J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 50
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Forof the ship relative to a horizontals(t)welater purposes265also introduce the angular velocityvector <o of the moving coordinate system:= a)(t)J,<o(9.2.3)and the angle abyFig. 9.2.1)(cf.a(0(9.2.4)=aj(r)dr.IJ oThe equationsof transformation from one coordinate system to theother arecos a +z sinY=yZ!X=x0(xc)cos az=(XXc)sin(ZZa -\-(ZZc)sina;y=Yx sin a +z cos a -\-Z cBy 0(X, Y, Z; t) we denote0(X,Y,Z;t)(9.2.6)~Xa+Jt c ;x=(Xc)cos a.the velocity potential and write+ 2 sin a + Xcos a= <p(x, y, z;;c,xy,sina+z cos+Zac;t)t).In addition to the transformation formulas for the coordinates,wealso need the formulas for the transformation of various derivatives.Onefinds without difficulty the following formulas:&x<Px==Y(9.2.7)Z<*+sinacos(p xIt is clear that grad 0(X, F, Z; t)a harmonic function in a?, y, z sincetisamorelittle= -sina<p y2culatey>z-f- q>z=gradiscos a.2<p(x, y, z; t)harmonict(sz(ftintwosets of equationsterms ofv(9.2.9)-f-(p(x, y, z; t);make(grad2Q2(s<p)Suppose now that F(X, F, Z;F(x cos a+..isTocal-<Ptan d.sin+a<pzzt=+Z C Thepossible to express Bernoulli's law+ <oz)<p x + cox<p +zisJ)., j/,xsina+.<p t=0.a boundary surface (fixedor moving) and set(9.2.10)X, F, Z.one has:I+ gy +it<p t<p v ytic>lastand that ydifficult; the result is+ coz)(px + cox<p +thisoneusesformula,yx +(To verify=the relations (9.2.5) together with s cos aX(9.2.8)in..;t)= f(x, y, z;t),WATER WAVES266=is the equation of the boundary surface relacoordinatemovingsystem.
The kinematic condition to besatisfied on such a boundary surface is that the particle derivativeso that/(a?, t/, z; t)tive to thedF/dt vanishes, and this leads to the boundary condition(9.2.11)<pJ 9+<p yf v+ <pj, -(*+ a*)/. +<*>*/*+=ftmoving coordinate system upon using the appropriatetransformation formulas. In particular, if yis therj(x, z\ t)equation of the free surface of the water, the appropriate kinematiccondition isrelative to the=(9.2.12)- yjj9+<P V- <Ptfz +(9+ coztyx - 0*09, - fy ==to be satisfied for yfree surface condition is of77.
(The dynamiccourse obtained for yfrom (9.2.9) by setting p0.)??turn next to the derivation of the appropriate conditions, both=Wekinematic and dynamic, on the ship's hull. To this endto introduce another moving coordinate system 0'isrigidly attached to the ship. Itisassumed that theit isconvenient#', y' 9 z'whichhull of the shiphas a vertical plane of symmetry (which also contains the center ofgravity of the ship); we locate the x' y' -plane in it (cf. Fig. 9.2.2) and#',suppose that the t/'-axis contains the center of gravity. The o'9t/',z'system, like the other moving system,issupposed to coincide(b)(a)Fig. 9.2.2a, b.Another moving coordinate systemwith the fixed system in the rest position of equilibrium.
The centerof gravity of the ship will thus be located at a definite point on thet/'-axis,say at distancey'from the origin o':of coordinates attached rigidly to the shipgravity has the coordinate (0, y c 0).in other words, theissystemsuch that the center of,we do not wish in general to carry out lineariHowever, since we shall in the end deal only with motionsIn the present sectionzations.THE MOTION OF A SHIP IN A SEAWAY267which involve small oscillations of the ship relative to the first movingcoordinate system ox, y, z it is convenient and saves time andspace to suppose even at this point that the angular displacementof the ship relative to the o#, y, z system is so small that it can9be treated as a vector 8:8-0^ +0J +(9.2.13)The transformation formulas,componentsicorrect3k.to first order terms in theupof 8, are then given by:'x9=+yyz'x3 (t/yc )y\y c= z + 62?/O^y-02*3ii-yc )with y c of course representing the ^-coordinate of the center ofgravity in the unprimed system.
It is assumed that y cy'e is a smallquantity of the same order as Q t and only linear terms in this quantityhave been retained. (The verification of (9.2.14) is easily carried out8 X r, for theby making use of the vector-product formula 8small displacement 8 of a rigid body under a small rotation 8.)The equation of the hull of the ship (assumed to be symmetricalwith respect to the #', j/'-plane) is now supposed given relative to the=primed system of coordinates/a oiK\yA\*Jt,\.*Jjfin theform:/Yr'ii'\^ 5 a^T S/>Vy'r4The equationof the hull relative to the obe written in the form(9.2.16)zx, y,+ Bv-OAy - y' - f (a, y) + [0# + [(y - y' - 0i* + OrtM*, y) = o,e)cwe postpone thisThe dynamical-3 (j/*'boundary^)]C.>now(*,y)o,are neglected.
The lefttinserted for / in (9.2.11) tocondition on the hull of the ship, butin (y cside of this equation couldyield the kinematics-system cane)when higher order termshand> O<y'c )andnow bestep until the next section.conditions on the ship's hull are obtained from theassumption that the ship is a rigid body in motion under the actionof the propeller thrust T, its weight Mg, and the pressure p of thewater on its hull. The principle of the motion of the center of gravityyields the condition(9.2.17)M~(*i+ yj) -Jpn dS+T-Mgj.WATER WAVES268By n we meanthe inward unit normal on the hull.=coordinate system o#, y, z is such that di/dt0, so that (9.2.17) can be written in the form(9.2.18)Msi- Mscok + My =pn dS(cjcok+T-Our movingand dj/dt =Mgj,momen-with p defined by (9.2.9).
The law of conservation of angulartum is taken in the form:(9.2.19)1(dt JM(R-Rc-RX (R)=(J sp(Rc)dm-Rc)n dS+(R T-Rx T.c)The crosses all indicate vector products. By R is meant the positionvector of the element of mass dm relative to the fixed coordinatesystem. R c (cf. Fig. 9.2.1) fixes the position of the e.g. and R Tlocates the point of application of the propeller thrust T. We introducer=(#, y, z)as the position vector ofany pointin the ship in themoving coordinate system and setso that=q(9.2.20)q is a vector from thee.g.r-toany pointR=R +(9.2.21)c+8(coy c j,+ 6)in the ship.TherelationX qthe angular velocity of the ship; thus (9.2.21)is simply the statement of a basic kinematic property of rigid bodies.By using the last two relations the dynamical condition (9.2.19) canbe expressed in terms of quantities measured with respect to theholds, since coismoving coordinate system(9.2.22)~J(r-y cj)oXx9[(coy, z, as follows:+ 9)X(r-y c j)]dnr0( rJ/cJ)X n dS+(R TRc)x T.Wehave now derived the basic equations for the motion of theship.
What would be wanted in general would be a velocity potential<p(x, y, z; t) satisfying (9.2.11) on the hull of the ship, conditions(9.2.9) (with p = 0) and (9.2.12) on the free surface of the water;and conditions (9.2.17) and (9.2.22), which involve 9? under integralsigns through the pressurepas givenby(9.2.9).Of course, the quan-THE MOTION OF A SHIP IN A SEAWAYtities fixingwaythat269the motion of the ship must also be determined in such aof the conditions are satisfied. In addition, there wouldallconditions and conditions at oo to be satisfied. Detailedconsideration of these conditions, and the complete formulation of thebeinitialproblem of determining y(x, y, z; t) under various conditions will bepostponed until later on since we wish to carry out a linearizationofallof the conditions formulated here.9.3.
Linearizationby a formal perturbation procedureBecause of the complicated nature of our conditions, it seems wise(as was indicated in sec. 1 of this chapter) to carry out the linearization by a formal development in order to make sure that all terms ofa given order are retained; this is all the more necessary since termsof different orders must be considered.
The linearization carried outhere is based on the assumption that the motion of the water relativeto the fixed coordinate system is a small oscillation about the restposition of equilibrium. It follows, in particular, that the elevation ofthe free surface of the water should be assumed to be small and, ofcourse, that themotion of the ship relative to thefirstmovingcoor-should be treated as a small oscillation.Wedo not, however, wish to consider the speed of the ship with respectto the fixed coordinate system to be a small quantity: it should ratherbe considered a finite quantity.
This brings with it the necessity torestrict the form of the ship so that its motion through the water doesnot cause disturbances so large as to violate our basic assumption;dinate system ox, y> zwe must assumethe ship to have the form of a thinclear that the velocity of such a disk-like shipof necessity maintain a direction that does not depart too muchthe plane of the thin disk if small disturbances only are to bein otherwords,disk.
In addition, itmustfromcreated.(9.3.1)isThus we assume that the equation of the ship'sz'=ph(x'9z'y'),>hullisgiven by0,a small dimensionless parameter, so that the ship is a thinftdisk symmetrical with respect to the x \ j/'-plane, and @h takes theplace of f in (9.2.15). (It has already been noted in the introductionto this chapter that this is not the most general way to describe thewithlshape of a disk that would be suitable for a linearization of the typecarried out here.
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