Potential of double-layer

Let us imagine in a not potentiant unspecified Q a Dipôle gravific formed of two specific masses of opposed signs, M₊ = +M_Q and M_- = - M_Q, located on both sides Q at a distance ½ h.

A priori , the concept of such a dipole gravific appears stripped of physical significance, since the gravitation is always exerted like an attraction force, not like a force of repulsion. That implies that in theory of the gravitation there do not exist negative masses, contrary to what occurs in the theories from electrostatics and the magnetostatic one, where the consideration of electric and magnetic dipoles is of an major importance. Nevertheless, the gravific concept of dipole has a great utility in physical geodesy where one is brought to consider excesses or defects of density compared to a standard value, and to reason like if these mass excesses or defects would constitute in fact of the additional masses causing in a point potentié P a “element of potential of double-layer” δV_2L which is superimposed on the potential gravific out of P generated by the distribution of standard mass.

The combined potential generated out of P by the two masses of the dipole is, By indicating by r₊ the distance from P with M₊ and by r_- the distance from P with M_-, the combined potential generated out of P by the two masses of the dipole is

δV_2L (P) = GM_-/r_- + GM₊/r₊ = GM_Q (1/r₊ - 1/r_-).

The function between brackets is obviously a function of the distance H between the masses M_- and M₊, and is cancelled with h. By indicating M_- the point occupied by the M_- mass, M₊ that occupied by the M₊ mass, we have

M₊P = M_-P - M_-M₊

R ₊ = R _- - h' ,

the direction N being the unit vector pointing of M_- with M₊.

When H → 0 one thus has

r₊ ≈ r_- - h' ⋅ R _-/r_- and 1/r₊ ≈ 1/r_- + h' ⋅ R _-/r_- ³,

i.e.

1/r₊ - 1/r_- = h' ⋅ R _-/r_- ³.

The vector quantity R _-/r_- ³ is the gradient of 1/r_- evaluated at the M_- point. The expression ⋅ R _-/r_- ³ thus represents the derivative of 1/r_- in the direction N . We will note in the continuation derivation in the direction N by d/dn, and we obtain consequently

δV_2L = G Γ D (1/r) /dn

while posing

R = r_-

and

Γ (Q) = M_Q h.

Γ (Q) is the dipole moment gravific at the point Q. One supposes that the masses M_- and M₊ indefinitely grow when H tends towards zero, so as to give at this time dipolar a finished value. The relation δV_2L = G Γ D (1/r) /dn provides the potential of such an elementary dipole.

One can represent a double-layer on a surface ∂B like two simple layers separated by a small distance h. From a mathematical point of view, one will regard this distance as infinitesimal. The unit normal external to surface ∂B, namely N , intersects the two layers in two points M_- and M₊ very close one to the other and having surface densities ϰ of the same magnitude, but to opposite signs. Thus, each couple of corresponding points M_-, M₊ forms a dipole having a density of dipole moment gravific

γ = dΓ/dσ.

By adding the contributions δV_2L (P) with all the dipoles which one supposes distributed continuously on surface ∂B, one obtains the potential of double lay down :

V_2L (P) = G ∫_∂B γ (Q) D (1/r_PQ) /dn dσ (Q).

This potential is continuous safe everywhere on surface ∂B. On this surface, we obtain two limit different for the potential, according to whether we approach the outside or the interior of surface. External side, the potential is worth

V_2Le (P₊) = - 2πG γ (P) + ∫_∂B γ (Q) D (1/r_PQ) /dn dσ (Q),

and on the interior side, it is worth

V_2Li (P_-) = +2πG γ (P) + ∫_∂B γ (Q) D (1/r_PQ) /dn dσ (Q),

so that it has a discontinuity of

_-⁺ = V_2Le (P₊) - V_2Li (P_-) = - 4πG γ (P)

when one passes at the point P interior outside surface ∂B.

The preceding relations for the potential of double-layer are similar to the relations providing the normal derivative of the potential of simple layer. But it is advisable to understand well that these two types of potentials associated with a material surface have, in fact, of the very dissimilar behaviors, to some extent complementary. Indeed, the potential of simple layer is continuous everywhere, and its normal derivative is discontinuous on material surface. On the other hand, the potential of double-layer is itself discontinuous on material surface, but its normal derivative is continuous. However, at long distance from material surface, the two potentials have similar behaviors, since they decrease like 1/r and are cancelled ad infinitum.

It is the existence of a discontinuity, in the potential of double-layer itself on the one hand, in the normal derivative of the potential of simple layer on the other hand, which makes the concepts at the beginning purely mathematical of potentials of simple layer and double-layer so interesting for the Géodésie, in particular in relation to the theorems of Green.

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