Suppose the depth to which the tubes are plunged into the water to be represented by k; then the verti cal pressure of the water at the orifice b of the straight tube, when the water is at rest, is in proportion to this depth, and causes the water in the tube to rise to the level of the surrounding water; but when the water moves with a velocity due to the height z, the par ticles no longer press equally in all directions, but have a greater tendency to motion in the direction of the current than in any other; so that the vertical pressure of the particles at the orifice b is less than before, and by the experiment is found to be propor tional to k —z.
In the application of the result of this experiment to a floating parallelopiped, whose sides are perpendicu lar, and whose upper and lower surfaces are parallel to the water's surface, the pressure of the water on the sides being horizontal, has no effect in supporting its weight; and the vertical pressure of the particles of the water on the lower surface, being less when in motion than when at rest, in the proportion of k to k —z, k being in this case the perpendicular distance of the lower side of the body from the water's sur face, and z as before, the height due to the \ elocity of the current, the parallelopiped will sink deeper in the running water. than in the still water, in the same proportion: that is, the perpendicular depth of the immersed part of the body will be k+z, having sunk deeper the distance z.
When the bent tube c d e is placed with its lower end in the direction of the stream h i, the effect is the same as with the straight tube; the particles of water at the orifice e, pressing less on the particles in the tube when the water is in motion than when at rest, the water in the tube is not equally supported; so that it sinks be low the level of the surrounding water, a distance found by the experiment to be equal to z, the height of the water in the tube being k— z. The effect is the same also when the lower end of the bent tube is placed perpendicularly to the current; but when placed with its orifice presented to the direction of the current, the particles of the water in motion exert a pressure at the orifice e, greater than they would when at rest, in con sequence of the velocity in the direction of their mo tion, which causes the confined water in the tube to rise above the level of the surrounding water, a height found by the experiment to be equal to z, the altitude of the water in the tube being k+z.
Now as the water rose a distance z above the level of the surrounding water, when the lower end of the bent tube was placed exactly in the direction opposed to the current, and fell the same distance z below the level of the surrounding water, when the lower end of the tube was placed perpendicularly to the direction of the current, there must be an angle at which the tube might be placed with respect to the direction of the current, at which the water in the tube would be at the same height as the surrounding water. Taking
any line v in the direction of the current to represent its velocity, which is wholly effective in raising the water in the tube, when placed in the opposite direc tion to the current the distance z, and which has the effect of depressing the water in the tube the same dis tance z; when placed perpendicularly to the direction of the current, the angle at which the tube must be placed, in order that such a part of this velocity may be effective in causing the water in the tube to rise ex actly to the level of the surrounding water, may be found by supposing that at this angle the effective part may be equal to v, which, by the resolution of the directions of the pressures, makes the angle at which the tube must be placed 60° with the direction of the current.
In the application of this reasoning to the determi nation of the vet tical pressure of the water in motion on a ship's body, the pressure on the fore and after parts of the body must be considered separately; the greatest transverse section called the midship section, being the division between these parts.
The expression representing the pressure of the water on the fore part will be composed of two terms, the one expressing the pressure on the part of the body where it is greater than it would be if the body were at rest, and the other the pressure on the pal t of the fore body, where it is less than it would be if the body were at rest. The line of division, which we will call the neutral line, being the line on the fore part of the n ship's body, at which the pressure of the water is ei ther increased nor diminished by the velocity of the water, will be a curved line, depending on the form of the ship's body, but always before the greatest trans verse section. In regular figures, its position and form may be determined either geometrically or an alvtically; but in ships, can be found only by trial and calculation. In the expression for the pressure of the water on the part of the body contained between the neutral line and the mioship, the pressure repre sented by the proportional depth k will be increased by a function of z; in the expression for the part of the fore body contained between the neutral line and the midship section, the pressure represented by the proportional depth k will be diminished by a function of z: and in the after body, k—z will be the element representing the pressure.