ",
  Cell[BoxData[
      \(TraditionalForm\`\[PartialD]\[Phi]\/\[PartialD]t\)]],
  "=-g\[Eta], at z=0."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(dynbc = \((D[\[Phi], t] == \(-g\)*\[Eta])\) /. z -> 0\)], "Input"],

Cell[BoxData[
    \(\(-I\)\ \((A + B)\)\ E\^\(I\ \((k\ x - t\ \[Omega])\)\)\ \[Omega] == 
      \(-C\)\ E\^\(I\ \((k\ x - t\ \[Omega])\)\)\ g\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "The top kinematic bottom boundary  ",
  Cell[BoxData[
      \(TraditionalForm\`\[PartialD]\[Eta]\/\[PartialD]t\)]],
  "=",
  Cell[BoxData[
      \(TraditionalForm\`\[PartialD]\[Phi]\/\[PartialD]z\)]],
  ", at z=0."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(kinbc = \((D[\[Phi], z] == D[\[Eta], t])\) /. z -> 0\)], "Input"],

Cell[BoxData[
    \(E\^\(I\ \((k\ x - t\ \[Omega])\)\)\ \((A\ k - B\ k)\) == 
      \(-I\)\ C\ E\^\(I\ \((k\ x - t\ \[Omega])\)\)\ \[Omega]\)], "Output"]
}, Open  ]],

Cell[TextData[
"The three boundary condition equationshave a trivial solution A=B=C=0. We \
seek the nontrivial solution that occurs when the eiqenvalue \[Omega] assumes \
certain values.  We first eliminate two of the unknowns A and B and reduce \
the three equations to one.  "], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(oneeq = Simplify[Eliminate[{bottombc, dynbc, kinbc}, {A, B}]]\)], 
  "Input"],

Cell[BoxData[
    \(\(-C\)\ E\^\(\(-H\)\ k + 2\ I\ \((k\ x - t\ \[Omega])\)\)\ 
        \((\(-\((\(-1\) + E\^\(2\ H\ k\))\)\)\ g\ k + 
            \((1 + E\^\(2\ H\ k\))\)\ \[Omega]\^2)\) == 0\)], "Output"]
}, Open  ]],

Cell[TextData[
"We have no use for the C=0 solution in the above equation, so we go on  to \
solve for \[Omega]:"], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(dis1 = Solve[oneeq, \[Omega]]\)], "Input"],

Cell[BoxData[
    \({{\[Omega] \[Rule] 
          \(-\(\(\@\(\(-1\) + E\^\(2\ H\ k\)\)\ \@g\ 
                  \@k\)\/\@\(1 + E\^\(2\ H\ k\)\)\)\)}, {
        \[Omega] \[Rule] 
          \(\@\(\(-1\) + E\^\(2\ H\ k\)\)\ \@g\ 
              \@k\)\/\@\(1 + E\^\(2\ H\ k\)\)}}\)], "Output"]
}, Open  ]],

Cell["\<\
I will next try converting the above rules to a more familiar \
dispersion relation:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(dis2 = \[Omega]^2 == \((\[Omega]^2 /. dis1[\([1, 1]\)])\)\)], "Input"],

Cell[BoxData[
    \(\[Omega]\^2 == 
      \(\((\(-1\) + E\^\(2\ H\ k\))\)\ g\ k\)\/\(1 + E\^\(2\ H\ k\)\)\)], 
  "Output"]
}, Open  ]],

Cell["\<\
That still does not look too familiar.  Let's beat on it some more:\
\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(dis3 = FullSimplify[ExpToTrig[dis2]]\)], "Input"],

Cell[BoxData[
    \(\[Omega]\^2 == g\ k\ Tanh[H\ k]\)], "Output"]
}, Open  ]],

Cell["\<\
What do the waves look like?  Let's eliminate C between two \
equations and find A and B in terms of C:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(AB = Flatten[Solve[{bottombc, kinbc}, {A, B}]]\)], "Input"],

Cell[BoxData[
    \({A \[Rule] 
        \(-\(\(I\ C\ E\^\(2\ H\ k\)\ 
                \[Omega]\)\/\(\((\(-1\) + E\^\(2\ H\ k\))\)\ k\)\)\), 
      B \[Rule] 
        \(-\(\(I\ C\ \[Omega]\)\/\(\((\(-1\) + E\^\(2\ H\ k\))\)\ k\)\)\)}
      \)], "Output"]
}, Open  ]],

Cell["Now find f(z):", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(f = FullSimplify[ExpToTrig[Factor[f /. AB]]]\)], "Input"],

Cell[BoxData[
    \(\(-\(\(I\ C\ \[Omega]\ Cosh[k\ \((H + z)\)]\ Csch[H\ k]\)\/k\)\)\)], 
  "Output"]
}, Open  ]],

Cell[TextData[
"Recall \[Phi] was defined with a delayed assignment, \[Phi]:=f*phase, so we \
can use \[Phi] and the above f will be used:"], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(w = D[\[Phi], z]\)], "Input"],

Cell[BoxData[
    \(\(-I\)\ C\ E\^\(I\ \((k\ x - t\ \[Omega])\)\)\ \[Omega]\ Csch[H\ k]\ 
      Sinh[k\ \((H + z)\)]\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(TextForm
    \`We\ have\ use\ for \[PartialD]\[Eta]\/\[PartialD]t\ \(also : \)\)], 
  "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(detadt = D[\[Eta], t]\)], "Input"],

Cell[BoxData[
    \(\(-I\)\ C\ E\^\(I\ \((k\ x - t\ \[Omega])\)\)\ \[Omega]\)], "Output"]
}, Open  ]],

Cell[BoxData[
    FormBox[
      RowBox[{"We", " ", "normalize", " ", "w", " ", "by", 
        RowBox[{
          FormBox[\(\[PartialD]\[Eta]\/\[PartialD]t\),
            "TextForm"], ":"}]}], TextForm]], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(wnorm = w/detadt\)], "Input"],

Cell[BoxData[
    \(Csch[H\ k]\ Sinh[k\ \((H + z)\)]\)], "Output"]
}, Open  ]],

Cell["\<\
We also normalize the height, expressing height by a dimensionless \
Z:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(wnondim = wnorm /. z -> Z*H\)], "Input"],

Cell[BoxData[
    \(Csch[H\ k]\ Sinh[k\ \((H + H\ Z)\)]\)], "Output"]
}, Open  ]],

Cell["\<\
We next make a plot of w(z) for a moderate wavelength k=7/H and a \
long wavelength k=1/H.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(ParametricPlot[{{wnondim /. k -> 7/H, Z}, {wnondim /. k -> 1/H, Z}}, 
      \n\t{Z, \(-1\), 0}, 
      PlotStyle -> {{RGBColor[1. , 0. , 0. ]}, {RGBColor[0. , 0. , 1. ]}}]; 
    \)\)], "Input"],

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We next make a plot of u(z) for a moderate wavelength k=7/H and a \
long wavelength k=1/H .But first we make
a dimensionless profile:\
\>", "Text"],

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      "Possible spelling error: new symbol name \"\!\(unondim\)\" is similar \
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