J.J. Stoker - Water waves. The mathematical theory with applications, страница 5

Treatment bythe method of stationary phase8.29.210269IIIShallow Water29110.1Introductory remarks and recapitulation of the basic equations10.2Integration of the differential equationsby293acteristicsof a simple wavePropagation of disturbances into291the method of char-10.3The notion10.4water of constant depthwavesintostill water of constantofdepressionPropagation305depth308Discontinuity, or shock, conditionsConstant shocks: bore, hydraulic jump, reflection from a rigid314wall326The breaking of a damThe solitary waveThe breaking of waves in shallow33310.510.610.710.810.9300stillwater.

Development of bores10.11 Gravity waves in the atmosphere. Simplified version of theproblem of the motion of cold and warm fronts10.1010.12 Supercritical steady flows in two dimensions. Flow aroundbends. Aerodynamic applications10.13 Linear shallow water theory. Tides. Seiches. Oscillations inharbors. Floating breakwaters342351374405414CONTENTSXXVIIIPAGECHAPTER11.Mathematical Hydraulics451open channelsSteady flows. A junction problemProgressing waves of fixed shape. Roll waves45240911.5Unsteady flows in open channels. The method of characteristicsNumerical methods for calculating solutions of the differentialequations for flow in open channelsFlood prediction in rivers.

Floods in models of the Ohio River47411.6and482Differential equations of flow in11.111.211.311.4its456461junction with the Mississippi RiverNumerical prediction of an actual flood in the Ohio, and at itsjunction with the Mississippi. Comparison of the predicted with1 1 .7408the observed floodsAppendix to Chapter 11.characteristicExpansion505PART12.Problemsinin the neighborhood of the firstIVwhich Free Surface Conditions are Satisfied Exactly.The Breakingof aDam.Levi-Civita's513TheoryMotion of water due to breaking of a dam, and related problemsexistence of periodic waves of finite amplitude ....12.2a Formulation of the problem12.112.2The51352252212.2b Outline of the procedure to be followed in proving the existenceof the function a)(%)526The12.2d The53712.2csolution of a class of linear problemssolution of the nonlinear boundary value problem529...Bibliography545Author Index561Subject Index563PARTICHAPTER1Basic Hydrodynamics1.1.The laws of conservation of momentum and massAs has been stated in the introduction, we deal exclusively in thisbook with flows in water (and air) which are of such a nature astomakeitunnecessary to take into account the effects of viscosityand compressibility.

As a consequence of the neglect of internalfriction, or in other words of neglect of shear stresses, it is wellknownthat the stress system* in the liquid is a state of uniformcompression at each point. The intensity of the compressive stressis called the pressure p.The equation of motion of a fluid particle can then be obtained onthe basis of Newton's law of conservation of momentum, as follows.A small rectangular element of the fluid is shown in Figure 1.1.1Fig. 1.1.1.

Pressureon afluid elementwith the pressure acting on the faces normal to thelaw for the ^-direction is then[(Po?-axis.Newton'sQ a (*)* We assume that the usual concepts of the general mechanics of continuousmedia are known.WATER WAVESwhich X is the external or body force component per unit massand a (x) is the acceleration component, both in the ^-direction, andQ is the density. The quantities p X, and a (x) are in general functionsof x y, z, and t. Here, as always, we shall use letter subscripts todenote differentiation, and this accounts for the symbol a (x} to denotein99the component of a vector in the ^-direction.limit in allowing dx, dy, dz to approach zeroof motion for the ^-direction in the formanalogous expressions for thetwo otherUpon passing to thewe obtain the equationpx+ qX =directions.Qa (x}9 andThus we have theequations of motionPx+%=-Pv + Y =(1.1.1)-P*or, invector form:grad p(1.1.2)+F=a,Qwith an obvious notation.

The body force F plays a very importantin fact the mainrole in our particular branch of hydrodynamicsresults of the theory are entirely conditioned by the presence of the(0,gravitational force Fg, 0), in which g represents the acceler-=ation of gravity. It should be observed thatweconsider the positivey-axis to be vertically upward, and the x, z-plane therefore to be horizontal(usually it will be taken as the undisturbed water surface). This con-vention regarding the disposition of the coordinate axes will be maintained, for the most part, throughout the book.The differential equations (1.1.1) are in what is called the Lagrangian form, in which one has in mind a direct description of the motionof each individual fluid particle as a function of the time.

It is moremost purposes to work with the equations of motion in theso-called Eulerian form. In this form of the equations one concentrates attention on the determination of the velocity distribution inthe region occupied by the fluid without trying to follow the motion ofthe individual fluid particles, but rather observing the velocityuseful fordistribution at fixed points in space as a function of the time.

InBASIC HYDRODYNAMICS5other words, the velocity field, with components u v w is to bedetermined as a function of the space variables and the time. Afterwards, if that is desired, the motion of the individual particles can999be obtained by integrating the system of ordinary differential equaxw in which the dot over the quantities x y,u, yv, zz means differentiation with respect to the time in following the=tions9motion of an individual9particle.-In order to restate the equations of motion (1.1.1) in terms of theEuler variables u, v, w and in order to carry out other importantoperations as well, it is necessary to calculate time derivatives of9various functions associated with a given fluid particle in followingthe motion of the particle.

For example, we need to calculate thetime derivative of the velocity of a particle in order to obtain theacceleration components occurring in (1.1.1), and quite a few otherquantities will occur later on for which such particle derivatives willbe needed. Suppose, then, that F(x y, z; t) is a function associatedwith a particle which follows the path given by the vector9xit=(x(t),y(t) 9 z(t));follows thatx=(x(t) 9 y(t) 9 z(t))=(u, v 9w)the velocity vector associated with the particle. For this particlethe arguments x y z of the function F are of course the functionsof t which characterize the motion of the particle; as a consequenceis99we have^=Fxx+ Fy y + Fand hence the operation of taking thezz+Ftparticle derivative d/dtisdefined as follows:(1.1.3)The1(at)=u()m+v(distinction between dF/dtnoted.)y+w(and dF/dt+).=Ft(),.should be carefully=Since the acceleration a of a particle is given by a(du/dt dv/dt 9v oftheofthearevinwhichvelocityw)components(udw/dt)9999WATER WAVES6the particle,isitfollows from (1.1.3) that thecomponent a (x)du/dtgiven bydu= uux + vuy + wu + uzatt,with similar expressions for the other components.

The equations ofmotion (1.1.1) are therefore given as follows in terms of the Eulervariables:Ut+ UUX + VUy + WU =1--Zpx,6(1.1.4)Vt+ ^^ + TO V +=WO*1-Pyg,eI0f+ U^ + U^y + WW =Zi-pzQwhen wespecify the external or body force to consist only of theforce of gravity.Equations (1.1.4) form a set of three nonlinear partial differentialequations for the five quantities u v, w g, and p. Since the fluid isassumed to be incompressible, the density Q can be taken as a known99At the same time, the assumption of incompressibility leadsto a relatively simple differential equation expressing the law ofconservation of mass, and this equation constitutes the needed fourthconstant.equation for the determination of the velocity components and thepressure. Perhaps the simplest way to derive the mass conservationlaw is to start from the relationwhich states that the mass flux outward through any fixed closedsurface enclosing a region in which no liquid is created or destroyedis zero.

(By v n we mean the velocity component taken positive in thedirection of the outward normal to the surface.) An application ofGauss's divergence theorem:(1.1.5)JJ ei>ndS=drJJJdiv ( e v)RSto the above integral leads to the relationfdiv (QV) drR=BASIC HYDRODYNAMICS7any arbitrary region R. It follows therefore that div (gv)everywhere, and since Q = constant, we have finallyfor=divv(1.1.6)ux=+v v + w =zas the expression of the law of conservation of mass.

The equation(1.1.6) is also frequently called the equation of continuity.(1.1.4) and (1.1.6) are sufficient, once appropriateand boundary conditions (to be discussed shortly) are imposed,to determine the velocity components u, v, w, and the pressure pEquationsinitialuniquely.1.2.Helmholtz's theoremBefore discussing boundary conditions it is preferable to formulate a few additional conservation laws which are consequencesof the assumptions made so far in particular of the assumption thatinternal fluid friction can be neglected.Thefirstof these laws to be discussedof circulation.The notionof circulationisis the law of conservationdefined as follows.

Considera closed curve C which moves with the fluid (that is, C consists==always of the same particles of the fluid). The circulationF(t)around C is defined by the line integralFr(t)(1.2.1)=j>udx+vdy+wdz=<pv s dscwhich v s is the velocity component of the fluid tangent to C,and ds is the element of arc length of C.

The curve C is consideredas given by the vector x(a, t) with a a parameter on C such that^ a ^ 1 and x(0, t) = x(l, /). We are thus operating in terms ofthe Lagrange system of variables rather than in terms of the Eulersystem, and fixing a value of a has the effect of picking out a specificparticle on C.iniWe maywrite F(t)=\vx a dainwhich vxaisa scalar productoand x a as,usual, refers to differentiation with respect to a.time derivativeFwe havethereforeFor theWATER WAVES81=f(t)From thea== v,xaJ(v+v-equation of motion (1.1.2) in the Lagrangian form withgrad (gy), and from xa = va the last(0, -g, 0)F==,equation yieldsi(1.2.2)f(t)n=Xa'grad p- gx agrad y+vvJ daoi-=~Po-&Va+-(vv) a IdaJ=0,and v coincide at a = and or = 1, and pThe last equation evidently states that in anonviscous fluid the circulation around any closed curve consisting ofthe same fluid particles is constant in time.

This is the theorem of Helmholtz. The assumption of zero viscosity entered into our derivationsince the values of p,and g are constants.j/,through the use of (1.1.2) as equation of motion.*In this book we are interested in the special case in which thecirculation for all closed curves is zero. This case is very importantin the applications because it occurs whenever the fluid is assumedto have been at rest or to have been moving with a constant velocityatsome particular time,F vanisheshencev = const, holds at that time, andThe cases in which the fluid motionso thatfor all time.begins from such states are obviously very important.The assumption that F vanishes for all closed curves has a numberof consequences which are basic for all that follows in this book.The first conclusion from JTfollows almost immediately from=Stokes's theorem:r=j>v ds(1.2.3)sc=dA,JJ(curl v) nsin which the surface integral is taken over any surface S spanning the1curve C.