J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 44
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Influence points corresponding to a given pointbasic conclusions of a qualitative character through use of the condition of stationary phase (8.2.14).proceed next to study analytic-Weshape of the disturbed water surface by determining thecurves of constant phase, and later on by determining the amplitudeof the waves. To calculate the curves of constant phase it is convenientto express the basic condition (8.2.14) of stationary phase in otherally theforms through introduction of the following quantitydimension of length:a(8.2.15)=2c 2w=(8.2.14)which has thec 2t 2.2rgFroma,one then findsct(8.2.16)(8.2.17)= a cos 0,r = acosand20,as equivalent expressions of the stationary phase condition.It would be possible to calculate the curves of constant phase forany given course of theship.Wecarry this out for the case of acir-cular course (this case has been treated by L.
N. Sretenski [S.15] ) anda straight course traversed at constant speed. The notation for the caseindicated in Figure 8.2.7, which should be compared with Figure 8.2.1. For the past position (x v z l ) of the ship we haveof the circular courseisWAVE PATTERN CREATED BY A MOVINGSHIP285P(x,z)Fig. 8.2.7. Case of a circular course(8.2.18)==l{xz =1R sin acos a)R(lwitha(8.2.19)HereR=ct/R.the turning radius of the ship, t the time required for it totravel from Q to O, and c is the constant speed of the ship. The coordinates of the point P, where the disturbance created by the ship isisto be found, are givenby=x(8.2.20)r cos (al{xz^=z -r^n (alinwhichrandare the distancethese equations we replace x l and(8.2.17) to obtainR sin ax+6+ 0))and angle noted on the figure.
In^ from (8.2.18) and make use ofa- cos 22cos (a+ 6)(8.2.21)z=R(lcos a)- cos 2 6 sin (aA+0).We wish to find the locus of points (x* z) such that the phase <p remainsWATER WAVES236i.e. such that the quantity a in (8.2.15) is constant (cf. theremarks following (8.2.9)). It is convenient to introduce the dimensionless parameter K throughfixed,K(8.2.22)=One then finds that the angle aa(8.2.23)=a/R.(cf.(8.2.19))isgiven byHCOS0,through use of (8.2.16). In terms of these quantities the relations(8.2.21) can be put in the following dimensionless form:x/R=sin (x cos 6)z/R=1-cos 2 6 cos (6+ K cos 0)- cos 2+(8.2.24)cos (x cos 0)2sin (0cos 0).These equations furnish the curves of stationary phase in terms ofas parameter. Each fixed value of x furnishes one such curve, sincefixing K (for a fixed turning radius R) is equivalent to fixing the phaseIn Figure 8.2.8 a few curves of constant phase, as well as the(p.Fig.
8.2.8.Wavecrests fora circular courseoutline of the region of disturbance, as calculated from (8.2.24), arcshown; the successive curves differ by 2n in phase. These curves shouldbe compared with the photographs of actual cases given in Figures8.2.4 and 8.2.5. One sees that the wave pattern is given correctly bythe theory, at least qualitatively. The agreement between theory andobservation is particularly striking in view of the manner in whichthe action of a ship has been idealized as a moving pressure point. Inparticular there are two distinct sets of waves apparent, in conformitywith the fact that we expect each point in the disturbed region toWAVE PATTERN CREATED BY A MOVINGSHIP237correspond to two influence points: one set which seems to emanatefrom the ship's bow, and another set which is arranged roughly atright angles to the ship's course.
These two systems of waves are calledthe diverging and the transverse systems, respectively.From (8.2.24) we can obtain the more important case of the ship-> oo while x ->waves for a straight course by lettingin such aRwaythatRxThe-> a (cf. (8.2.22)).x=-(2 cosresultiscos 3 0)(8.2.25)z--cos 2sin2for the curves of constant phase. In Figure 8.2.9 the results of calculations from these equations are shown. These should once more becompared with Figure 8.2.4, which shows an actual case. Again theagreementisstriking in a qualitative way. Actually, the agreementFig.
8.2.9.Wavecrests for a straight coursewould be still better if the two systems of waves the diverging andtransverse systems had been drawn in Figure 8.2.9 with a relativephase difference: the photograph indicates that the crests of the twosystems do not join with acommontangent at the boundary of theregion of disturbance. We shall see shortly that a closer examinationof our approximate solution shows the two systems of waves to have aphase difference there. Itgeneral observationisworth while to verifymade earlier,i.e.in the present case athat the curves of constant phaseWATER WAVES238are orthogonal to the lines drawn back to the corresponding influencepoints. One finds from (8.2.25):_=dd_-(3sin2-2I)sin0(8.2.26)=-(3sin2-2ddI)cos0.=I/tan 0, which (cf.
Figure 8.2.10) means that thecurves of constant phase are indeed orthogonal to the lines drawn toHence dz/dxz>Fig. 8.2.10. Construction of curves of constant phase=10* at which 3 sin 2the influence points. The valuesare singular points of the curves; they correspond to points P at theboundary of the influence region where the influence points Q l andcoincide. Evidently there are cusps at these points. One sees alsothat the diverging set of waves (for z0, say) is obtained whenwhilethetransverse waves corresvaries in the range 0*fg jr/2,Q2>^pondto values ofin the range0^0^ 0*. Inaddition,we observethat to any point on the ship's course there corresponds (for0)only one influence point (of type Q 2 ) and it does not coincide with thepoint P.
(One sees, in fact, that the diverging wave does not occur onthe ship's course.) This is a fact that is needed to justify the application of theasmethod of stationary phasewe have remarkedto points on the ship's course,earlier in this section (cf. also the precedingsection).In order to complete our discussion we must consider the amplitudeof the surface waves, as given by our approximation, as well as theshape of the curves of constant phase. To this end we must calculatey>and d2(p/dt 2 (and even d3y>/dt*)for such values oftas satisfy theWAVE PATTERN CREATED BY A MOVINGstationary phase condition dq>/dtof section 6.5 and section 6.8.^? =(8.2.27)vindt 2view ofsuch that(8.2.11).dqp/dt=Wewe know from the discussion(8.2.10) we find easily0, asFrom(l2r \-2r dneed the value of d3<p/dt3 at pointsit is readily found to be given byshall alsod 2(p/dt 2=d3q>(8.2.28)0;d3r2=gt-2We wisht.239SHIPto express our results in terms of the parameterc cos 6 from (8.2.13) we haveSince drjdtinstead ofsmOn.c(8.2.29)d*2dtthe speed of the ship, now assumed to be constant.
In orderto calculate dO/dt we introduce the angles /? and r indicated in Figurewithc,8.2.11.Wehave=n(/?+T),and hence2>-tP(x,z)TheFig. 8.2.11.anglesin which s refers to the arc length of C.radius of curvature of C; and since /?find=d88.2.31/dsd%\ - - -If/ - ^ -i^r Lv2ftandTBut dr/ds/(s-arc tan (z-vz,)dB\\-1^1/18,withz^)l(x=singindsjr912thex) weWATER WAVES240since the quantity in the square brackets is the vector product of rand t/|t|.
The expression for d 2qp/dt 2 given by (8.2.27) can now beexpressed in terms of 6 and r as follows:(8.2.32)'VnZ=JJ.L 12r [I-2 tan 23 sin 2-R-sin 01Jsin(1-sin 2 0)/t~2p.COS 2as one can easily verify. The quantity a is defined by (8.2.15), andthe relation (8.2.17), in addition to those immediately above, hasbeen used.
The points on the boundary of the region of disturbancecould be determined analytically, as follows: the set of all influence=the one-parameter family of circles given by dy/dt%(x,z,t)0, and the region of disturbance is bounded by the envelope ofinthese circles, i.e. by the points at which d 2(p/dt 2d%/dtpointsis==addition tojf=0.=In the case of a straight course traversed at con-=oo thatstant speed, for example, we .see from (8.2.32) for R3 sin 2a result found above,then has the value 0* given by 1where the value0* also was seen to characterize cusps on the loci==of constant phase. From the form of the relation (8.2.32) one can conclude that the only courses for which the pattern of waves behind theship follows it without change (i.e.
follows it like a rigid body) arethose for whichconst.; and thus only the straight and the circular courses have this property.R=we have to consider the amplitude j\(x, z) of the waves givenourbyapproximate solution. The contribution of a point t Q of stationary phase to (8.2.7) is given by (cf. (6.5.2)):Finally,,(,(8.2.33))= y(r, 0)I\?L_\* X* M)i)|?>"( r 0)1 /in which (r, 0) are polar coordinates which locate the point of stationary phase on the course C relative to the point (#, z) (cf. Figure8.2.1).
The sign of the termn/4t is to be taken the same as that of=d 2q?/dt 2 In principle, the surface elevation can be calculated forany course, but the results are not very tractable except for thesimplest case; i.e. a straight course. We confine our discussion ofq>".amplitudes, therefore, to this case in what follows.haveFrom(8.2.32)weWAVE PATTERN CREATED BY A MOVINGSHIP241rin'OX(8.2.34)2r \dt*coss2We know that there are two values of/callthem=8land2at each0: one belonging forpoint in the disturbed region for which dcpjdtarc sin l/\/8 to the transverse system, the other forOl0*0* ^S 2n l% to the diverging system of waves.
In the former case^~^<d 2<p/dt 2is positive; in the latter case negative. (At the boundary ofthe region of disturbance, where 99"0, the formula (8.2.33) is notThiscasewillbeknow.aswedealtwith later.) For points invalid,=we have,the interior of the region of disturbancetherefore,ri(x, z)(8.2.35)V|?"(r 2 ,0 2 )|y>====2C 2 t 2 /2r i9 and^a i cos 0^, a t2c^p /gfc^J/r} (cf. (8.2.15), (8.2.8)) at the points of stationary phase,may write (8.2.35) in the formSince r ict tcos>trt--=we(8.2.36)\IV1secaj3 sin 21X|3V|!~32sin 2 02|of waves are thus seen, as was stated above, to havea relative phase difference of n/2 at any point where a x2 Theirthefromandthatondieoutlikeship,l/\/a tgoing awayamplitudesmeans that they die out like the inverse square root of the distanceThe two systems=from the ship.
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