Observation point

It is well established that GO approximation leads to aceurate results if both the source and the observation points are not close to the interface. In practice, this means that both points must be distant from the interface of at least one wavelength. This condition is always fulfilled for the source point. For the field-point, the accuracy of the solution is ensured if the above condition is completed, as shown by comparing exact and approximate results [5]. 

Of partieular interest are those surfaees where AFM has provided eomplementary infonnation or revealed surfaee stnieture whieh eould not be obtained by STM. One obvious applieation is the imaging of insulators sueh as NaCl(OOl) [120]. In this ease it was possible to observe point defeets and themially aetivated atomie jump proeesses, although it was not possible to assign the observed maxima to anion or eation. 

Cooling towers are commonly used for water cooling, but they can also be used for heat recovery from outlet air. If the water temperature is higher than the dewpoint of the air, water will cool in the tower. Cooling is caused by vaporization on the surface of the water drops. The vaporization energy comes from the inner energy of the water and in a certain phase, when the water temperature is lower than the dry bulb temperature of the air, also from the airflow. When the water temperature drops to near the air wet bulb temperature at the observation point,... 

Saint-Venant stated that two different loadings that are statically equivalent produce the same stresses and deformations at a distance sufficiently far removed from the area of application of the loadings. Thus, if two statically equivalent loadings are applied and the observation point is near the end where the loading is applied, then the stresses and deformations will be different for each loading. Hence the name Saint-Venant end effects. 

The aforementioned inconsistencies between the paralinear model and actual observations point to the possibility that there is a different mechanism altogether. The common feature of these metals, and their distinction from cerium, is their facility for dissolving oxygen. The relationship between this process and an oxidation rate which changes from parabolic to a linear value was first established by Wallwork and Jenkins from work on the oxidation of titanium. These authors were able to determine the oxygen distribution in the metal phase by microhardness traverses across metallographic sections comparison of the results with the oxidation kinetics showed that the rate became linear when the metal surface reached oxygen... 

Po is the far-field air density r designates the distance separating the compact flame and the observation point... 

If the observation point is in the far-field and if the source region is compact, the previous expression becomes... 

The rate of phosphoprotein formation in the presence of 5 mM CaCl2 was only slightly affected by mild photooxidation in the presence of Rose Bengal, but the hydrolysis of phosphoenzyme intermediate was inhibited sufficiently to account for the inhibition of ATP hydrolysis [359]. The extent of inhibition was similar whether the turnover of E P was followed after chelation of Ca with EGTA, or after the addition of large excess of unlabeled ATP. These observations point to the participation of functionally important histidine residues in the hydrolysis of phosphoprotein intermediate [359]. 

Assuming that the distribution of masses inside the volume V is given, this vector function g p) depends only on the coordinates of the observation point p, and by definition it is a field. It is appropriate to treat the masses in the volume V as sources of the field g p). In other words, these masses generate the field at any point of the space, and this field may be supposed to exist whether a mass is present or absent at this point. When we place an elementary mass at some point p, it becomes subject to a force equal to... 

Repeating these calculations with different pairs of gx(x) we may increase the accuracy of the evaluation of h. Next, making use of the value of this component at any point, the mass m is evaluated. In the case when only the vertical component is known, the determination of the position of mass and its value is similar. Here it is appropriate to notice the following. Inasmuch as an arbitrary body, located at a large distance from an observation point p, creates a field, known always with some error, often it cannot be practically distinguished from that of an elementary particle, and for this reason we are able to determine only the product of volume and density, mass, but each of them remains unknown. It is the first illustration of the fact that the solution of the inverse problem in gravity, as well as in other geophysical methods, is an ill-posed one, because some parameters of a body... 

As was pointed out earlier. Equation (1.6) allows us to find the attraction field everywhere, but it requires a volume integration, that in general is a rather cumbersome procedure. Fortunately, in many cases the calculation of the field g(p) can be greatly simplified. First, consider an elementary mass with density 6 q), located in the volume AV. Now let us start to increase the density and decrease the volume in such a way that the mass remains the same. By definition, these changes do not make a noticeable influence on the field because the observation point p is far away. In the limit, when... 

Here, q> and j/ are continuous functions with continuous first and second derivatives, p is an observation point where the potential is determined, and q is an arbitrary point. 

Here Lqp is the distance between points q and p. Note that G q, p) is called a Green s function. There are an infinite number of such functions and all of them have a singularity at the observation point p. Inasmuch as the second Green s formula has been derived assuming that singularities of the functions U and G are absent within volume V, we cannot directly use this function G in Equation (1.99). To avoid this obstacle, let us surround the point by a small spherical surface S and apply Equation (1.99) to the volume enclosed by surfaces S and S, as is shown in Fig. 1.9. Further we will be mainly interested by only cases, when masses are absent inside the volume V, that is. 

That is, functions U q) and dU q)ldn on the surface S approach their values at the observation point, respectively. Also from Fig. 1.9 it follows that the normal n and radius vector r on this surface are opposite to each other, and therefore for points on this surface we have... 

Then, applying the mean value theorem, the surface integral around the observation point p can be represented as... 

Here q is the point of the boundary surface and p the observation point. 

Here. s is a point located somewhere inside the sphere along the line Op, p and sq the distances between points q, p and q, s, respectively, and a. a constant. Now we demonstrate that for each observation point p it is possible to find such a point and coefficient a, that the Green s function is equal to zero at all points of the sphere. In order to prove it, we first choose the position of the point from the condition ... 

Here m is the mass enclosed by the surface of integration, S. As follows from Equation (1.126), the field outside coincides with that caused by a point source with the same mass, M, located at the sphere center. This is a well-known result, which is hard to predict. In fact, this behavior occurs regardless of how close the observation point is to the sphere, and it results from the superposition of fields, caused by elementary masses. This is rather an exception, since in general the field differs from that generated by an elementary mass. 

Certainly, Equations (1.127 and 1.128) can be derived directly from Equation (1.6), but it requires a rather cumbersome integration. This example also allows us to illustrate the fact that the gravitational field has a finite value inside any mass. With this purpose in mind, imagine that the observation point p is located at the center of a small and homogeneous sphere. Fig. 1.12a. Then, the total field can be represented as a sum ... 

Suppose that masses are distributed within a plane layer whose thickness is much smaller than the distance from these masses to the observation point. Fig. 1.13a. In other words, the distance between the observation point and any point of the elementary volume is practically the same. Taking into account this fact, we can replace this layer by a plane surface with the same mass, located somewhere at the middle of the layer. Fig. 1.13b. Inasmuch as every elementary volume contains the mass dm — 5(q)hdS,its distribution on the surface can be described by dm — a(q)dS, where... 

First, introduce a Cartesian system of coordinates with its origin at the middle of the layer and z-axis directed perpendicular to its surface. Let us note that the layer has infinite extension along the a and y axes, (Fig. 1.14a). At the beginning, suppose that the observation point is located outside the layer, that is, z >h/2. Then we mentally divide the layer into many thin layers which in turn are replaced by a system of plane surfaces with the density a — 5Ah, where Ah is the thickness of the elementary layer. Taking into account the infinite extension of the surfaces, the solid angle under which they are seen does not depend on the position of the observation point and equals either —2n or 2%. Correspondingly, each plane surface creates the same field ... 

It is interesting to notice that these formulas are used to calculate the Bouguer correction. Now we will study the attraction field inside a layer when the coordinate of the observation point z satisfies the condition, (Fig.. Ah) z0. Then, the total field can be presented as a sum of two fields one of them is caused by masses with thickness equal h/2 — z,and the second one is due to masses in the rest of the layer, having the thickness z + A/2, (Fig. 1.14b). In accordance with Equation (1.146) these fields are... 

The behavior of the field g caused by masses of the layer is shown in Fig. 1.14c. Thus, for negative values of z the field component g inside the layer is positive, since the masses in the upper part of the layer create a field along the z-axis, and this attraction prevails over the effect due to masses located below the observation point. At the middle of the layer, where z = 0, the field is equal to zero. Of course, every elementary mass of the layer generates a field at the plane z = 0, but due to symmetry the total field is equal to zero. For positive values of z the field has opposite direction, and its magnitude increases linearly with an increase of z. As follows from Equations (1.146-1.148) the field changes as a continuous function at the layer boundaries. 

One more feature of the field behavior is worth noting. Inasmuch as the layer has infinite extension in horizontal planes, the distribution of masses possesses axial symmetry with respect to any line parallel to the z-axis that passes through the observation point. For this reason, it is always possible to find two elementary masses such that the tangential component of the field caused by them is equal to zero. Respectively, the field due to all masses of the layer has only a normal component gz. 

In studying the attraction and gravitational fields of the earth it is very useful to represent the potential in terms of Legendre s functions and this is mainly related to the fact that the earth is almost spherical and the position of observation points is... 

Here R is the distance between an observation point p and the center of mass, where the origin of the spherical system of coordinates is located. R is the distance of an elementary mass from the origin Lqp is the distance between this mass and the point P-... 

Thus, measuring the spring extension we can evaluate the gravitational field. However, the main goal of the static gravimeter is to determine the change of the field as a function of the observation point. Suppose that g2 and g are field magnitudes at two points. Then, in accordance with Equation (3.101), we have... 

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