E C G Fig. 51. X B from D to ; and if the depressing force be now withdrawn, the water will fall from н with a velocity corresponding to its height above G, and will be carried by its momentum above o to E, just as the ball of a pendulum ascends in its arc by the momentum it possesses-and the water will continue to oscillate up and down H like the ball of a pendulum, until it is finally n brought to rest by friction. If the tube be of equal bore throughout and be bisected in o, then as the accelerating force is the difference in the masses of the two unequal columns divided by their sum, the accelerating force will be represented by E G divided by o A B D, or what is the same thing, by EA BF; or it will be proportional to the half of this, or to E o divided by o a o. The time of the oscillation or the time in which the surface of the water will fall from the highest to the lowest point, is equal to that in which a pendulum of the length o A O makes one vibration. Hence the time in which the surface will pass from the highest point to the lowest, and to the highest again, will be that in which a pendulum of the length o A o will make two vibrations, or it will be that in which a pendulum of four times that length makes one vibration, or a centrifugal pendulum of the height equal to o Ao makes one revolution. These relations equally hold, if we suppose the same kind of motion which exists in the water to be produced by a piston at o; and the side of the ship may be supposed to be such a piston, and if properly formed, the ship will impart sideways to the water precisely the same kind of motion which exists in the case here illustrated. If a sheet of paper be drawn vertically behind a pendulum furnished with a tracing point, then if the pendulum be stationary, the tracing point will draw a straight line represented by the dotted lino fig. 52. But if the pendulum be put into motion, then the tracer will describe the waving line ABOD SHARPNESS SHOULD VARY WITH SPEED. 421 where the point A answers to the stem of a ship, the point в to the midship frame, and the point o to the stern; and the paper will pass from A to o during the time the pendulum makes two oscillations. Since the pendulum has to make two oscillations while the vessel passes through a distance equal to her own length, the combined motions of the tracer and pencil will delineate the proper form for the side of the vessel; and if made in this form the particles of water will have the same motion as the ball of a pendulum, which motion enables the water to be moved with the minimum of loss. It will be useful, however, to take a particular case to show in what manner the proper form may be practically determined. Fig. 53. D A P Suppose A c, fig. 53, to represent the keel of a vessel-which we may take at 200 feet long and 40 feet wide -and which is intended to maintain a speed of 10 statute miles per hour, or 880 feet per minute. Now as the vessel has to pass through her length, or from a to o, during the time that the pendulum P makes double beat, or to pass from A to B, which is 100 feet, during the time the pendulum make a single beat, there will be 880 divided by 100, or 8.8 vibrations of the pendulum per minute; and the rod of the pendulum must be of such length as to produce that number of vibrations. Now to determine the length of the rod of a pendulum which shall perform any given number of vibrations per minute, we divide the constant number 375.36 by the number of vibrations per minute, and the square of the quotient is the length in inches. Hence 375 36 divided by 8.8 = 42.6, the square a Ja B で of which is 1814.76 inches or 151.23 feet, and a pendulum 151.23 feet long beating in an arc 20 feet long with the paper travelling at a speed of 880 feet per minute, will describe the line A B 0, which will be the proper water-line for the side of a ship if the cross-section be rectangular; and whatever the form of cross section this figure will equally determine the proper area of cross-section at each successive frame. If instead of moving at 10 miles an hour, the vessel has only to move at the rate of 5 miles an hour, the figure described will be that represented by Da E, and the breadths bb in the longer figure and b′ b' in the shorter are the same, both being equal to half the breadth at a a. The rod of the pendulum PP passes through the point b, and the pendulum vibrates from the plane of the keel to the plane of the side, so that the chord of the arc in which the vibration is performed is equal to half the breadth of the vessel, while the versed sine or height through which the pendulum falls at each beat, will be equal to the height of the wave at the midship frame. To find the versed sine of the arc, we divide the square of half the chord by twice the length of the pendulum. The chord being 20 feet the half of it is 10 feet; and the pendulum being 151.23 feet long the double of it is 302:46 feet, and 100 divided by 302·46 : 33 feet or 3.96 inches. The height of the wave at the midship frame, in a vessel formed in the manner indicated, will accordingly be 3.96 inches, or rather this would be the height if the water were moved without friction, so that practically the height will be somewhat greater than is here indicated. If we increase the speed of the vessel, or increase the breadth, the hydrostatic resistance will increase very rapidly. Thus, if the speed of the vessel be increased to 20 miles an hour, or 1,760 feet per minute, the pendulum will require to make 17.6 beats per minute, and its length will be 375.36 divided by 17·6 = 21·3, the square of which is 453.69 inches, or 37.8 feet. Now, 100 divided by 37·8 = 2·6 feet, which will be the height of the wave at the midship frame in this case, and the hydrostatic pressure will be the half of this, or equivalent to 1.3 feet of water acting on the breadth of the vessel. In like manner, successive additions to the breath of the vessel without increasing the length add rapidly to the hydrostatic resistance, as they involve the necessity of the oscillating particles ascending higher and higher in the arc to enable the vessel to pass. FRICTION OF WATER 423 FRICTION OF WATER. It remains to consider the friction of water upon the bottom of the vessel, and this is by much the most important part of the resistance which ships have to encounter. Beaufoy made a number of experiments to ascertain the amount of this resistance by drawing a long and a short plank through the water: and, by taking the difference of their resistances and the difference of their surfaces, he concluded that the friction per square foot of plank was, at one nautical mile per hour, 014 lbs.; at two nautical miles per hour, 0472 lbs.; at three, 0948 lbs.; four, 153 lbs.; five, 2264 lbs.; six, 3086 lbs.; seven, 4002 lbs. ; and eight, 5008 lbs. At two nautical miles an hour, the force required to overcome the friction was found to vary as the 1·825 power of the velocity, and at eight nautical miles an hour as the 1-713 power. Other experimentalists have deduced the amount of friction from the diminished discharge of water flowing through pipes. If there were no friction in a pipe, the velocity of the issuing water should be equal to the ultimate velocity of a body falling by gravity from the level of the head to the level of the orifice.* But as the velocity is found by the diminished discharge to be only that due to a much smaller height, the difference is set down as the measure of the power consumed by friction. This mode of estimating the friction is not applicable to the determination of the friction of a ship; for, in the first place, the discharge is a measure not of the maximum, but of the mean velocity; and, in the second place, there is every reason to believe that the friction per square foot on the bottom of the ship is quite different near the bow from what it is near the stern. As the water adheres to the bottom there will be a filin of water in contact with the ship, which will be gradually put *There is sometimes misconception on this subject, arising from a neglect of the difference between the ultimate and mean velocities of a falling body. Thus, if water flows from a small hole in the side of a cistern, the water will issue with the ultimate velocity which a heavy body would acquire by falling from the level of the head to the level of the orifice, which, if the height be 16 feet, will be 82 feet per second. The mean velocity of falling, however, is only 16 feet per second, so that the ultimate velocity is twice the mean velccity. into motion by the friction; and the longer the vessel is the less will be the friction upon a square foot of surface at the sternseeing that such square foot of surface has not to encounter stationary water, but water which is moving with a certain velocity in the direction of the vessel. The film of water moving with the vessel will become thicker and thicker as it passes towards the stern, and it will rise towards the surface by reason of the virtual reduction of weight consequent upon the motion. The whole of the power, therefore, expended in friction is not lost, as the power expended in the front part of the vessel will reduce the friction of the after part; added to which, the rising current which the friction produces may be made to aid the progress of the ship, if we give to the after-body of the ship such a configuration as to be propelled onward by this rising current. Finally, when the screw is the propelling instrument, the slip of the screw will be reduced, and may even in some cases be rendered negative, by the circumstance of the screw working in this current; and whatever brings this current to rest will use up the power in it, and so far recover the power which has been expended in overcoming the friction. In my investigations respecting the physical phenomena of the river Indus in India, I observed that the water not only ran faster in the middle of the stream, but that it also stood higher in the middle, so that a transverse section of the river would exhibit the surface as a convex line. At the centre of the river the stream is very rapid, but it is slow at the sides, so that boats ascending the river keep as close as possible to either bank; and in some parts at the side there is an ascending current forming an eddy. I further observed, that not merely were there rapid and considerable changes in the velocity, which I imputed partly to the agency of the wind in deflecting the most rapid part of the current to the one side or the other of the river, but there were also diurnal tides; or, in other words, the stream ran more swiftly in the afternoon than in the early morning. This had been long before observed, and was imputed to the heat of the sun melting the snows in the mountains more during the day than during the night. But although such an effect might be |