ance will assume serious proportions relatively to the other factors of the resistance. Very similar remarks apply to the relations which should be secured between the length of the "run" of a ship-upon which the length and natural speed of the stern wave depend —and the maximum speed at which the ship is to be driven. Unless the length of the run is sufficient, a serious increase in the wave-making resistance may be looked for as the full speed is approached. With the same lengths of entrance and run, very different forms and proportions of length to breadth may have to be associated, in order to fulfil changed conditions. Two ships, for example, may have nearly the same area of immersed skin, but in the one the proportion of length to breadth may be much smaller than in the other; and although the lengths of entrance and run correspond in the two cases, the water-lines of the shorter ship may be bluffer than those of the longer ship. It would then be reasonable to suppose that the frictional and eddy-making resistances of the two ships would be practically equal for any assigned speed; but the wave-making resistances might differ considerably, the shorter, bluffer ship creating much greater disturbance. And this increase in wave-making resistance might have to be incurred in order that other conditions, even more important than those relating to diminution of resistance, might be fulfilled. Hence it appears that in ships having lengths of entrance and run amply sufficient to fulfil the condition stated above, differences in forms and proportions of the bows and sterns may produce corresponding differences in the heights of the bow and stern waves, as well as in the forces required to create and maintain such waves, these forces varying with the heights. At present, however, we are not concerned with this general question of the variation of resistance produced by changes in the form and proportions, but simply desire to explain the important practical rule that certain minimum lengths of entrance and run proportioned to the full speeds should be provided, and can be provided, in nearly all ships, no matter what forms and proportions may have been determined upon in order to fulfil their special conditions of service. Unless these minimum lengths are secured, the wave-making resistance at the full speeds must assume undue importance. The laws which govern the wave-making resistance are not yet fully understood; but the following explanation of the rapid increase in the magnitude of that resistance, which takes place when a certain limit of speed is exceeded, appears reasonable. If the lengths of entrance and run are sufficient for the intended full speed, the bow and stern waves, once formed, will have such a natural speed that they can travel with the ship, causing but little additional resistance by the expenditure of power required to maintain them. The case is parallel to that of the deep-sea waves maintaining their speeds while they travel long distances; with this difference, that the bow and stern waves have to be maintained at their full heights, at the expense of a virtual increase in the resistance of the ship. On the other hand, if the lengths of entrance and run are insufficient for the intended full speed, the natural speed of the bow and stern waves will be less than the full speed of the ship, and in order that the waves may accompany her, their speed will have to be accelerated. Hence it follows that, instead of travelling with the ship, these slower-moving waves diverge from her path, carrying off into still water the energy she has impressed upon them. The ship has therefore to be continually creating new waves as she advances, the power thus expended being a virtual increase in the resistance, often of a serious character. This cause of increased resistance begins to operate only when the speed of the ship exceeds the natural speed of the waves corresponding to the lengths of entrance and run; it may therefore be anticipated that in a vessel, defectively formed as to the length of entrance or length of run, the law of her resistance will undergo a sudden change when her speed passes beyond that of the natural speed of her bow and stern waves. Numerous experiments have confirmed the general accuracy of this view of the subject; and one or two illustrations will be given hereafter. Mr. Scott Russell first drew attention to the importance of wave-making resistance, and its relation to the length of entrance and run. His researches on this subject furnished the data upon which his well-known "wave-line theory" of constructing ships was based.* This theory has not greatly influenced the practice of naval architects, nor is it generally accepted as an accurate representation of the phenomena of resistance; but it has the great merit of having enforced the importance of, and given practical rules for, proportioning the lengths of the entrance and run to the intended speeds of ships; and these rules deserve mention. The lengths of entrance and run are measured as before described. The entrance is also termed the "fore body "; the run, the "after body"; and if amidships there is a certain length of constant cross-sectional form, it is termed the "middle body." The length of the entrance, it is considered, should be equal to the length of the "wave of translation," of which the natural speed equals the maximum speed for which the ship is designed. This wave of translation differs from a trochoidal wave in being wholly situated above the still-water level, travelling as a heap of water, and not having hollows depressed below the still-water level. But for deep water, and for the small heights which waves attain when travelling with ships, no error of practical importance is involved in estimating the period and speed of waves of translation by the rules previously given for trochoidal waves. shallow water there would be a necessity for considering the waves of translation separately, and also for altering the rules given for the trochoidal deep-sea waves; but into these special circumstances it is not necessary to enter, since they In For particulars of this theory, the Transactions of the Institution see Mr. Russell's work on Naval of Naval Architects. Architecture; also vols. i. and ii, of are important only in vessels designed for river or shallowwater service, and scarcely affect sea-going ships. Treating the wave of translation as a trochoidal wave in the relation of its length and velocity, the rules of Mr. Scott Russell may be stated in the following simple form:-Let V be the maximum speed of the ship (in knots per hour); L1 be the length of entrance appropriate to the speed V, and L2 the length of run (both lengths being expressed in feet): then For example, let V=15 knots, then, to avoid undue wavemaking, the theory prescribes : Length of entrance = 0·562 × 152 = 126 feet; = 0·375 × 152 = 84 feet. With these dimensions might be associated any required length of middle body, the additional resistance for the assigned speed being chiefly due to friction on the enlarged immersed surface.* Of these two rules, that relating to the length of run is thought to have the greatest practical importance, many successful vessels having had a less length of entrance than that prescribed by the formula; whereas vessels with shorter runs than the formula prescribes have done badly. As a matter of fact, however, sea-going vessels usually have greater lengths both of entrance and run, in proportion to their maximum speeds, than are required by these rules; and instead of having the run only two-thirds as long as the entrance, the lengths of entrance and run are commonly equal, or nearly so. It will be observed from the preceding formula that whence V2 = 1·067 (L1 × L2); and V = 1·03 ✓ L1+L2 (nearly). *See further on this subject the recent experiments of Mr. Froude, mentioned at page 450. So far as can be seen at present, this last equation enables a fair approximation to be made to the speed (V) at which a small increase in speed causes an increase in resistance altogether disproportionate to that which would accompany an equal increase in speed when the vessel was moving more slowly. Putting the equation in this form allows for any variations which may be desirable in practice in the ratio of the length of entrance to that of run; although neither of these can become very short in proportion to the speed without producing increased resistance. Suppose, for instance, that the common practice is adhered to, and the lengths of entrance and run made equal to one another: it may be desired to know what are the lengths appropriate to a speed of 16 knots. Here L2+L2 = 0·937 × (16)2 = 240 feet (nearly). The Inconstant, which has a total length of 337 feet, and a speed of 16 knots on the measured mile, might therefore have about 90 feet of parallel middle body, without encountering this undue growth of wave-making resistance; but other considerations might make it preferable to avoid having any parallel middle body. Professor Rankine suggested another mode of approaching the investigation of the speed at which wave-making resistance begins to grow at a very disproportionate rate. He proposed to express the natural speed of advance of the waves raised by a ship in terms of the "virtual depth of disturbance" which she causes, and considered that this virtual depth must bear some relation to the " mean depth of immersion" of the part of the ship which creates the wave. Taking the after body of a ship, for example, he would estimate its displacement (in cubic feet), and divide this by the area of the water section of that part (in square feet), the quotient expressing the mean depth of immersion. But he owned that experiment alone can decide what the relation may be between this mean depth and the natural speed of the waves created. From a few calcula |