Revisiting Aether Electrodynamics

Submitted to Proceedings A Revisiting aether electrodynamics Journal: Proceedings A Manuscript ID Draft Article Type: Research Fo Date Submitted by the n/a Author: Complete List of Authors: KOTCHERGENKO, IHOR; Instituto Militar de Engenharia Mathematical physics < MATHEMATICS, mathematical physics < PHYSICS, Wave motion < PHYSICS ether and electrodynamics, areolar strain concept, arther simile, areolar strain, elasticity ev Keywords: rR Subject: Subject Category: Physics iew ly On http://mc.manuscriptcentral.com/prsa Page 1 of 7 Author-supplied statements Relevant information will appear here if provided. Ethics Does your article include research that required ethical approval or permits?: This article does not present research with ethical considerations Statement (if applicable): CUST_IF_YES_ETHICS :No data available. Data It is a condition of publication that data, code and materials supporting your paper are made publicly available. Does your paper present new data?: My paper has no data Fo Statement (if applicable): CUST_IF_YES_DATA :No data available. Conflict of interest ev rR I/We declare we have no competing interests Statement (if applicable): CUST_STATE_CONFLICT :No data available. Authors’ contributions On I am the only author on this paper iew Statement (if applicable): CUST_AUTHOR_CONTRIBUTIONS_TEXT :No data available. ly 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Submitted to Proceedings A http://mc.manuscriptcentral.com/prsa Submitted to Proceedings A Revisiting aether electrodynamics Ihor D. Kotchergenko Retired Professor, IME-Instituto Militar de Engenharia Rio de Janeiro, Brazil, kotcihor@gmail.com SUMMARY 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 theory, and for details of the mathematics involved read from beginning. ev rR Fo Key words: ether and electrodynamics, areolar strain concept. 1 2 INTRODUCTION iew Whittaker in his book [12], “A History of the Theories of Aether and Electricity” says that Thomson in 1856 mentions “that magnetism possesses a rotatory character”. At the same book it is registered that Maxwell expressed “that light consist in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.”, and also “induced me to regard magnetism as a phenomenon of rotation and electric currents as phenomena of translation.” The writer presents an electromagnetic theory showing that “magnetism is a phenomenon of rotation and electric currents are phenomena of translation” which are produced through a combination of harmonic lateral displacements and rotations and that the resulting shear of the space generates the magnetic field while the speed of the lateral displacements generates the electric field. Fluids and vortices are not involved. This understanding of the intrinsic properties of the space should promote a deeper insight into physics, specially on Casimir effect. ly On 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Page 2 of 7 The application of the areolar theory of elasticity to wave problems was presented at Kotchergenko [18]; It must be emphasized that previous mechanical models of electromagnetic waves didn’t detected that a dipole was present in each wavelength , (see Fig. 3). The dipole appearance in the areolar model of shear waves is the reason for producing the present model. Previous mechanical models for electromagnetic waves appealed to inviscid fluid vortices to create the dipole required for the existence of a magnetic field. There are no reasons to regard that it is not a simple mechanical simile but just the mathematical description of the real world. http://mc.manuscriptcentral.com/prsa Page 3 of 7 2.0 2.1 OUTLINE OF THE THEORY The areolar strain concept Let a plane region of the variables z  x  i y and z  x  i y be mapped in a one-to-one manner onto the plane of the displacements u( x, y) and v( x, y) by means of the transformation w( z, z )  u( x, y)  i v( x, y) . The areolar strain is defined as the gradient of the vector field w( z, z ) , through the Riemann derivative [13, 14]:   lim z  z 0 w w dz  dz w  w0 z  z z  z0 dz Fig. 1: Gradient of a plane vector field or  w w i 2  e z z (1) Fo where the polar form has been used for the ratio dz | dz | ei    ei 2 dz | dz | ei  The 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 U  1 / 2 Cij  i  j dV , can be converted into its canonic form rR The last expression presupposes that z tends to z 0 , maintaining the direction  as z tends to z0. Equation (1) is obtained making U  1 2  [ 1 2(C11  C12 ) 2  1 2 (C11  C12 ) 2  C13 2 ]dV (4) u u v v dx  dy , dv  dx  dy dx dy x y w w du  i dv  (dx  i dy )  (dx  i dy ) z z du  with w u v v u  (  )  i (  )    i 2 z x y x y 2 w u v v u  (  )  i(  )    i  z x y x y (2) ly On 2 iew ev 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Submitted to Proceedings A Fig. 2: Fundamental modes of the plane strain Equations (2) are known as Kolosov-Wirtinger [13] derivatives. Written in the complex form the areolar strain given by Eq. (1) reads   1 2 (  i 2)  1 2 (  i  ) e2i (3) 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 . 2.2 Compatibility equations If z 0 and z are two points pertaining to the complex plane, their relative displacement is given by w  w0    dz . C Since w  w0 is independent of the path of integration dw  w w dz  dz z z (5) is a total differential. Consequentially, the displacement field must comply with the condition of continuity  w  w ( ) ( )0 z z z z (6) Separating the real and imaginary parts of this equation, the following compatibility equations are obtained: http://mc.manuscriptcentral.com/prsa Submitted to Proceedings A   (2   )  (   )  0 y x equations   (2   )  (   )  0 y x 2.3 Equilibrium equations .with For isotropic material, 1 (C11 C 12 )     , 1 (C11 C12 )   and C   . Thus 13 2 2 the work expression given by Eq. (4) reduces to U     0 x y     0 y x (7) 1 [(   ) 2   ( 2   2 )]dV 2  (8)    (   ) )0  (  x x y    (   )  )0  ( x y y T  i 2.5 Gain of area The gain of area, during a plane deformation, is dA    or (9) rR Taking into account that dA    ev 2.4   [(   ) ]  2 [  (  i )]  0 z z Traction vector (11) 1 (   ) dz   (  i )dz i C (17) The symbol det J stands for the determinant of the Jacobean of the mapping w( z, z )  u( x, y)  iv( x, y) . Figure 1 shows that a finite “Cartesian rotation”  has a significant influence into the change of area of a plane elastic body. (12) 2.6 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) results the traction vector on a boundary C results T  (   )   (  i )e 2 w w     det J dz z u   u 1  x y  1  dA   v  v  1   x y  (   )2 2i 2 (16) ly    [(   ) ]  [  (  i )]dxdy  z z 1   2  ( 2   2 ) 4 On Applying Green’s formula in the complex form (e.g. Courant [1]) to eqn. (11), 2  4 iew homogeneous equilibrium equations can be cast into the complex form: 2 2 dA is obtained from the determinant  (10) (  i  )   2u ( x, y )  i  2 v( x, y ) z where the symbol  2 stands for the Laplacian operator, Lamé’s 2 (14) (15) With this concept, traction became an analytic function. This approach has important reflections in contour mapping applications. where  and  are Lame’s elastic material constants. The Euler equations for this functional are Fo 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Page 4 of 7 (13) 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 ds | dz | , then n ds  i dz  1 dz . The real part of T i is the normal stress while the imaginary part is the shear at the contour. The shear stress component is always rotated minus 90 degrees with respect to normal stress. In Cartesian coordinates the equilibrium equation obeys Cauchy-Riemann    2 (u  i v )   2 (  i  )   0 z z t 2 (18) Compatibility equations (7) can be cast into the following complex form   (  2i  )  (  i  )  0 z z (19) Substitution of the second term of eqn. (19) into the second term of eqn. (18), gives (  2  )2    2 (u  i v )  i 4  0 z z t 2 (20) If body forces can be disregarded, the equilibrium equation is also a Cauchy-Riemann equation. Derivative in z results http://mc.manuscriptcentral.com/prsa Page 5 of 7 (  2 )2 As 4  2  2  2 w i4  2 0 zz zz t z (21) 2 w 1  2 and  (  i 2 ) , separating the real and z 2 zz imaginary parts, the following irrotational and equivoluminal wave equations:are obtained 2   2 0   2  t 2 2   (22)   2 0  t 2 As the two other fundamental modes (23)  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 Fo  w w i 2(0  2 )  e 0 z z For a P-wave, as   0 , Eq. (3) gives 2.7 iew v  0 and y u 0 y respectively, but an inspection of Fig. 2 allows a deeper insight. The first condition gives zero radial strain in Y direction while the second condition gives zero circumferential strain in that same direction as it implies that the amplitude  / 2 of the rotation with respect to the X-axis, resulting from the  shear mode, is neutralized by the amplitude of the rotation  . As a result, the strain amplitudes in the radial and circumferential Y directions remains zero while the strain amplitude in the radial X direction will be 1   1    while the strain amplitude in 2 condition 2 X direction reaches  / 2     . The 2  22 cos[ 2 ( s  cT t )]sin 22 (29)  2  22 cos[ 2 ( s  cT t )]cos 22 (30) As for S-wave,   0 , the areolar strain in the direction  2 , according to eq. (3) will be 1  2  [2 i 2  (2  i  2 )e 2 i  ] 2 2 (31) Substitution of equations (29) and (30) gives  2  i 22 cos[ 2 ( s  cT t )] (30) 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. Integration of eqn. (30) along s, gives its displacement    gives rise to a P-wave while the condition 2   gives rise to an S-wave, both traveling in the X direction. (28) Equation (26) gives ly (27) For    , eqn. (24) gives    and 2   . Actually 2 circumferential 2  2 cos[ 2 ( s  cT t )] On w w  0 z z the and traveling at the direction  2 can be put in the form (26)  . Grafting    into eqn. (24), results 2 these equations simply mean that 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 S-wave, having amplitude 2 (25) 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 Fig, 3 – Reflection of a P-wave (24) ev     sin 2 0    cos 2 0 and for S-wave, as   0 ,    2 cos 2 0   2 sin 2 0 rR 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Submitted to Proceedings A w2  i 22 sin[ 2 ( s  cT t )] /  2 and derivative in time, the velocity http://mc.manuscriptcentral.com/prsa (31) Submitted to Proceedings A w2  i 22cT cos[ 2 ( s  cT t )] travels in the direction of axis X at a speed c. Substitute Lame’s shear constant by the vacuum permeability μ0 . (32)  x y ( x, t )  0 2  y cos[ ( x  c t )] 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  2    (2  i  2 )e 2 i ( 2  ) 2 B z ( x , t )   x y ( x, t ) Bz ( x, t )  μ 0 2 Ω z cos   x  c t   (33) w y ( x, t )  2  z sin[ ( x  c t )] /  (34) E y ( x, t )  A (35) rR E y x B   iew (2  i  2 )(2  i  2 )  22   22  22 (38) 1  2  ( 2   2 )  0 , what is an indirect proof that the 4 guide waves equations 25 and 26 are correct. A The Electromagnetic wave 1  0 c2 (39) be the vector field of rotations of the aether in the {x,y,z} reference frame, where  z is the modulus of the harmonic rotation in the z-axis. We replace the notation for the rotation  used in the areolar theory for  in order to avoid confusion with the frequency. Take the shear stress of the solenoidal elastic wave from Eq. 35 supposing that it (46) (47) (48) E y 0 (49) 1 2z cos[ ( x  c t )] c 0 (50) t Calculate the speed of the wave with equation E y ( x , t )  c Bz [ x , t ] xA  {( y Az   z Ay ), ( z Ax   x Az ), ( x Ay   y Ax )} Ω(z, t)   0, 0, z  (45) . Then E y ( x, t )  c Let  x B   0 0 ly resulting We will need to frequently consult the form of the curl of a vector A = {Ax, Ay, Az}. (44) Now we determine the constant A using Maxwell’s equation Maxwell’s equatios in vacuum 3.0 Bz ( x, t ) x x B  0  2 z sin[ ( x  c t )] On Observe that the gain of area, given by Eq. 14, in the equivoluminal condition dA=0 and   0 is reduced to 3 t (37) where 2 represents the amplitude of the S-wave. The total shear strain is given by the modulus of the complex shear strain. From Eq. 34, t  A c 2 2 z sin[ ( x  c t )] ev 1 (22 ) 2 cT  2 w y ( x, t ) E y  A c 2z cos[ ( x  c t )] (36) The mean flux of energy of the equivoluminal wave will then be 2  (43) where A is a constant to be determined. Now Multiplying by the transversal velocity given by eqn. (32) gives the flux of energy  2  (22 ) 2 cT  cos2 [ 2 ( s  cT t )] (42) Define the Y-component of the vector electric field as gives  2  i  22 cos[ 2 ( s  cT t )] (41) Now take the transversal Y displacement of the elastic wave from Eq 31 Substitution of eqn. (25), rewritten in the form  2  22 cos 2 2 2  22 sin 2 2 (40) Define the z component of the magnetic field as  Fo 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Page 6 of 7 1 (51) (52)  0 0 Hence x E  E y ( x , t ) x   satisfying Maxwell’s equation http://mc.manuscriptcentral.com/prsa 0 0 2 z sin[ ( x  c t )] (53) Page 7 of 7 x E   Bz [ x, t ] t Coulombs (54) Fy  Coulombs [ The divergences of the vector fields Poynting vector is obtained from 0 Sx  0 1 E B 0 y z (57) (2Ω z )2 cos   x  c t  with mean value 0 (58) 0 Ω 2z or REFERENCES (60) Ω2z Nm  2 m Coulombs s m2 s N m2 (61) Coulombs m/s (62) ly and also S x  2 c 0 2z (63) that gives z  3.3 Amp m (64) Lorentz Force Let Vy be the velocity of translation of a charge q in the Y direction under an electromagnetic field, then Fy  q ( E y  v y Bz ) Fy  2 q c  vy c2  0 On results z  [1] Courant R. and Hilbert, D., Methods of Mathematical Physics, Vol. II, Wiley: New York, pp. 350-351, 1989. [2] Achenbach, J. D., Wave Propagation in Elastic Solids, Elsevier, 1999. [3] Elmore, W. C. and Held, M. A., Physics of Waves, Dover. [4] Ewing, W. M., Jadetzky, W. S, and Press, F., Elastic Waves in Layered Media, McGraw-Hill. [5] Eringen, A. C., and Suhubi, E. S., Elastodynamics, Academic Press. [6] Kolsky, H., Stress Waves in Solids, Dover, 1963. [7] Graff, K.L., Wave Motion in Elastic Solids. The Clarendon Press, 1975. [8] Miklowitz, J, Elastic Waves and Waveguides, North-Holland, 1978 [9] Morse, P. M, and Feshbach, H., Methods of Theoretical Physics, McGraw-Hill, 1953. [10] Muskhelishvili, N. I., Some Basic Problems of the Theory of Elasticity, Noordhoff, Grotingen, 1963. [11] Sokolnikoff, I. S., Mathematical Theory of Elasticity, McGraw-Hill, 1956. [12] Whittaker, E. T., A History of the Theory of Aether and Electricity, Longman, Green, 1910. [13] Mitrinovic, D.S., Keckic, J.D., From the History of Nonanalitic Functions, Series: Mathématiques e Physique, No. 274-371, Publications de La Faculté D’Electrotechnique de L’Université à Belgrade, 1969, (open access). [14] Miloge Rajovic, Dragan Dimitrovski, The Clairaut and Lagrange Areolar Equation, Kragujevac J. Math. 24 (2002) 123–133. (open access). [15] Kotchergenko, I.D., Applications of Generalized Analytic Functions to Elasticity, CMM-2005-Computer Methods in Mechanics, Polish Academy of Sciences, June 2005. [16] Kotchergenko, I.D., The Areolar Strain Concept, Mechanics of Solids, Structures and Fluids, Volume 12, ASME, 2008. [17] Kotchergenko, I.D., The Areolar Strain Concept Applied to Elasticity, WIT Transactions on Modeling and Simulation, Vol. 46, 2007, WIT Press. (open access) [18] Kotchergenko, I.D., The Areolar Strain Approach for grazing waves, , WIT Transactions on Modeling and Simulation, Vol. 55, 2013, WIT Press, (open access). iew Equating with the dimension of Poyting vector  y cos[ ( x  c t )] ]  N (67) (59) ev 2 Ω 2z Sx  c 0 rR Sx  2 2 2 2 ( m / s ) Coulombs The areolar strain concept proves to be a very efficient tool for the study wave phenomena. Elmore and Held, [3], calculated the power flow in a vibrating rope and further, when presenting the plane equivoluminal waves, declared that it is impossible to calculate the power flow of these waves without taking into account the rotation of the elastic environment. For the power flow on the rope it was required to take into account the rotation caused by bending. As the areolar strain concept solidly incorporates the rotation into its definition, the writer made an attempt to obtain the result. A few years later, perceiving that both shear and translation could be described simply by the rotation of the areolar strain, and that each wavelength has a dipole, saw the opportunity to present this mechanical model for electromagnetic waves. Everything described by just a single rotation as postulate at Occan’s eraser. Poynting vector Sx  ms 2 7 Conclusions B   0,0, Bz  are null. 3.2 m/s N m2 E   0, E y ,0 and Fo 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Submitted to Proceedings A (65) (66) http://mc.manuscriptcentral.com/prsa Submitted to Proceedings A [19] Kotchergenko, I.D., The Tri-harmonic Plate Bending Equation, WIT Transactions on Modeling and Simulation, Vol. 59, 2015, WIT Press, (open access). [20] Kotchergenko, I. D., Power Flow in Equivoluminal Elastic Waves, ICM 2018-International Conference of Mathematicians, Rio de Janeiro. iew ev rR Fo ly On 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 http://mc.manuscriptcentral.com/prsa Page 8 of 7