J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 43
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The #-axis is taken along thetangent to the course C, but is taken positive in the direction oppositeat theto the direction of travel of the ship, Since we have taken tofthetheisthetintime; asnegativereallyorigin(8.2.1)parameterasa consequence the tangent vector t to C at a point Q(x l9 t/ t )given bybutit is=isin the direction opposite to that of the ship in traversing the courseWATER WAVES226C.The speed c(t) of the ship is the length of the vector t and is given byThe point P(x,wavesisz) is the point atto be computed;=r(8.2.3)The angleit iswhich the amplitude of the surfacelocatedx l9(xindicated on the figurevectors r andby means of the vectorzisr:%).the angle(^n) between thet.As we have statedthe surface elevationr)(% z) at P(x, z)dueto a point imtheelevationsby integratingpulse moving along C.
The effect of an impulse at the point Q is asssumed to be given by the approximate formula (6.5.15), in which,isearlier,9to be determinedhowever, we omit a constant multiplier whichisunessential for thediscussion to follow:3rj(r;t)~(8.2.4)t---tf?sin 2_.r44rIn other words, we assume that the formula (8.1.1) for the surfacehas been approximated by two successive applicationsr)of the method of stationary phase.
This formula yields the effectat time t and at a point distant r from the point where the impulseelevationwas applied at the time=t0; it therefore applies in the present situa-tion withr*(8.2.5)=(x- ^) +-2(z^)2,does indeed represent the length of time elapsed since the"ship" passed the point Q on its way to its present position at O. Theintegrated effect of all the point impulses is therefore given bysincetn(x(8.2.6)9z)=CT l3A;-4Jo ret 2sin51dt,4ra certain constant.
For points on the ship's course, wheresome value tin the intervaltrg tT, this integralevidently does not exist. However, it has been shown in the precedingsection that neighborhoods of such points can be ignored in calculatingwithr=fcfor=^approximately provided that they are not points of stationary phase.This condition will be met in general, and hence we may imagine thata small interval about a point where r(t )has been excluded fromrj=WAVE PATTERN CREATED BY A MOVING227SHIPthe range of integration in case we wish the wave amplitude at a pointship's course. We write the integral in the formon the(8.2.7)f=ri(x t z)rV (*)* w)dt,Joand take the imaginarybyThe functionpart.\p(t)and the phase<p(t)are given(8.2.8)y(t)(8.2.9)<p(t)= kQt*/r*= gt*/4r.Weproceed to make the calculations called for in applying thestationary phase method.
In the integral given by (8.2.7) no largeparameter multiplying the phase is put explicitly in evidence; however, from the discussion of the preceding section we know that the2approximation will be good if the dimensionless quantity gJ?/4cwith R the distance from the ship, is large. It could also be verifiedthat (8.2.6) would result if the integrations in (8.1.6) on ft and werefirst approximated by stationary phase followed by a re-introduction,Weof the original variables.therefore beginby calculatingdq>/dt:=(8.2.10)'v4\rdtHence the condition of stationary phase,dtpjdt=0,leads to the im-portant relation(8.2.11)<^dtThe quantity dr/dt is next(8.2.5); we find (cf. Figure=inwhichc(t) isr=*.tcalculated for the ship's course using8.2.1):cr cos 0,tonce more the speed of the ship.
Thusdr(8.2.13)=ccos0,dtwhich is a rather obvious result geometrically. Combining (8.2.11)and (8.2.13) yields the stationary phase condition in the form(8.2.14)r=WATER WAVES228We recall once more the significance of this relation: for a fixed pointyields those points Q f on C which are the sole points effective(within the order of the approximation considered) in creating thethe contributions from all other points being, indisturbance atP(#, y)itPout through mutual interference. It is helpful to introduce the term influence points for the points fy determined in thisway relative to a point P at which the surface elevation of the wateris to be calculated.effect, cancelledTheclusion at onceway(cf.makesit possible to draw an interesting conwhichcan be interpreted in the following(8.2.14),Figure 8.2.2): At point Q the speed c of the ship and / arelast observationfromFig.
8.2.2. Points influencedknown. The relationwith respect to Q, ofby a given pointQ(8.2.14) then yields the polar coordinates (r, 0),for which Q is the influence point inpointsallPPthe sense of the stationary phase approximation. Such pointsevidently lie on a circle with a diameter tangent to the course C ofthe ship at Q, and Q is at one end of the diameter. The center of thecircle is located on the tangent line from Q in the direction towardwhich the ship moves (i.e. in the directiont). We repeat that thePon the circle just described are the only points for which Qpointsis a point of stationary phase of theintegral (8.2.7), and consequentlythe contribution of the impulse applied at Q vanishes (within theorder considered by us) for all points except those on the circle. Itnow becomes obvious that the disturbance created by the ship doesnot affect the whole surface of the wpter, since only those points areWAVE PATTERN CREATED BY A MOVINGaffected whichlieon one or more of theSHIP229circles of influence of all pointson theQship's course.
In other words, the surface waves created bythe moving ship will be confined to the region covered by all the influence circles, and thus to the region bounded by the envelope of thisone-parameter family of curves. This makes it possible to constructgraphically the outline of the disturbed region for any given coursetraversed at any given speed: one need only draw the circles in themanner indicated at a sufficient number of points Q and then sketchthe envelope. Two such cases, one of them a straight course traversedat constant speed, the other a circular course, are shown in Figure8.2.3. In the case of the straight course it is clear that the envelope(b)(a)Fig. 8.2.3.Region of disturbance(a) Circular course (b) Straight coursea pair of straight lines; the disturbance is confined to a sector ofarc sin 1/31928', as one readily seessemi-angle r given by ris==from Figure 8.2.3.
This is already an interesting result: it says thatthe waves following the ship not only are confined to such a sectorbut that the angle of the sector is independent of the speed of theship as long as the speed is constant. If the speed were not constantalong a straight course, the region of disturbance would be boundedby curved lines, and its shape would also change with the time. It is,of course, not true that the disturbance is exactly zero outside theregion of disturbance as we have defined it here; but rather it issmall of a different order from the disturbance inside that region.The observations of actual movingships bear out this conclusion ina quite startling way, as one sees from Figures 8.2.4 and 8.2.5.The discussion of the region of disturbance has furnished us with acertain amount of interesting information, but we wish to know a gooddeal more.
In particular, we wish to determine the character of theWATER WAVES230wave pattern created by the ship and the amplitude of the waves.For these purposes a more thoroughgoing analysis is necessary, and itwillbe carried outlater.In the special case of a straight course traversed at constant speedpossible to draw quite a few additional conclusions through further discussion of the condition (8.2.14) of stationary phase.
In theabove discussion we asked for the points P influenced by a givenpoint Q on the ship's course. We now reverse the question and ask forit isFig. 8.2.4. Ships in a straight courseWAVE PATTERN CREATED BY A MOVING231SHIPthe location ofall influence points Q* that correspond to a given pointP. This question can be answered in our special case by making another simple geometrical construction (cf. Lamb [L.3], p.
435), asindicated in Figure 8.2.6. In this figure O represents the location ofFig. 8.2.5a.Aship in a circular coursePthe point for which the influence points are to be determined. The construction is made as follows: A circle through P withcenter on OP and diameter half the length of OP is constructed; itsintersections with the ship's course are denoted by S l and S 2 Fromthe ship,.the latter points lines are drawn to P and segments orthogonal toare drawn to their intersections Q t and Q 2 on the ship'sthem atPcourse.The points Q l and Q 2 are thedesired influence points.To provethat the construction yields the desired result requires only a verification thatdoes indeed lie on the influence circles determined by thePpoints Q! andQ2in themanner explained above.
Consider the pointWATER WAVES282PQ =for example. Since the angle S l90, it follows that alas diameter contains the point P. The segmentscircle with1contoareatbothsinceareandbyright anglesparallellQ 19RSS^S^;PQsidering the triangleOPQaone now sees that the segmentOS lisjustFig. 8.2.5b. Ships in curved courseshalf the length of OQ 1? and that is all that is necessary to show that theas diameter is the influence circle for Q r Thus therehavingcircleS^are in general two influence points or no influence points, the latteroutside the influence region; the trancase corresponding to pointsis on the boundary of the region of influencesition occurs whenas diameter is tangent8.2.6 having(i.e.
when the circle of FigurePPPRto the courseOQ 2of the ship),two influence points Q x andQ2and onesees that in this limit case thecoalesce. Consequently onemight wellexpect that the amplitude of the waves at the boundary of the regionof disturbance will be higher than at other places, and this phenomenon is indeed one of the prominent features always observed physical-WAVE PATTERN CREATED BY A MOVINGSHIP233In addition, the direction of the curves of constant phase a waveexample can be determined graphically by theabove construction: one expects these curves to be orthogonal to thely.crest, or trough, forFig.
8.2.5c. Aircraft carrierslinesPQlandmaneuvering (from Life Magazine)PQ 2 drawn back from a point P to each of the points ofinfluence corresponding to P.Thatthisisindeed the case will be seenPlater, but it is evidently a consequence of the fact that the wave atis the sum of two circular waves, one generated at Qi and the other atWATER WAVES234wave pattern behind the ship is made up ofwaves another fact that is a matter of common observation and which is well shown in Figures 8.2.4 and 8.2.5.We have been able to draw a considerable number of interesting andQ 2 Thus we-twosee that thedifferent trains ofFig. 8.2.6.
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