Revisiting Arther Electrodynamics

ihor kotchergenko

Revisiting Arther Electrodynamics

ihor kotchergenko

PROCEDINGS OF THR ROYAL SOCIETY a

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At ICM 2018-International Conference of Mathematicians the writer presented a poster showing that shear waves perform a deformation that combines harmonic transversal displacements and rotations. Each wavelength presents two rotations in opposite directions. Whittaker in his book “A History of the Theories of Aether and Electricity” says that Maxwell expressed that “induced me to regard magnetism as a phenomenon of rotation and electric currents as phenomena of translation”. The writer shows precisely this, applying the areolar theory of elasticity, which enables expressing both shear and displacements through rotations only. The writer hopes that this theory will provide aids for new experiments and deeper interpretation of electromagnetic phenomena. Several results for elastic shear waves in solids are available and may be useful. A presentation of the areolar theory of elasticity, witch is behind the exposition is included for completeness. The writer recommends that the reader start reading section 3, which presents the electromagnetic model, and for details of the mathematics involved, read from beginning.

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The last expression presupposes that z tends to 0 z , maintaining the direction  as z tends to z0. Equation 1

Equations (2) are known as Kolosov-Wirtinger [13] derivatives. Written in the complex form the areolar strain given by Eq. (1) reads

The real part of the areolar strain represents a radial strain while the imaginary part represents either, a circumferential strain or a rotation, see Fig. 2. In this conception, the plane strain is decomposed into two orthogonal complex strain components .

Figure 2

Fundamental modes of the plane strain

, can be converted into its canonic form

is a total differential. Consequentially, the displacement field must comply with the condition of continuity ) (

Separating the real and imaginary parts of this equation, the following compatibility equations are obtained:

Equilibrium equations

For isotropic material,

. Thus the work expression given by Eq. 4

where  and  are Lame's elastic material constants. The Euler equations for this functional are

where the symbol 2  stands for the Laplacian operator, Lamé's homogeneous equilibrium equations can be cast into the complex form:

Traction vector

Applying Green's formula in the complex form (e.g.

the traction vector on a boundary C results

where  points towards the direction of the vector element of arc dz of the closed contour curve C in the counterclockwise direction. Observe that if n is the unit outward vector, normal to the element of arc (14) .with

Ti   (15) With this concept, traction became an analytic function. This approach has important reflections in contour mapping applications.

Gain of area

The gain of area, during a plane deformation, is

The symbol det J stands for the determinant of the Jacobean of the mapping

. Figure 1 shows that a finite "Cartesian rotation"  has a significant influence into the change of area of a plane elastic body.

Figure 1

Gradient of a plane vector fieldThe second complex term is the complex shear. The total shear is obtained through its modulus. . The components of the areolar strain are orthogonal as the quadratic form contained in the integrand of the work expression

Plane waves in infinite media

Plane waves in homogeneous infinite media are shortly commented, using the approach of superimposing the fundamental strain modes shown in Fig 2. Addition of the inertial term to Lame's equation, (11)

Compatibility equations (7) can be cast into the following complex form

Substitution of the second term of eqn. (19) into the second term of eqn. (18), gives

If body forces can be disregarded, the equilibrium equation is also a Cauchy-Riemann equation. As the two other fundamental modes  and  must vibrate together with either mode  or  , in order to comply with the compatibility equation 6, similar wave equations governs these modes Plane waves, displacing in a direction forming an angle 0  with the X-axis, comply with the condition of zero strain in the direction normal to its path, in order to avoid spreading waves in this direction, hence In a plane wave displacing in the X direction, the in situ strain in the Y direction must not be disturbed in order to avoid wave spreading in this direction. Looking at Fig. 2, we see that the amplitude of vibration of mode  must be neutralized by the amplitude of mode 

Flux of energy of the shear wave

The flux of energy of irrotational (P-waves) is exposed on several books, however the same does not occurs with the equivoluminal (S-waves

The imaginary number is indicating that this is a circumferential strain in the directions of the path s , resulting from the superposition of rotation and shear modes. This strain highlights the mechanism of shear wave propagation. The displacement in the direction of the wave path produced by the circumferential strain is neutralized by the displacement produced by the rotation. On the other hand the strain produced by shear modes in the transverse direction is neutralized by the rotation, avoiding wave propagation in that direction. This superposition can be visualized in fig. 3. (32) In equations (31) and (32) the imaginary number indicates that displacements and velocities are transversal to the wave path. Hence in the shear wave no displacement occurs in the direction of the wave path and the propagation of the wave is due only to the pulsation of the areola in this direction without displacement of its center. The shear at the wave front, according with the imaginary part of Eq. 13 is

where A is a constant to be determined. Now

Lorentz Force

Let Vy be the velocity of translation of a charge q in the Y direction under an electromagnetic field, then

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Revisiting Arther Electrodynamics

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