J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 73
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In particular, one might learn something about thekind of perturbations that are necessary to initiate motions of thetype observed, and under what circumstances the motions can bemaintained.In carrying out the derivation of the differential equations of ourtheory according to the plan outlined above, we calculatenumber ofTheintegrals.-(10.11.27):r6JFromthesehdy=-honwefirstderivebyrjr*(yJi~^differentiations with respect to= oM<5 -n) -*%*>i*nC 68iv x dyJ1= -vx (di_215??'r*Jr\a_other set of relations:Jfirstof these arise from (10.11.26) andii_""x andtan-LONG WAVESINSHALLOW WATER403(In deriving these relations, it is necessary to observe that the lowerlimit T] is a function of x and t.) One additional relation is needed, asfollows:r*1^(hu) x dyIvyuT)h dy\%}=\|&1ri-I-1_+ hux )(6 -?)--~ hur} x(h xu.We nowintegrate both sides of the equations (10.11.16) with respect to y from 77 to d, make use of the above integrals, note thatuThe result is theu(x, t) is independent of j/, and divide by dTJ.=equations11_kh+ MM, + - kh x - - i -n x -*t.-(10.11.29)<5-Av,v2_jo12khI^Q'\77u-uh x-h,=-0.with k a constant replacing the quantity g(lQ'/'Q).
These equations,fourformaofwithsystempartial differentialtogether(10.11.28),andfourfunctions7Lfortheu, 77.By analogy with gasequationsdynamics and the nonlinear shallow water theory, it is convenient tointroduce a new dependent quantity c (which will turn out to be the,propagation speed of wavelets) through the relation--\h.(10.11.30)is less than Q, and this holds since the warmthan the cold air. In terms of this new quantity theequations (10.11.28) and (10.11.29) take the formThe quantityc is real if Q'air is lighterut+uu x+ 2rr,vtTJ X+ uv x =(10.11.31)=-2AItisnow/1uQ'u\\ ,6+ 2uc x =+ ur/ x =cu xrj tAi",cvdr)i".easy to write the equations (10.11.31) in the characteristicWATER WAVES404form simply by replacing theand by their difference.
The(10 11 32)xt+ urj xfirstand third equations byresult--4c= --o=theirsumis:2/2A [u-o'\ri\u'\,JQv.As onesees, the equations are in characteristic form: the characteristiccurves satisfy the differential equationsdx(10.11.33)dt=dxu +c,=dxuc,=u,dtdtand each of the equations (10.11.32) contains only derivatives in thedirection of one of these curves. The characteristic curves defined byu are taken twice. Thus one sees that the quantity c is indeeddxjdtentitled to be called a propagation speed, and small disturbances canbe expected to propagate with this speed in both directions relative tothe stream of velocity u. (In the theory by Whitham, in which themotion in each vertical plane yconst, is treated separately, the=propagation or sound speed of small disturbances is given by Vkh.The sound speed in the theory given here thus represents a kind ofaverage with respect to y of the sound speeds of Whitham's theory.
)Since the propagation speed depends on the height of the discontinuity surface, it is clear that the possibility of motions leading tobreaking is inherent in this theory.Once the dynamical equations have been formulated in characteristicformitbecomes possible tosee rather easilywhatsort of subsi-diary initial and boundary conditions are reasonable. In fact, thereare many possibilities in this respect.
One such possibility is theit is assumed thatfollowing. At time tconst, (as in a stationary front), but that rj t= 0, h =u = const.,= f(x) over a segment ofr)the r-axis. In other words, it is assumed that a transverse impulse isgiven to the stationary front over a portion of its length. The sub'sequent motionrically.Anotheruniquely determined and can be calculated nume=possibility is to prescribe a stationary front at tisLONG WAVES IN SHALLOW WATERforxat x405> 0, say, and then to give the values of all dependent quantities*= as arbitrary functions of the time; to prescribe a boundaryi.e.condition which allows energy to be introduced gradually into thesystem. One might visualize this case as one in which, for example,cold air is being added or withdrawn at a particular point (xin=the present case). This again yields a problem with a uniquely determined solution, and various possibilities are being explored numerically.was stated above that the most objectionable feature of thepresent theory is the assumption of a fixed vertical barrier back of theItThere is, however, a different way of looking at the problem asa whole which may mitigate this restriction somewhat.
One might tryto consider the motion of the entire cap of cold air that lies over thefront.polar region, using polar coordinates (0, y) (with the latitude angle,say). One might then consider motions once more that dependand t by getting rid of the dependence on 6same type of assumptions (linear behavior in 0,say) as above. Here the North Pole itself would take the place of thevertical barrier (v0!). The result is again a system of nonlinearthistimewith variable coefficients. Of course, it wouldequationsbe necessary to begin with a stationary flow in which the motion takesessentially only onthrough use of theq>place along the parallels of latitude.All in all, the ideas presented here would seem to yield theoriesflexible enough to permit a good deal of freedom with regard to initialand other conditions, so that one might hope to gain some insight intothe complicated dynamics of frontal motions by carrying out numerical solutions in well-chosen special cases.10.12.
Supercritical steady flows in two dimensions. Flow around bends.Aerodynamic applicationsThea slight misnomer, since the flows inquestion are really three-dimensional in nature; however, since weconsider them here only in terms of the shallow water theory, thetitleof this sectiondepth dimensiontwo componentsisThus thevelocity is characterized by thehorizontal(u, w) in theplane (the x, 2-plane); andthese quantities, together with the depth h of the water at any pointconstitute the quantities to be determined in any given problem. Byis leftout.* In the numerical cases so far considered we have hadc< ueven on the /-axis all four dependent quantities can be prescribed.\||so that\WATER WAVES406specializing the general equations (2.4.18), (2.4.19), (2.4.20), of theshallow water theory as derived in Chapter 2 for the case of a steadyflow, the differential equations relevant for this section result.
Theycan also be derived readily from first principles, as follows: Assumingthat the hydrostatic pressure law holds and that the fluid starts fromrest (or any other motion in which the vertical component of thevelocity of the water is zero) it follows that the vertical component ofthe velocity remains zero and that u and w are independent of the vertical coordinate. The law of continuity can thus be readily derived fora vertical column; for a steady flow it isWe+(hu) x(10.12.1)=(hw) zassume that the flows we study arewx =uz(10.12.2)0.irrotational,0.The Bernoulli law then holds and can be written(u(10.12.3)In these equations his2and hence that+ w + 2gh =2)in theformconst.the depth of the water at any point.
By usingu and w, and introducing the quan-(10.12.3) to express h in terms oftity c by the relationc2(10.12.4)we obtainghthe equation2(10.12.5)=(c-u 2 )u xuw(w xand this equation together withu and w through (10.12.4) and+u +z)(10.12.2),w 2 )wz2(cwithc=- 0,defined in terms of(10.12.3), constitute a pair of firstorder partial differential equations for the determination of u(x, z)and w(x, z).The theory of theselatter equations can be developed, as in thecases treated previously in this chapter, by using the method ofcharacteristics, provided that the quantity c remains always lessthe flow speed everywhere,c2(10.12.6)The flowi.e.thanprovided that<u +w22.then said to be supercritical.
(In hydraulics the contrastsupercritical is commonly expressed as tranquil-shooting.) Only then do real characteristics exist. We shall not developthis theory here, but rather indicate some of the problems which havebeen treated by using the theory. Complete expositions of the charissubcriticalLONG WAVES IN SHALLOW WATERacteristic theory can be found in thein Chapter IV of the book by Courant407paper by Preiswerk [P.16], andand Friedrichs [C.9]. The theoryof course, again perfectly analogous to the theory of steady twodimensional supersonic flows in gas dynamics.is,Fig.
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