J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 48
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In particular, itwould be necessary to examine the higher order terms in the condi-surfaceafter all, the conditions (9.1.13), which arealso used by Michell and Havelock (and everyone else, for thatand not on the actual displaced positionmatter), are satisfied at y =of the free surface. One way to obtain a more accurate theory wouldbe, of course, to carry out the perturbation scheme outlined here totions at the freehigher order terms.Still another point hascome up for frequent discussion (cf., forandLundeexample,Wigley [L.18]) with reference to the boundarycondition on the hull. Itisfairlycommonin the literature to refer toships of Miehell's type, by which is meant ships which are slendernot only in the fore-and-aft direction, but which are also slender inthe cross-sections at right angles to this direction-(b)Fig.
9.1.3a, b. Ships with fullthat h y,inFig. 9.1.3) soy(o)bottom(cf.our notation,issmall.and with narrow mid-sectionsThusships with a rather broadwith a full mid-section,arc often considered as ships to which the present theory does notapply. On the other hand, there are experimental results (cf. Havelock(cf.
Fig. 9.1.3a), or, as it is also put,[H.7]) which indicate that the theory is just as accurate for shipsfull mid-section as it is for ships of Miehell's type. When theproblem is examined from the point of view taken here, i.e. as awith aproblem to be solved by a development with respect to a parametercharacterizing the slenderness of the ship, the difference in the twocases would seem to be that ships with a full mid-section should beregarded as slender in both draft and beam, (otherwise no linearization basedon assuming small disturbancesin thewater would beWATER WAVES258reasonable), while a ship of Michell's type is one in which the draft isfinite and the beam is small.
In the former case a development different from the one given above would result: the mass and momentsof inertia would be of second order, for instance, rather than firston we shall have occasion to mention other possible waysof introducing the development parameter.continue by pointing out a number of conclusions, in additionto those already given, which can be inferred from our equationsorder. LaterWewithout solving them. Consider, for example, the equations (9.1.22)and (9.1.23) for the heave y l and the pitching oscillation 31 and make,the assumption thatf(9.1.24)xhdx=(which means that the horizontal section of the ship at the water linehas the e.g.
of its area on the same vertical as that of the whole ship).If this condition is satisfied it is immediately seen that the oscillationsseen to31 and yl are not coupled. Furthermore, these equations arehave the form+ AJft = p(0- q(t)31 + AJ0 81(9.1.25)*/i(9.1.26)with(9.1.27)Af=r~**=(9.1.28)r%6(yAformAcos (Ajis+ry'Lt-*--It follows that resonance*of the~)hdA-\-~\x 2 hdx_Ji:-=L.harmonic componentcomponent of the formpossible ifp(t) has aB)orq(t)a+5): in other words, one could expect exceptionallyif the speed of the ship and the seaway were to beoscillationsheavysuch as to lead to forced oscillations having frequencies close to thesevalues.
One observes also that these resonant frequencies can becos (A 2 <computed without reference to the motion of the sea or thethe quantities A 15 A 2 depend only on the shape of the hull.**ship:*The term resonance is used here in the strict sense, i.e. that an infiniteamplitude is theoretically possible at the resonant frequency.** Theequation (9.1.27) can be interpreted in the following way: it furnishesthe frequency of free vibration of a system with one degree of freedom in whichthe restoring force is proportional to the weight of water displaced by a cylinderof cross-section area 2L hdx when it is immersed vertically in water to a depth y v.THE MOTION OF A SHIP IN A SEAWAY259In spite of the fact that the linear theory presented here must beused with caution in relation to the actual practical problems concerning ships in motion, it still seems likely that such resonant frequencies would be significant if they happened to occur as harmoniccomponents in the terms p(t) or q(t) with appreciable amplitudes.Suppose, for instance, that the ship is moving in a sea-way thatconsists of a single train of simple harmonic progressing plane waveswith circular frequency a which have their crests at right angles tothe course of the ship.
If the speed of the ship is s one finds that thecircular excitation frequency of the disturbances caused by suchwaves, as viewed from the moving coordinate system (#, t/, z) that isused in the discussion here, is a* a 2 /g, since o 2 /g is 2n times the+reciprocal of the wave length of the wave train.
Thus if A x or A 2 shouldhappen to lie near this value, a heavy oscillation might be expected.One can also seethat a change of course to one quartering the waves atwouldleadto a circular excitation frequency a+s Q cos y a 2 /gangle yand naturally this would have an effect on the amplitude of the response.It has already been stated that the theory presented here is closelyrelated to the theory published by Haskind [H.4] in 1946, and it wasindicated that the two theories differ in some respects. We have notmade a comparison of the two theories in the general case, which wouldnot be easy to do, but it is possible to make a comparison rather easilyin the special case treated by Haskind in detail. This is the specialcase dealt with in the second of his two papers in which the ship isassumed to oscillate only in the vertical plane as would be possibleif the seaway consisted of trains of plane waves all having their crestsat right angles to the course of the ship.
Thus only the quantities y^t)and 31 (/), which are denoted in Haskind's paper by (0 and y>(t), are ofinterest. Haskind finds differential equations of second order for thesequantities, but these equations are not the same as the correspondingequations (9.1.22), (9.1.23) above. One observes that (9.1.22) contains as its only derivative the second derivative/}^ and (9.1.23) contains as its sole derivative a term withfirstderivative terms atall,31 ;in otherand the couplingwords there are noarises solelythroughthe undiffcrentiated terms.
Haskind's equations are quite differentsince first and second derivatives of both dependent functions occurboth of the two equations; thus Haskind, on the basis of his theory,can speak, for example, of damping terms, while the theory presentedhere yields no such terms. On the basis of the theory presented so farthere should be no damping terms of this order for the followinginWATER WAVES260reasons: In the absence of frictional resistances, the only way inwhich energy can be dissipated is through the transport of energy toinfinity by means of out-going progressing waves. However, we havealready given valid reasons for the fact that those oscillations of theship which consist solely of displacements parallel to the verticalplane produce waves in the water with amplitudes that are of higherorder than those considered in thefirstapproximation.
Thus no suchdissipation of energy should occur.* In any case, our theory has thisfact as one of its consequences. Haskind [H.4] also says, and we quotefrom the translation of his paper (sec page 59): "Thus, for a ship., only in the absencesymmetric with respect to its midship sectionof translatory motion, i.e., for s0, are the heaving and pitchingoscillations independent." This statement does not hold in our version..=of the theory.if,and onlyif,As oneochdxsees^from (9.1.22) and0,whetherSQ(9.1.23) coupling occursvanishes or not.
In addition,Haskind obtains no resonant frequenciesin these displacements because of the presence of first-derivative terms in his equation; theauthor feels that such resonant frequencies may well be an importantfeature of the problem. Thus it seems likely that Haskind's theoryfrom that presented here because he includes a number ofterms which are of higher order than those retained here. Of course, itdoes not matter too much if some terms of higher order are includedin a perturbation theory, at least if all the terms of lowest order arereally present: at worst, one might be deceived in giving too muchsignificance to such higher order terms.The fact that the theory presented so far leads to the conclusiondiffersthat no damping of the pitching, surging, and heaving oscillationsis naturally animportant fact in relation to the practical problems.
Unfortunately, actual hulls of ships seem in many cases to beoccursway that damping terms in the heaving and pitching oscillations are numerically of the same order as other terms inthe equations of motion of a ship. (At least, there seems to be experidesigned in such amental evidence from model studies see the paper by KorvinKrukovsky and Lewis [K.I 6] which bears out this statement.)Consequently, one must conclude that either actual ships are not* Itis, however, important to state explicitly that there would be dampingof the rolling, yawing, and swaying oscillations, since these motions create waveshaving amplitudes of the order retained in the first approximation, and thusenergy would be carried off to infinity as a consequence of such motions.THE MOTION OF A SHIP IN A SKA WAY261sufficiently slender for the lowest order theory developed here to applywith accuracy, or that important physical factors such as turbulence,viscosity, etc., have effects so large that they cannot be safely neglected.
If it is the second factor that is decisive, rather than the lossof energy due to the creation of waves through pitching and heaving,it is clear that only a basic theory different from the one proposedhere would s6rve to include such effects. If, however, the dampinghas its origin in the creation of gravity waves we need not be entirelyhelpless in dealing with it in terms of the sort of theory contemplatedhere.
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