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T. Gladd", "Subtitle", CellChangeTimes->{{3.691434597547227*^9, 3.691434612619321*^9}}], Cell[TextData[{ StyleBox["Initialization: ", FontWeight->"Bold"], "Be sure the file ", StyleBox["NTGUtilityFunctions.m", FontSlant->"Italic"], " is in the same directory as that from which this notebook was loaded. Then \ execute the cell immediately below by mousing left on the cell bar to the \ right of that cell and then typing \[OpenCurlyDoubleQuote]shift\ \[CloseCurlyDoubleQuote] + \[OpenCurlyDoubleQuote]enter\ \[CloseCurlyDoubleQuote]. Respond \[OpenCurlyDoubleQuote]Yes\ \[CloseCurlyDoubleQuote] in response to the query to evaluate initialization \ cells." }], "Text", CellChangeTimes->{{3.69023148371922*^9, 3.6902314907825966`*^9}, { 3.690231568228395*^9, 3.690231573307065*^9}, {3.690231629378787*^9, 3.6902317034648433`*^9}, {3.6902318285880823`*^9, 3.6902318790622425`*^9}, {3.690561803160874*^9, 3.690561837304328*^9}, 3.6914346883149395`*^9, {3.691434721455573*^9, 3.691434844326234*^9}, { 3.6914349066404066`*^9, 3.6914349142663536`*^9}, {3.691434956254755*^9, 3.6914349884966564`*^9}, {3.7330653691000385`*^9, 3.733065381155084*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"SetDirectory", "[", RowBox[{"NotebookDirectory", "[", "]"}], "]"}], ";"}], " ", RowBox[{"(*", " ", RowBox[{ "set", " ", "directory", " ", "where", " ", "source", " ", "files", " ", "are", " ", "located"}], " ", "*)"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Get", "[", "\"\\"", "]"}], ";"}], " ", RowBox[{"(*", " ", RowBox[{"Load", " ", "utilities", " ", "package"}], " ", "*)"}]}]}], "Input", InitializationCell->True, CellChangeTimes->{{3.690231304262511*^9, 3.6902313300154333`*^9}, { 3.6902313821303463`*^9, 3.6902313878808374`*^9}, {3.690231423337756*^9, 3.6902314287291145`*^9}, {3.6902315145507984`*^9, 3.6902315306304874`*^9}, {3.6905614204501295`*^9, 3.6905616087617702`*^9}, {3.690561669084352*^9, 3.6905616745378094`*^9}, { 3.690561776716816*^9, 3.6905617961927767`*^9}, 3.6911973050047274`*^9, { 3.733065402347458*^9, 3.733065404123144*^9}, {3.733066601122711*^9, 3.733066602035139*^9}}], Cell[CellGroupData[{ Cell["1 Introduction", "Section", CellChangeTimes->{{3.738762259560293*^9, 3.7387622623402653`*^9}, { 3.7389664639410887`*^9, 3.7389664705488825`*^9}, {3.7400683294273586`*^9, 3.740068329906178*^9}}], Cell["\<\ The distinctive wave/wake pattern behind a moving ship is a familiar one and, \ remarkably, the pattern is similar whether the moving object is a a ship at \ sea or a duck in a pond. 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Below, I use Mathematica to work through some \ classical analysis related to these waves. Since the calculations are \ complicated and involve several approximations, the detailed step-by-step \ sequence of results that Mathematica makes possible may be of interest to \ students and nonspecialist researchers. \n\nI follow a procedure similar to \ Kelvin. Kelvin calculated the linear response of water to a point disturbance \ in deep water (imagine the ripples generated from a pebble dropped in water). \ He modeled a ship as a moving point disturbance. The cumulative response \ observed behind a moving ship is the integral of the disturbances along the \ path of the ship. In general, waves will propagate radially outward from each \ point of disturbance with different wave lengths and phase velocities. A key \ point in the analysis is that an observer on the ship sees a wave pattern \ that is stationary in time.\n\nThe seminal document is ", StyleBox["On the waves produced by a single impulse in water of any depth, \ or in a dispersive medium", FontSlant->"Italic"], ". Proceedings of the Royal Society of London, Ser. A. Vol. 42, 1887, pp. \ 80-85. Lord Kelvin.\n \nThe history of the study of ship waves is \ complicated. I won\[CloseCurlyQuote]t attempt a review. I will generally \ follow the treatment given in Chapter 8 of ", StyleBox["Water Waves: The Mathematical Theory with Applications", FontSlant->"Italic"], ", J. J. Stoker, 1957. Lord Kelvin is credited with inventing the method of \ stationary phase while working on this problem. In fact, in the course of the \ analysis, the method of stationary phase is invoked three times! \n \nThe \ importance of this problem for applied mathematics is underscored by the fact \ that the cover picture of The ", StyleBox["Princeton Companion to Applied Mathematics", FontSlant->"Italic"], " is an image of a Kelvin ship wave. \n\nIn order to get quickly to the \ heart of the subject, I will start with a particular approximate formula for \ the linear response of water to a point disturbance in deep water. Assuming \ that a ship can be modeled as a moving version of this point disturbance \ enables one to deduce the characteristic wave features behind the ship. \ Insight is also gained into the detailed physics associated with the \ generation of the wave pattern.\n\nIn a second section, I will derive the \ particular formula for the linear response from a more fundamental result. \ This is a longish calculation involving a double integral that is both \ analytically and numerically difficult. Along the way I will provide various \ visualizations that clarify aspects of the calculation.\n\nOverall this is a \ calculation that, if you know where you are going, can be carried out with \ pen and paper in 20 or 30 pages of calculation. However, it is easy to go \ astray. As I do with the notebooks on this website I try to demonstrate that \ all of the substantive calculations can be carried out with Mathematica. 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Cell[TextData[{ "The ship moves along the X-axis in the negative direction at constant speed \ V. The analysis is performed in the reference frame of the moving ship. The \ ship\[CloseCurlyQuote]s position is parameterized by the positive quantity s. \ The point S denotes the position of the ship {", Cell[BoxData[ FormBox[ SubscriptBox["X", "s"], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ SubscriptBox["Y", "s"], TraditionalForm]]], "} = {", Cell[BoxData[ FormBox[ SubscriptBox["X", "s"], TraditionalForm]]], "(s) = {V s, 0}. The point O denotes the position of the ship when s = 0. \ Thus, the parameter s corresponds to the positive interval of time required \ for the ship to move from S to O, while moving at constant speed V. The \ point P(X, Y) is a point being observed from O, when s = 0. The quantity ", Cell[BoxData[ FormBox[ OverscriptBox["r", RowBox[{"\[Rule]", " "}]], TraditionalForm]]], "is a vector from S to P and \[Alpha] is the angle between the velocity \ vector of the ship and ", Cell[BoxData[ FormBox[ OverscriptBox["r", "\[Rule]"], TraditionalForm]]], ". \n\nThe quantity of interest is the cumulative disturbance at P(X, Y)" }], "Text", CellChangeTimes->{{3.7395595328916893`*^9, 3.7395595832255487`*^9}, { 3.7404299402390223`*^9, 3.740429944389101*^9}, {3.7404300017888317`*^9, 3.7404300138452168`*^9}, {3.740492921198847*^9, 3.740492932890585*^9}, 3.7404929830754437`*^9}], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[Delta]z", "(", "P", ")"}], " ", "=", " ", RowBox[{ SubsuperscriptBox["\[Integral]", "0", SubscriptBox["s", "max"]], RowBox[{"ds", " ", RowBox[{"\[Delta]z", "(", RowBox[{ RowBox[{ OverscriptBox["r", RowBox[{"\[Rule]", " "}]], "(", "s", ")"}], ",", " ", "s"}], ")"}]}]}]}], TraditionalForm]], "DisplayFormulaNumbered", CellChangeTimes->{{3.7395596003490906`*^9, 3.7395597051349173`*^9}, { 3.7395616675091066`*^9, 3.7395616751926765`*^9}, {3.7395617089926205`*^9, 3.7395617166516747`*^9}, {3.740062571326746*^9, 3.740062571486442*^9}}], Cell[TextData[{ "where \[Delta]z is some infinitesimal change in the level of water and ", Cell[BoxData[ FormBox[ SubscriptBox["s", "max"], TraditionalForm]]], " is some arbitrary large interval during which the ship has been moving and \ generating waves. The basic physical model for ship-generated waves is that \ of the waves generated from a point source. Another way of imagining the \ model is \[LongDash] an experimenter sits on a platform above still deep \ water that is flowing past with speed V. The experimenter continuously drops \ pebbles into the water and observes the stationary wave pattern that forms in \ the downstream flow of water." }], "Text", CellChangeTimes->{{3.7395597320984116`*^9, 3.739559805352681*^9}, { 3.7395600463237524`*^9, 3.73956008886506*^9}, {3.7398277074948416`*^9, 3.739827782569106*^9}, {3.74006254808667*^9, 3.7400626205095778`*^9}, { 3.7400626854868402`*^9, 3.740062689869407*^9}, {3.740062869903385*^9, 3.740063080924843*^9}, {3.7400631208578777`*^9, 3.7400631964734592`*^9}, { 3.740063243585683*^9, 3.7400633148747034`*^9}, {3.7400633636089864`*^9, 3.7400633713114653`*^9}, {3.7400634083098145`*^9, 3.740063773633354*^9}, 3.7400639185978336`*^9, {3.7403177610818977`*^9, 3.74031792921122*^9}, { 3.7404300548267317`*^9, 3.7404300854968615`*^9}, {3.740493027468704*^9, 3.7404930286744776`*^9}}], Cell["\<\ The general linear response of water near the surface to an instantaneous \ point source is\ \>", "Text", CellChangeTimes->{{3.740063841120469*^9, 3.7400638623707027`*^9}, { 3.740063961672617*^9, 3.7400639824881377`*^9}, {3.7404301076825705`*^9, 3.740430109512695*^9}}], Cell[BoxData[ RowBox[{" ", FormBox[ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[Delta]z", RowBox[{"inital", " ", "value", " ", "problem"}]], "(", RowBox[{"r", ",", "t"}], ")"}], " ", "=", " ", RowBox[{ RowBox[{"-", " ", FractionBox["1", RowBox[{"2", " ", "\[Pi]", " ", "\[Rho]", " ", SqrtBox["g"]}]]}], RowBox[{ SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[{"dk", " ", SuperscriptBox["k", RowBox[{"3", "/", "2"}]], " ", RowBox[{ SubscriptBox["J", "0"], "(", RowBox[{"k", " ", "r"}], ")"}], RowBox[{"sin", "(", RowBox[{ SubscriptBox["\[Omega]", "k"], "t"}], ")"}]}]}]}]}], "\[IndentingNewLine]", RowBox[{ SubscriptBox["\[Omega]", "k"], " ", "=", " ", SqrtBox[ RowBox[{"k", " ", "g"}]]}]}], TraditionalForm], " "}]], "DisplayFormulaNumbered", CellChangeTimes->{{3.7395605440968432`*^9, 3.739560659231826*^9}, { 3.739561653056039*^9, 3.739561657868281*^9}, {3.739828046294095*^9, 3.7398280670057497`*^9}, {3.740158285420536*^9, 3.7401583262624545`*^9}}], Cell[TextData[{ "where g is the gravitational acceleration, \[Rho] is the density of water, \ and ", Cell[BoxData[ FormBox[ SubscriptBox["J", "0"], TraditionalForm]]], "(x) is a Bessel function of order 0. The wave frequency ", Cell[BoxData[ FormBox[ SubscriptBox["\[Omega]", "k"], TraditionalForm]]], " is appropriate for deep water (the derivation of the dispersion equation \ for water waves is presented in another notebook). The integrand of this \ expression becomes highly oscillatory as k increases (product of two \ different oscillating functions each depending on k) and \ it\[CloseCurlyQuote]s valuation is challenging even using computational \ methods. A classic reference for equation (2) is ", StyleBox["Hydrodynamics", FontSlant->"Italic"], ", Lamb, 1932 (still available as a Dover book).\n\nIn Section 2 below, the \ following approximate form for equation (2) will be calculated. 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The calculation not only produces an approximate value \ for \[Delta]z(r, t), but also provides insight about the wave pattern. I \ start by defining some functions for future convenience\ \>", "Text", CellChangeTimes->{{3.73963519316181*^9, 3.739635218546471*^9}, { 3.7398282365322495`*^9, 3.7398284207893496`*^9}, {3.7400644087508783`*^9, 3.740064502295653*^9}, {3.7400645506585326`*^9, 3.7400645569597187`*^9}, { 3.7403231495148487`*^9, 3.7403232321987333`*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"Clear", "[", RowBox[{"\[CapitalPsi]", ",", " ", "\[CapitalPhi]"}], "]"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[CapitalPsi]", "[", RowBox[{"r_", ",", " ", "s_"}], "]"}], " ", ":=", " ", FractionBox[ SuperscriptBox["s", "3"], SuperscriptBox["r", "4"]]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"\[CapitalPhi]", "[", RowBox[{"r_", ",", " ", "s_"}], "]"}], " ", ":=", " ", FractionBox[ RowBox[{"g", " ", SuperscriptBox["s", "2"]}], RowBox[{"4", " ", "r"}]]}], ";"}]}], "Input", CellChangeTimes->{{3.739635090357792*^9, 3.7396351855853252`*^9}}], Cell[TextData[{ "The constant phase condition, ", Cell[BoxData[ FormBox[ RowBox[{"d\[CapitalPhi]", "/", "dr"}], TraditionalForm]]], " = 0 results in a differential equation" }], "Text", CellChangeTimes->{{3.7396354413376465`*^9, 3.7396354758607693`*^9}, { 3.7398284366837587`*^9, 3.7398284968243084`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"w2", "[", "1", "]"}], " ", "=", " ", RowBox[{ RowBox[{"Sol", "[", RowBox[{ RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{"\[CapitalPhi]", "[", RowBox[{ RowBox[{"r", "[", "s", "]"}], ",", " ", "s"}], "]"}], ",", " ", "s"}], "]"}], " ", "\[Equal]", " ", "0"}], ",", " ", RowBox[{ SuperscriptBox["r", "\[Prime]", MultilineFunction->None], "[", "s", "]"}]}], "]"}], " ", "//", " ", "RE"}]}]], "Input", CellChangeTimes->{{3.7396352283564887`*^9, 3.7396352977152967`*^9}, { 3.7396353800551033`*^9, 3.7396353816954565`*^9}, {3.740064562889596*^9, 3.740064563041421*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["r", "\[Prime]", MultilineFunction->None], "[", "s", "]"}], "\[Equal]", 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Cell[BoxData[ RowBox[{ RowBox[{"w2", "[", "2", "]"}], " ", "=", " ", RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"w2", "[", "1", "]"}], ",", " ", RowBox[{ RowBox[{"r", "[", "s0", "]"}], " ", "\[Equal]", " ", "r0"}]}], "}"}], ",", " ", RowBox[{"r", "[", "s", "]"}], ",", " ", "s"}], "]"}], "\[LeftDoubleBracket]", RowBox[{"1", ",", "1"}], "\[RightDoubleBracket]"}], " "}]], "Input", CellChangeTimes->{{3.7396353689014535`*^9, 3.7396354240448337`*^9}, { 3.7400645954197164`*^9, 3.740064600152437*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"r", "[", "s", "]"}], "\[Rule]", FractionBox[ RowBox[{"r0", " ", SuperscriptBox["s", "2"]}], SuperscriptBox["s0", "2"]]}]], "Output", CellChangeTimes->{{3.7396354179836483`*^9, 3.739635424669578*^9}, { 3.73982867121766*^9, 3.739828689505571*^9}, 3.739897400563429*^9, 3.7399775512614202`*^9, 3.740064601188931*^9, 3.7400684017024336`*^9, 3.740323271438752*^9, 3.74048502083083*^9}] }, Open ]], Cell[TextData[{ "At this point, there is no 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The point P has been chosen so that OP is orthogonal to PS. The \ gray circle is constructed and it\[CloseCurlyQuote]s intersections with \ x-axis is calculated. There are either two points of intersection ", Cell[BoxData[ FormBox[ SubscriptBox["C", "1"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ SubscriptBox["C", "2"], TraditionalForm]]], ", or none at all. The points ", Cell[BoxData[ FormBox[ SubscriptBox["Q", "1"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ SubscriptBox["Q", "2"], TraditionalForm]]], " are constructed so that ", Cell[BoxData[ FormBox[ SubscriptBox["C", "1"], TraditionalForm]]], "P \[UpTee] ", Cell[BoxData[ FormBox[ SubscriptBox["Q", "1"], TraditionalForm]]], "P and ", Cell[BoxData[ FormBox[ SubscriptBox["C", "2"], TraditionalForm]]], "P \[UpTee] ", Cell[BoxData[ FormBox[ SubscriptBox["Q", "2"], TraditionalForm]]], "P. These points are specific \[OpenCurlyDoubleQuote]influence points\ \[CloseCurlyDoubleQuote] that give rise to different types of waves." }], "Text", CellChangeTimes->{{3.7396425532193403`*^9, 3.7396428131256876`*^9}, { 3.739642860442646*^9, 3.73964288554611*^9}, {3.7396429288326006`*^9, 3.7396429616375213`*^9}, {3.739831742568749*^9, 3.739831804436878*^9}, { 3.739831846684826*^9, 3.739831847739786*^9}}], Cell["\<\ The next step is the calculate the phase fronts in the frame of Figure 1: to \ be specific, X = X(\[Alpha]), Y = Y(\[Alpha]). 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For \[Alpha] > ", Cell[BoxData[ FormBox[ SubscriptBox["\[Alpha]", "cusp"], TraditionalForm]]], " Y is decreasing back to 0. " }], "Text", CellChangeTimes->{{3.739833244662139*^9, 3.7398333749596105`*^9}, { 3.7398336791681833`*^9, 3.739833690821333*^9}}], Cell["\<\ Consider the point on Figure 4 where the blue curve intersects the green \ curve. It can be determined numerically.\ \>", "Text", CellChangeTimes->{{3.7398323037297893`*^9, 3.739832337467084*^9}, { 3.73983340469392*^9, 3.739833448460862*^9}, 3.7398337888716516`*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"w2", "[", "18", "]"}], " ", "=", " ", RowBox[{"With", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"a2", " ", "=", " ", "2"}], ",", " ", RowBox[{"a3", " ", "=", " ", "3"}]}], "}"}], ",", RowBox[{"FindRoot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"\[CapitalPhi]Curve", "[", RowBox[{"\[Alpha]1", ",", " ", "a2"}], "]"}], "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}], " ", "\[Equal]", " ", RowBox[{ RowBox[{"\[CapitalPhi]Curve", "[", RowBox[{"\[Alpha]2", ",", " ", "a3"}], "]"}], "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}]}], ",", RowBox[{ RowBox[{ RowBox[{"\[CapitalPhi]Curve", "[", RowBox[{"\[Alpha]1", ",", " ", "a2"}], "]"}], "\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}], " ", "\[Equal]", " ", RowBox[{ RowBox[{"\[CapitalPhi]Curve", "[", RowBox[{"\[Alpha]2", ",", " ", "a3"}], "]"}], "\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}]}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"\[Alpha]1", ",", " ", RowBox[{"0.615", " ", "-", " ", "0.2"}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"\[Alpha]2", ",", " ", RowBox[{"0.615", " ", "+", " ", "0.2"}]}], "}"}]}], "}"}]}], "]"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.7398334897900076`*^9, 3.739833508763277*^9}, { 3.7398335696859164`*^9, 3.7398336027068205`*^9}, {3.7398337417950153`*^9, 3.739833780845315*^9}, {3.7399789004930477`*^9, 3.739978901093972*^9}, { 3.7400692934373407`*^9, 3.740069293655057*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"\[Alpha]1", "\[Rule]", "0.19242443123007671`"}], ",", RowBox[{"\[Alpha]2", "\[Rule]", "1.1992265122205843`"}]}], "}"}]], "Output", CellChangeTimes->{3.739833730134018*^9, 3.739833781910302*^9, 3.739897452316889*^9, 3.7399789019954147`*^9, 3.740069295807599*^9, 3.740324290478239*^9, 3.740486868722517*^9}] }, Open ]], Cell[TextData[{ "where I have assisted the root finder by guessing ", Cell[BoxData[ FormBox[ SubscriptBox["\[Alpha]", "1"], TraditionalForm]]], " < ", Cell[BoxData[ FormBox[ SubscriptBox["\[Alpha]", "cusp"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ SubscriptBox["\[Alpha]", "2"], TraditionalForm]]], " > ", Cell[BoxData[ FormBox[ SubscriptBox["\[Alpha]", "cusp"], TraditionalForm]]], ".\n\nNext I construct the tangent to the green curve at the intersection \ point (the dashed green line). 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\[LongDash] that is, the \ points along the ship\[CloseCurlyQuote]s path that strongly influence the \ disturbance at point P. 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In the photographs above, both of these wave patterns can be observed." }], "Text", CellChangeTimes->{{3.739834109301592*^9, 3.739834174401248*^9}, { 3.7398342370986805`*^9, 3.7398353189970274`*^9}, 3.739895940667658*^9, { 3.7400591917717977`*^9, 3.740059344218747*^9}, {3.7400593855184526`*^9, 3.7400594273773193`*^9}, 3.7403243583976145`*^9, {3.7404869231739626`*^9, 3.7404869568069944`*^9}}], Cell["Returning to equation (4) for the linear response", "Text", CellChangeTimes->{{3.7399841239204035`*^9, 3.7399841439420347`*^9}}], Cell[BoxData[{ FormBox[ RowBox[{ RowBox[{ RowBox[{"\[Delta]z", "(", RowBox[{"r", ",", " ", "t"}], ")"}], " ", "=", RowBox[{ RowBox[{"\[Kappa]", " ", RowBox[{ SubsuperscriptBox["\[Integral]", "0", SubscriptBox["S", "max"]], RowBox[{"ds", " ", FractionBox[ SuperscriptBox["s", "3"], RowBox[{" ", SuperscriptBox["r", "4"]}]], RowBox[{"exp", "(", RowBox[{"\[ImaginaryI]", FractionBox[ RowBox[{"g", " ", SuperscriptBox["s", "2"]}], RowBox[{"4", " ", "r"}]]}], ")"}]}]}]}], " ", "\[Congruent]", " ", RowBox[{"\[Kappa]", " ", RowBox[{ SubsuperscriptBox["\[Integral]", "0", SubscriptBox["S", "max"]], RowBox[{"ds", " ", RowBox[{"\[CapitalPsi]", "(", RowBox[{"r", ",", " ", "s"}], ")"}], SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", RowBox[{"\[CapitalPhi]", "(", RowBox[{"r", ",", " ", "s"}], ")"}]}]]}]}]}]}]}], " "}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{" ", RowBox[{"\[TildeFullEqual]", " ", RowBox[{"\[Kappa]", " ", RowBox[{"\[CapitalPsi]", "(", RowBox[{ SubscriptBox["r", "in"], ",", " ", SubscriptBox["s", "in"]}], ")"}], SqrtBox[ FractionBox[ RowBox[{"2", " ", "\[Pi]"}], RowBox[{"\[VerticalSeparator]", RowBox[{ RowBox[{ RowBox[{"\[CapitalPhi]", "''"}], RowBox[{"(", RowBox[{ SubscriptBox["r", "in"], ",", " ", SubscriptBox["s", "in"]}], ")"}]}], "|"}]}]]], SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", RowBox[{"\[CapitalPhi]", "(", RowBox[{ SubscriptBox["r", "in"], ",", " ", SubscriptBox["s", "in"]}], ")"}], " "}]], SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", RowBox[{"sign", "(", RowBox[{ RowBox[{"\[CapitalPhi]", "''"}], RowBox[{"(", RowBox[{ SubscriptBox["r", "in"], ",", " ", SubscriptBox["s", "in"]}], ")"}], RowBox[{"\[Pi]", "/", "4"}]}], ")"}]}]]}]}]}], TraditionalForm]}], "DisplayFormulaNumbered", CellChangeTimes->{{3.7395596003490906`*^9, 3.7395597051349173`*^9}, { 3.7395613877664557`*^9, 3.73956157350377*^9}, {3.7395616154491196`*^9, 3.739561646535474*^9}, {3.7395617297558646`*^9, 3.7395617348609996`*^9}, { 3.7398279899884233`*^9, 3.7398280108905544`*^9}, 3.739984152368581*^9, { 3.739984202840082*^9, 3.739984443637562*^9}, {3.740072640776719*^9, 3.740072643147396*^9}}], Cell[TextData[{ "where the second line is the result of applying the stationary phase \ approximation. The formula is derived in Appendix C. \n\nThe observable \ response at point P is the consequence of waves emanating from ", Cell[BoxData[ FormBox[ SubscriptBox["Q", "1"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ SubscriptBox["Q", "2"], TraditionalForm]]], ". For the explicit stationary phase evaluation, it is more convenient to \ use r and \[Alpha] as the variables. 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OverHat[$CellContext`\[CapitalPhi]]][ $CellContext`r[$CellContext`sP], $CellContext`sP]] == -($CellContext`\ \[Alpha][$CellContext`s] > $CellContext`\[Alpha]Cusp)}], Editable->False]], "Output", CellChangeTimes->{3.7404891765069904`*^9}] }, Open ]], Cell["\<\ As illustrated in Appendix C, the method of stationary phase usually involves \ a formal large parameter. In the calculations above, there is no such formal \ large parameter. However, Stoker introduces some dimensionless variables\ \>", "Text", CellChangeTimes->{{3.7403573936826696`*^9, 3.7403574628828897`*^9}, { 3.74048716509212*^9, 3.7404872217486334`*^9}, {3.7404935588698444`*^9, 3.7404935596517916`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"w2", "[", "19", "]"}], " ", "=", " ", RowBox[{ RowBox[{ RowBox[{"\[CapitalPhi]", "[", RowBox[{"r", ",", " ", "s"}], "]"}], " ", "/.", " ", RowBox[{"r", " ", "\[Rule]", " ", RowBox[{"\[ScriptCapitalR]", " ", "R0"}]}]}], " ", "/.", " ", RowBox[{"s", " ", "\[Rule]", " ", RowBox[{"\[ScriptCapitalS]", " ", "S0"}]}]}]}]], "Input", CellChangeTimes->{{3.7403575216542883`*^9, 3.7403576164584885`*^9}, { 3.740357974483139*^9, 3.7403579780083323`*^9}, {3.7403581186129055`*^9, 3.740358118910636*^9}}], Cell[BoxData[ FractionBox[ RowBox[{"g", " ", SuperscriptBox["S0", "2"], " ", SuperscriptBox["\[ScriptCapitalS]", "2"]}], RowBox[{"4", " ", "R0", " ", "\[ScriptCapitalR]"}]]], "Output", CellChangeTimes->{{3.7403575274184837`*^9, 3.7403575456642513`*^9}, 3.740357617421995*^9, 3.7403583130552297`*^9, 3.7404872261358633`*^9}] }, Open ]], Cell["\<\ where \[ScriptCapitalR] and \[ScriptCapitalS] are dimensionless forms for \ space and time variation. The only natural scale length in the problem is R0, \ namely the distance from O to P. There is no natural time scale but S0 can be \ chosen such that R0/S0 = V, the speed of the ship. Then \ \>", "Text", CellChangeTimes->{{3.740357939346146*^9, 3.7403581537663507`*^9}, { 3.7404872369020786`*^9, 3.740487284621492*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"w2", "[", "20", "]"}], " ", "=", " ", RowBox[{ RowBox[{"w2", "[", "19", "]"}], " ", "/.", " ", RowBox[{"S0", " ", "\[Rule]", " ", RowBox[{"R0", "/", "V"}]}]}]}]], "Input", CellChangeTimes->{{3.7403581548487315`*^9, 3.7403581984320755`*^9}, { 3.7403582481537805`*^9, 3.740358255155203*^9}}], Cell[BoxData[ FractionBox[ RowBox[{"g", " ", "R0", " ", SuperscriptBox["\[ScriptCapitalS]", "2"]}], RowBox[{"4", " ", SuperscriptBox["V", "2"], " ", "\[ScriptCapitalR]"}]]], "Output", CellChangeTimes->{3.740358200149349*^9, 3.7403582567139797`*^9, 3.7403583373019857`*^9, 3.740487292942252*^9}] }, Open ]], Cell["Defining the dimensionless parameter \[ScriptCapitalK]", "Text", CellChangeTimes->{{3.7403583672637415`*^9, 3.7403583992271624`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"def", "[", "\[ScriptCapitalK]", "]"}], " ", "=", " ", RowBox[{"\[ScriptCapitalK]", " ", "\[Equal]", " ", FractionBox[ RowBox[{"g", " ", "R0"}], RowBox[{"4", " ", SuperscriptBox["V", "2"]}]]}]}]], "Input", CellChangeTimes->{{3.740358401837761*^9, 3.740358429556061*^9}, { 3.740358525044627*^9, 3.7403585253550425`*^9}}], Cell[BoxData[ RowBox[{"\[ScriptCapitalK]", "\[Equal]", FractionBox[ RowBox[{"g", " ", "R0"}], RowBox[{"4", " ", SuperscriptBox["V", "2"]}]]}]], "Output", CellChangeTimes->{ 3.7403584367021933`*^9, {3.74035850772546*^9, 3.7403585258269787`*^9}, 3.7404872982231665`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"w2", "[", "21", "]"}], " ", "=", " ", RowBox[{ RowBox[{"\[CapitalPhi]", " ", "\[Equal]", " ", RowBox[{"w2", "[", "20", "]"}]}], " ", "/.", " ", RowBox[{"Sol", "[", RowBox[{ RowBox[{"def", "[", "\[ScriptCapitalK]", "]"}], ",", " ", "g"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.740358442524375*^9, 3.7403584987312307`*^9}, { 3.7403585532339034`*^9, 3.740358554923907*^9}}], Cell[BoxData[ RowBox[{"\[CapitalPhi]", "\[Equal]", FractionBox[ RowBox[{"\[ScriptCapitalK]", " ", SuperscriptBox["\[ScriptCapitalS]", "2"]}], "\[ScriptCapitalR]"]}]], "Output", CellChangeTimes->{{3.740358481572688*^9, 3.740358555974944*^9}, 3.7404873020129843`*^9}] }, Open ]], Cell["\<\ and it is clear that \[ScriptCapitalK] >> 1 is required for the stationary \ phase approximation to be accurate. This corresponds to an observation point \ P far from the ship. When looking at the wake close to a moving ship, the \ waves, of course, are not stationary and the physics is much more \ complicated. The shape of the ship is important, the method of propulsion is \ important, the existing distortion of the water is important, and \ nonlinearities abound. \ \>", "Text", CellChangeTimes->{{3.740358561266618*^9, 3.7403586339994564`*^9}, { 3.7404873414126406`*^9, 3.7404873605724115`*^9}, {3.740493632387273*^9, 3.740493776235653*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Appendix A: Point disturbance in deep water (k integral)", "Section", CellChangeTimes->{{3.738283635944289*^9, 3.738283637325591*^9}, { 3.738768227652459*^9, 3.7387682477431903`*^9}, {3.7387682829465055`*^9, 3.7387682875298653`*^9}, {3.738768408614607*^9, 3.7387684092942796`*^9}, { 3.740069641383979*^9, 3.74006968497173*^9}}], Cell["\<\ In cylindrical coordinates, the linear response to a point disturbance at the \ surface of water, e.g, dropping a pebble in a pond, is\ \>", "Text", CellChangeTimes->{{3.7387682625365987`*^9, 3.7387683532058735`*^9}, { 3.7401582253570957`*^9, 3.740158240875105*^9}}], Cell["\<\ Linear response to a point disturbance at the surface of a fluid (Stoker \ 6.4.23)\ \>", "Text", CellChangeTimes->{{3.738759372920413*^9, 3.738759401684229*^9}, { 3.73876107331835*^9, 3.7387610782160625`*^9}, {3.73876113592841*^9, 3.7387611382217317`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"wA1", "[", "1", "]"}], " ", "=", " ", RowBox[{ RowBox[{"\[Delta]z", "[", RowBox[{"r", ",", " ", "t"}], "]"}], " ", "\[Equal]", " ", RowBox[{ RowBox[{"-", FractionBox["1", RowBox[{"2", " ", "\[Pi]", " ", "\[Rho]", " ", SqrtBox["g"]}]]}], RowBox[{"Intk", "[", RowBox[{ SuperscriptBox["k", RowBox[{"3", "/", "2"}]], RowBox[{"J0", "[", RowBox[{"k", " ", "r"}], "]"}], " ", RowBox[{"Sin", "[", RowBox[{ SqrtBox[ RowBox[{"g", " ", "k"}]], "t"}], "]"}]}], "]"}]}]}]}]], "Input", CellChangeTimes->{{3.7387592441186275`*^9, 3.738759310959074*^9}, 3.7401582564668894`*^9}], Cell[BoxData[ RowBox[{ RowBox[{"\[Delta]z", "[", RowBox[{"r", ",", "t"}], "]"}], "\[Equal]", RowBox[{"-", FractionBox[ RowBox[{"Intk", "[", RowBox[{ SuperscriptBox["k", RowBox[{"3", "/", "2"}]], " ", RowBox[{"J0", "[", RowBox[{"k", " ", "r"}], "]"}], " ", RowBox[{"Sin", "[", RowBox[{ SqrtBox[ RowBox[{"g", " ", "k"}]], " ", "t"}], "]"}]}], "]"}], RowBox[{"2", " ", SqrtBox["g"], " ", "\[Pi]", " ", "\[Rho]"}]]}]}]], "Output", CellChangeTimes->{3.738759355710822*^9, 3.7387623746459413`*^9, 3.7387644883155637`*^9, 3.738847117977986*^9, 3.738852891671709*^9, 3.740160062272953*^9, 3.7402566872095795`*^9, 3.7404873918039017`*^9}] }, Open ]], Cell[TextData[{ "where \[Delta]z is the surface elevation, g is gravitational acceleration, \ \[Rho] is density. This result is derived in many places, e.g. Stoker ", StyleBox["Water Waves", FontSlant->"Italic"], ". Basically, it follows from solving the initial value problem for the wave \ equation in cylindrical coordinates for a delta function source in space and \ time. \n\nThe term Intk is a shorthand notation for an integral" }], "Text", CellChangeTimes->{{3.7387684283941336`*^9, 3.738768600903097*^9}, { 3.738768648778822*^9, 3.738768649811736*^9}, {3.7388471348904233`*^9, 3.738847200807437*^9}, {3.7388472604294167`*^9, 3.738847289954626*^9}, { 3.74015993834109*^9, 3.740159938856985*^9}, {3.740487415396821*^9, 3.740487437259405*^9}}], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"Intk", "[", RowBox[{"f", "(", "k", ")"}], "]"}], " ", "\[Congruent]", " ", RowBox[{ SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[{ RowBox[{"f", "(", "k", ")"}], RowBox[{"\[DifferentialD]", "k"}]}]}]}], TraditionalForm]], "Input", CellChangeTimes->{{3.7387669033415203`*^9, 3.738766960409154*^9}, { 3.740160335203081*^9, 3.740160336732689*^9}}], Cell["\<\ I use Intk instead of Integrate because I want to perform manipulations \ without triggering the actual process of integration. I prefer this simple \ notational device to Mathematica functions such as Hold or Inactivate.\ \>", "Text", CellChangeTimes->{{3.738847312107377*^9, 3.738847544611094*^9}}], Cell["\<\ For parameters of physical interest, the integrand becomes highly \ oscillatory. I illustrate for some nominal physical parameters.\ \>", "Text", CellChangeTimes->{{3.738762204028685*^9, 3.7387622271371183`*^9}, { 3.738847632713465*^9, 3.738847655096161*^9}, {3.740159785493432*^9, 3.7401598083882604`*^9}}], Cell[BoxData[ GraphicsBox[{{{}, {}, {GrayLevel[0], Thickness[Large], Opacity[1.], LineBox[CompressedData[" 1:eJwUV3c8lu8Xtvfee/PyGlGUSu6jLVLJqEilIURSKbsio5CtJCHZvmZC6n6N svfO3rL3Hj+/v57P+TzP57qvc53z3Nc54uYP9e9SkJGRBTKSkf3/+cnNnEbl Tqom/5lI5Hk1+FhZs5iusZgv4q/2q/Q20UW7f355iIl9QCFEJ4mZMkvEEpyX ydARh+LOqEo3sTqhTyR/UWqxZPQ99koSTYQnUpi9E7BrkYEkaVvyKpEf6mGJ 3tlqz0ZcKejs25ggJJVAf3Bj8xv6Y71oaeEeih4ce2qzKlqANucJzy95hKOc lv74pRNFSHihlOKwbwTasNbtnrfAaFiK6aKCZwSKIe/2XT1VjLTHv2dcdAlH J2uDpFfaS9B+sZtU036haPz9meIlyzLk8Te+KX0gCPnd2TZd3PyN4v/bHI+6 54+UlXPW5v3L0QfhsoOp3l6oefN+6JxoJYoLtHW+aOWOHMpFlGezqhCT0RLH JbLHSCCkpXr6RA2aCBmYIjbeQr/M3lhMtdYio0zKEhdqLWROBMpJi3pk+EFN +DCnGKbYdKpfcmlAHc/WUHbkEdTSbyi8dKoRyX/RE6bnMkcJf5StF1ma0Bkx 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None}}, FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}}, GridLines->{None, None}, GridLinesStyle->Directive[ GrayLevel[0.5, 0.4]], ImagePadding->All, LabelStyle->Directive[Bold, "Helvetica"], Method->{ "DefaultBoundaryStyle" -> Automatic, "DefaultMeshStyle" -> AbsolutePointSize[6], "ScalingFunctions" -> None, "CoordinatesToolOptions" -> {"DisplayFunction" -> ({ (Part[{{Identity, Identity}, {Identity, Identity}}, 1, 2][#]& )[ Part[#, 1]], (Part[{{Identity, Identity}, {Identity, Identity}}, 2, 2][#]& )[ Part[#, 2]]}& ), "CopiedValueFunction" -> ({ (Part[{{Identity, Identity}, {Identity, Identity}}, 1, 2][#]& )[ Part[#, 1]], (Part[{{Identity, Identity}, {Identity, Identity}}, 2, 2][#]& )[ Part[#, 2]]}& )}}, PlotLabel->FormBox[ StyleBox[ InterpretationBox[ "\"Integrand for s = \\!\\(\\*FormBox[\\\"20\\\", \ TraditionalForm]\\)sec, r = \\!\\(\\*FormBox[\\\"2\\\", \ TraditionalForm]\\)m\"", StringForm["Integrand for s = ``sec, r = ``m", 20, 2], Editable -> False], Bold, FontFamily -> "Helvetica", StripOnInput -> False], TraditionalForm], PlotRange->{All, All}, PlotRangeClipping->True, PlotRangePadding->{{ Scaled[0.02], Scaled[0.02]}, { Scaled[0.05], Scaled[0.05]}}, Ticks->{Automatic, Automatic}]], "Figure", CellChangeTimes->{{3.7401596512596908`*^9, 3.740159677816592*^9}, { 3.740159747861*^9, 3.740159774197302*^9}, 3.7404874970165844`*^9}], Cell["Figure A1", "FigureCaption", CellChangeTimes->{{3.740159820830551*^9, 3.7401598477575407`*^9}}], Cell["\<\ This integral is evaluated using the method of stationary phase. In fact, \ Lord Kelvin (William Thompson) developed that method of approximation while \ attacking this problem. https://en.wikipedia.org/wiki/Stationary_phase_approximation\ \>", "Text", CellChangeTimes->{{3.7387646956743965`*^9, 3.7387647714738064`*^9}, { 3.7387649495973854`*^9, 3.7387649680371103`*^9}}], Cell["\<\ The classical form of integrals for which this approximation is appropriate is\ \>", "Text", CellChangeTimes->{{3.7387649820443764`*^9, 3.7387650097882833`*^9}}], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"I", "(", "k", ")"}], " ", "=", " ", RowBox[{"\[Integral]", RowBox[{"dx", " ", RowBox[{"g", "(", "x", ")"}], " ", SuperscriptBox["e", RowBox[{"I", " ", "p", " ", RowBox[{"f", "(", "x", ")"}]}]]}]}]}], TraditionalForm]], "Input", CellChangeTimes->{{3.738765018631689*^9, 3.738765079472074*^9}, { 3.7388476950549393`*^9, 3.7388476962149367`*^9}, {3.7404876094081163`*^9, 3.740487611053675*^9}}], Cell["\<\ The idea is that the integrand is highly oscillatory except when the phase \ f(x) is relatively \[OpenCurlyDoubleQuote]stationary\[CloseCurlyDoubleQuote]. \ The point of stationary phase is the value of x for which \ f\[CloseCurlyQuote](x) = 0, and the integral can be approximated by \ restricting the range of integration to the region near the point of \ stationary phase. The accuracy increases as p becomes are and the region of \ stationary phase is more sharply localized I perform a series of transformation that cast wA1[1] into the classical \ form. 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