Level surfaces

Nitrtir, n. (-ous) nitride. Cf. Chloriir. Nitrylsaure,/. nitrylic acid (nitrous acid). Niveau, n. level, -flache, /. level surface, -flasche,/. leveling bottle, leveling vessel (as for a gas buret), -rohr, n., -rohre, /. level tube, leveling tube, -stufe, /. (of electrons) energy level. ... 

Wasser-festigkeit, /. waterproofnesa. -fiache, /. water level surface of water sheet of water, -flasche, /. water bottle, -fleck, m. water stain,... 

Sub-base selected imported granular material to form a stable level surface on which to construct the slab. Where lean-mix concrete is not used the surface is normally blinded with sharp sand ... 

In contrast chemical and electrolytic polishing enables a smooth level surface to be produced without any residual stress being developed in the surface because the surface is removed by dissolution at relatively low chemical potential and at relatively low rates is such a way that metallic surface asperities are preferentially removed. For this to be most effective the solution properties must be optimised and the pretreatment must leave an essentially bare metal surface for attack by the electrolyte. 

Orange Peel the pock-marked appearance, in particular of a sprayed film, resembling the skin of an orange due to the failure of the film to flow out to a level surface. (See also spray mottle.)... 

Electropolishing surface finishing of a metal by making it the anode in an appropriate solution, whereby a bright and level surface showing specular reflectivity is obtained. 

On the level surface S, surrounding this volume, we have... 

By definition, the reaction field gs at any point p is normal to its level surface at this point and for this reason the direction of gs is called the vertical that corresponds to the direction of a plumb line. 

As was pointed out in the Chapter 1, it is very useful to study a scalar field with the help of equipotential or level surfaces. At each point of such a surface the potential is constant, and correspondingly its equation is... 

Here Uq is the value of the potential on the surface. It is proper to notice that potentials of the attraction field and the centrifugal force usually vary on the level surface of the gravitational field. Changing the value of the constant, Uq, we obtain different level surfaces, including one which coincides in the ocean with the free undisturbed surface of the water and, as was pointed out earlier, this is called the geoid. As follows from Equation (2.73) the projection of the field g on any direction / is related to the potential U by... 

To study geometric features of a potential field in detail, consider the curvature of the level surfaces. As is well known, the curvature of a curve y — f(x) is defined as... 

In the same manner we obtain an expression for the curvature of the function z = z(y), which is the intersection of the level surface at the point p with the zy-plane ... 

It is proper to point out again that the simplicity of Equations (2.87 and 2.88) is related to the fact that the plane xOy is tangent to the level surface at the point p and, correspondingly, the plumb line coincides with the z-axis. 

This equation was derived by Bruns, and it establishes a relation between the derivative of the field along the vertical and the mean curvature of the level surface. 

The similarity with the case of the level surface is obvious. From Equation (2.91) we obtain for points in the vicinity of p... 

We see that the gradient of the density and that of the gravitational field are parallel to each other. This means that at each point the field g has a direction along which the maximal rate of a change of density occurs. The same result can be formulated differently. Inasmuch as the gradient of the density is normal to the surfaces where 5 is constant, we conclude that the level surfaces U = constant and 5 — constant have the same shape. For instance, if the density remains constant on the spheroidal surfaces, then the level surfaces of the potential of the gravitational field are also spheroidal. It is obvious that the surface of the fluid Earth is equip-otential otherwise there will be tangential component of the field g, which has to cause a motion of the fluid. But this contradicts the condition of the hydrostatic equilibrium. 

Taking into account the fact that on the surface of the ellipsoid e = eq and d — dg we arrive at the obvious result, namely, at the level surface the tangential component of the gravitational field vanishes, — 0. Next, consider the component Again differentiating Equation (2.153) we obtain... 

By definition, we have for the field magnitude on this level surface ... 

This formula was introduced by the Italian geodesist Somigliana in 1929 and defines the behavior of the gravitational field on the level surface of the spheroid for any distribution of density inside, as long as the outer surface remains equipotential. 

This equation describes any level surface of the potential U of the gravitational field y, where x, y are coordinates of a point on the surface, while C is the value of the potential. At the same time, the potential of the attraction field varies on this surface. Our next step is to represent the left hand side of Equation (2.195) in the spherical system of coordinates and then, using Equation (2.192), obtain the equation of the equipotential surface, which coincides with the outer surface of the earth spheroid. As was shown earlier, the potential related to a rotation is... 

The Equation (2.231) describes the potential of the gravitational field at points where R>a. In order to obtain the equation of the level surface of the potential which coincides with the surface of our model of the earth, we have to let / = C in Equation (2.231) and specify the potential at one point of this surface. With this purpose in mind assume that R — a and tp = 0. Substitution of these values in Equation (2.231) gives... 

Correspondingly, the equation of the level surface of the potential which closely describes the earth s figure is... 

As we know, the behavior of the potential W p) of the gravitational field can be illustrated with the help of the equipotential or level surfaces... 

As we know, the quantity Ah is called the elevation or an orthometric height of the point b with respect to point b. In particular, if both points are located at the same level surface, this elevation is zero. If we take a different path, for instance, aa b, Equation (2.255) gives a height of the point a with respect to a, which can differ from the first one. 

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