J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 83
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In this case the equations of(11.1.6) and (11.1.8) become, for acontinuity and motion given byrectangular channel of fixed breadth and slope:m oo^f^(*>(11.3.3)|U)y :+ yv =^with S the slope of the channel andfirstdefined, as before,byequation of (11.3.3) can be integrated to yield(11.3.5)(vas one readily verifies,TheSf^Sf =(11.3.4)The0,cU)y=D=const.and the second equation then takes the formorder differential equation (11.3.6) has a great variety ofsolutions, which have been studied extensively, for example by Thomas [T.I], but most of them are not very interesting from the physicalfirstpoint of view. However, one type of solution of (11.3.6) is particularlyinteresting from the point of view of the applications, and we therefore proceed to discuss it briefly. The solution in question furnishesthe so-called monoclinal rising flood wave in a uniform channel (seethe articlebyGilcrest in thebook of Rouse [R.ll,p.
644]). This, asname suggests, is a progressing wave the profile of which tends todifferent constant values (and the flow velocity v also to different con-itsstant values) downstream and upstream, with the lower depth downstream, connected by a steadily falling portion, as indicated schematically in Fig. 11.3.1. In thiswave the propagation speedU islargerMATHEMATICAL HYDRAULICS463than the flow velocityv. It is always apossible type of solution of(11 .3.6) if the speed of propagation of the wave relative to the flow isFig. 11.3.1. Monoclinal rising floodsubcritical,i.e. if(7v)2is lesswavethan gy, in which case the coefficientof the first derivative term in (11.3.6) is seen to be positive.
This canbe verified along the following lines. The differential equation can besolved explicitly for f as a function of y\-f(11.3.7)Jv*with the integrand I(y) defined in the obvious manner; here y* is the0. The function I(y) has the generalvalue of y corresponding toFig.
11.8.2.The integrandI(y) for a monoclinalwaveWATER WAVES464form shownin Fig. 11.3.2ifthe propagation speedU and the constantD in (11.8.5) are chosen properly. The main point that the curve hastwo vertical asymptotes at yy and y = y between which I(y)isQnegative.
By->oo as y -* t/ whilef -*is that I(y) becomes infinite at+Thiswaytitylchoosing y* between yisis,oo as,in fact, the case;j/we canand y l we can hope thaty -> y x all that is necessary:and y l of sufficiently high order.select values ofandin such aDUand y l through setting the quant/(11.3.6) equal to zero, i.e.
by choosing D and U such thatthat I(y) becomes infinite atSjS in(11.3.8)and y l these are a pair oftaking a square root) which determine U andFor given positive values oftions (afterz/linear equa-D uniquely.Anelementary discussion of the possible solutions of these equationsshows that U must be positive and D negative, and this means, as wosee from (11.3.5), that U is larger than v i.e. the propagation speed ofthe wave is greater than the flow speed.By taking the numerical data for the model of the Ohio given in thepreceding section and assuming the depth far upstream to be 40 feet,far downstream 20 feet, it was found that the corresponding monoclinal flood wave in the Ohio would propagate with a speed of 5miles/hour. The shape of the wave will be given later in sec.
6 of thischapter, where it will be compared with an unsteady wave obtained bygradually raising the level in the Ohio at one point from 20 feet to40 feet, then holding the level fixed there at the latter value, and calculating the downstream motion which results. We shall see that themotion tends to the monoclinal flood wave obtained in the manner9just now described.
Thus the unsteady wave tends to move eventuallyat a speed of about 5 miles/hour, while on the other hand, as we knowfrom Chapter 10 (and will discuss again later on in this chapter), thepropagation speed of small disturbances relative to the stream is <\/gyand hence is considerably larger in the present case, i.e.
of the orderof 15 to 25 miles/hour. This important and interesting point will bediscussed in sec. 6 below.MATHEMATICAL HYDRAULICSFig. 11.8.8. Roll waves, looking465down stream (The Grunnbach, Switzerland)WATER WAVES466We turn next to another type of progressing waves in a uniformchannel which can be described with the aid of the differential equation (11.3.6), i.e. the type of wave called a roll wave. A famous example of such waves is shown in Fig. 11.3.3, which is a photograph takenfrom a book of Cornish [C.7], and printed here by the courtesy of theCambridge University Press. As one sees, these waves consist of aseries of bores (cf.flow.The sketch ofChapter 10.7) separated by stretches of smoothFig. 11.3.4 indicates this more specifically.
SuchFig. 11.3.4. Rollwaveswaves frequently occurin sufficiently steep channels as, for example,or in open channels such as that of Fig. 11.3.3.Roll-waves sometimes manifest themselves in quite unwanted places,as for example in the Los Angeles River in California. The run-off fromspill-ways indamsthe steep drainage area of this river is carried through the city of LosAngeles by a concrete spill- way; in the brief rainy season a largeamountis carried off at high velocity. It sometimes happenswaves occur with amplitudes high enough to cause spillingover the banks, though a uniform flow carrying the same total amountof water would be confined to the banks.
The phenomenon of rollwaves thus has some interest from a practical as well as from a theore-thatof waterrollpoint of view; we proceed to give a brief treatment of it in theremainder of this section following the paper of Dressier [D.12]. Indoing so, we follow Dressier in taking what is called the Ch^zy formula for the resistance rather than Manning's formula, as has beendone up to now. The Chzy formula gives the quantity S f the followingticaldefinition:a "roughness coefficient" and R is, as before, the hyFor a very broad rectangular channel, the only casewe consider, Ry. Under these circumstances the differentialinwhichr2isdraulic radius.=equation (11.3.6) takes the formMATHEMATICAL HYDRAULICS467sS _^(Uy+D)\Uy+D\o"dy-_Z) 2qas can be readily seen.It is natural to inquire first ofall whether (11.3.10) admits of soluwhich are continuous periodic functions ofsince this is theofwemotionseek.Theregeneral typeare, however, no such periodicand continuous solutions (cf.
the previously cited paper of Thomas[T.I]) of the equations; in fact, since the right hand side of (11.3.10)can be expressed as the quotient of cubic polynomials in y thetypes of functions which arise on integrating it are linear combinations of the powers, the logarithm, and the arc tangent function andone hardly expects to find periodic functions on inverting solutionsC(j/) of this type. This fact, together with observations of roll waves ofthe kind shown in Fig. 11.3.3, leads one to wonder whether there mighttionsnot be discontinuous periodic solutions of (11.3.10) with discontinuithe form of bores, which should be fitted in so that the discontinuity or shock conditions described in sec.
6 of the preceding chapter * are satisfied. This Dressier shows to be the case; he also gives aties incomplete quantitative analysis of the variouspossibilities.Thestarting point of the investigation is the observation, due toThomas [T.I], that only quite special types of solutions of (11.3.10)comewave problem has been formulated interms of a periodic distribution of bores. In fact, we know from Chapter 10 that the flow relative to a bore must be subcritical behind abore but supercritical in front of it; consequently there must be an intermediate point of depth j/ say, (cf. Fig. 11.3.4) where the smoothin question once the roll,flow has thecritical speed,i.e.K-(11.3.11)where2tf)= g2/>the speed of the bore, coincides with the propagation speedof the wave. At such a point the denominator on the right hand sideof (11.3.10) vanishes, since(vU)y and hence dy/d would beunless the numeratorobservationstotheinfinite therecontrarysinceC7,D=of the righthand9side also vanishes at that point.Therighthandsidecan now be written as a quotient of cubic polynomials, and we know* The shock conditions were derived inChapter 10 under the assumptionthat no resistances were present.
As one would expect, the resistance terms playno role in shock conditions, as Dressier [D.12] verifies in his paper.468WATER WAVESthat numerator and denominator have y Q as a common root; it followsthat a factor yyQ can be cancelled and the differential equationthen can be put in the formafter a little algebraic manipulation. Since the denominatorright hand side is positive and since we seek solutions foron thewhichdy/d is everywhere (cf. Fig.
11.3.4) positive, it follows in particularthat the quadratic in the numerator must be positive for yy Q Thisleads to the following necessary condition for the formation of roll=.waves(11.3.13)4r 2< 5,obtained by using (11.3.11 ) and other relations. A practically identicalinequality was derived by Thomas on the basis of the same type ofreasoning. The inequality states that the channel roughness, which is2larger or smaller with r , must not be too great in relation to the steepness of the channel, and this corroborates observations by Rouse[R.10] that roll waves can be prevented by making a channel sufficiently rough. Dressier also shows in his paper that it is important forthe formation of roll waves that the friction force for the same rough-and velocity should increase when the depth decreases;he finds, in fact, that roll waves would not occur if the hydraulic radius R in the Chzy formula (11.3.9) were to be assumed independentness coefficientof the depth y.Dressier goes on in his paper toshow that smoothsolutions of(11.3.12) can be pieced together through bores in such a way that theconditions referring to continuity of mass and momentum across thediscontinuity are satisfied as well as the inequality requiring a lossrather than a gain in energy.
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