Patent Publication Number: US-11379635-B2

Title: Method of solving closed form exact solution for the field generated by a finite-length harmonic linear current source in a whole space

Description:
I. FIELD OF THE INVENTION 
     The present invention belongs to the artificial source electromagnetic exploration in frequency domain, and in particular, to a method of solving closed form exact solution for the field generated by a finite-length harmonic linear current source in a whole space. 
     II. BACKGROUND OF THE INVENTION 
     In the field of geophysical electromagnetic exploration, a processing method of electric dipole approximation is adopted for a harmonic linear current source: the source current is considered as being uniformly distributed, and the linear current source is considered as a point source, so as to obtain a closed form solution. Such a dipole approximation formula is suitable for use in a far zone. 
     With the development of exploration methods, in a process of advancing of an observation point from a far zone to a near zone through a middle zone, it is necessary to obtain exact solution for the field generated by a finite-length harmonic linear current source as the theoretical basis for the development of all zone exploration. However, integral expression of exact solution for the field generated by the finite-length harmonic linear current source in assumed uniform distribution cannot realize closed form quadrature. Although closed evaluation may be carried out for integration of exact solution for the field generated by a finite-length current source in sinusoidal distribution, for which the terminal current is set to 0, sinusoidal expression is inconsistent with the actual current distribution of the geophysical harmonic linear current source. 
     III. SUMMARY OF THE INVENTION 
     In accordance with the shortages and defects of the prior art, the present invention provides a method of solving closed form exact solution for the field generated by a finite-length harmonic linear current source in a whole space, wherein the key is cosinoidal expression of the uniform current distribution with the purpose of carrying out elementary quadrature for the integral formula of a field containing a source point position vector, and the closed form exact solution for the field generated by a finite-length harmonic linear current source in the whole space is not only suitable for the all zone, but also can express essential attributes of the harmonic current being distributed along the line. 
     The technical solution adopted in the present invention is as following: 
     a method of solving closed form exact solution for the field generated by a finite-length harmonic linear current source in a whole space includes: 
     step 1, listing integral formula for vector magnetic potential containing a source point position vector of a finite-length harmonic linear current source; 
     step 2, performing a cosine processing for the uniform current; 
     step 3, expressing current in the integral formula for vector magnetic potential containing the source point position vector by a cosine function; and 
     step 4, solving the integral containing the source point position vector to obtain the closed form exact solution for the field generated by the finite-length harmonic linear current source in the whole space. 
     Further, taking a cylindrical-coordinate system, a middle point of the linear current source coincides with coordinates origin O and is placed along the z axis; as shown in  FIG. 1 , in a uniform, linear, isotropic and time-invariant unbounded medium, a vector magnetic potential formula A(r) of the harmonic linear current source containing the source point position vector is listed: 
                     A   ⁡     (   r   )       =         z   ^     ⁢       A   z     ⁡     (     ρ   ,   z     )         =         z   ^     ⁢           ⁢     μ     4   ⁢           ⁢   π       ⁢       ∫       -   l     ⁢     /     ⁢   2       l   ⁢     /     ⁢   2       ⁢           I   ⁡     (     z   ′     )       ⁢     e       -   jk     ⁢          r   -     r   ′                         r   -     r   ′              ⁢     dz   ′           =         z   ^     ⁢     μ     4   ⁢           ⁢   π       ⁢         ∫       -   l     ⁢     /     ⁢   2         l   ⁢     /     ⁢   2       ⁢           I   ⁡     (     z   ′     )       ⁢     e       -   jk     ⁢         ρ   2     +       (     z   -     z   ′       )     2                     ρ   2     +       (     z   -     z   ′       )     2           ⁢     dz   ′           =       z   ^     ⁢           ⁢     μ     4   ⁢           ⁢   π       ⁢         ∫       -   l     ⁢     /     ⁢   2         l   ⁢     /     ⁢   2       ⁢           I   ⁡     (     z   ′     )       ⁢     e     -   jkR         R     ⁢     dz   ′                         (     1   ⁢   a     )               
where A z (ρ,z) is the z component of the vector magnetic potential A(r); due to the symmetry, A z  is only the function of (ρ,z) among three coordinate variables (ρ,φ,z) of the cylindrical-coordinate system, {circumflex over (z)} is the unit vector along the z-axis direction, r is the field point position vector, r′ is the source point position vector, z′ is the source point position coordinate, y is the permeability, l is the length of the linear current source, I(z′) is the current distribution function, j is the imaginary unit, R is the distance from the source point to the field point, k is the wave number, and p is the radial distance in the cylindrical-coordinate system.
 
 k =√{square root over ((ω 2   με−j ωμσ)}=√{square root over (2π f μ(2π fε−j σ))}  (1b)
 
where ω is the angular frequency (in the unit of rad/s) of the source current, f is the frequency (in the unit of Hz) of the source current, ε is the permittivity (in the unit of F/m), and σ is the conductivity (in the unit of S/m).
 
     Further, the uniform current is subjected to cosine processing as following: setting the current at the middle point of the linear current source to I 0 ; 
                       I   ⁡     (     z   ′     )       =       I   0     ⁢     cos   ⁡     (     kz   ′     )           ,       -     l   2       ≤     z   ′     ≤       l   2     ⁢     ;                 (   2   )               
where I 0  is the peak value of the harmonic current.
 
     Further, the formula (2) is put into the formula (1a) to obtain the integral expression containing the source point position vector: 
     
       
         
           
             
               
                 
                   
                     A 
                     ⁡ 
                     
                       ( 
                       r 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         z 
                         ^ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           μ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             I 
                             0 
                           
                         
                         
                           4 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                         
                       
                       ⁢ 
                       
                         
                           
                             ∫ 
                             
                               l 
                               ⁢ 
                               
                                 / 
                               
                               ⁢ 
                               2 
                             
                           
                           
                             
                               - 
                               l 
                             
                             ⁢ 
                             
                               / 
                             
                             ⁢ 
                             2 
                           
                         
                         ⁢ 
                         
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 kz 
                                 ′ 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               e 
                               
                                 
                                   - 
                                   jk 
                                 
                                 ⁢ 
                                 
                                    
                                   
                                     r 
                                     - 
                                     
                                       r 
                                       ′ 
                                     
                                   
                                    
                                 
                               
                             
                             
                                
                               
                                 r 
                                 - 
                                 
                                   r 
                                   ′ 
                                 
                               
                                
                             
                           
                           ⁢ 
                           
                             dz 
                             ′ 
                           
                         
                       
                     
                     = 
                     
                       
                         z 
                         ^ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           μ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             I 
                             0 
                           
                         
                         
                           4 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                         
                       
                       ⁢ 
                       
                         
                           
                             ∫ 
                             
                               l 
                               ⁢ 
                               
                                 / 
                               
                               ⁢ 
                               2 
                             
                           
                           
                             
                               - 
                               l 
                             
                             ⁢ 
                             
                               / 
                             
                             ⁢ 
                             2 
                           
                         
                         ⁢ 
                         
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 kz 
                                 ′ 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               e 
                               
                                 - 
                                 jkR 
                               
                             
                             R 
                           
                           ⁢ 
                           
                             dz 
                             ′ 
                           
                           ⁢ 
                           
                             ; 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     Further, a process for obtaining the exact solutions includes: 
     step 4.1, the relation formula between the vector magnetic potential A(r) containing the source point position vector and the magnetic field intensity H(r) is expanded in the cylindrical-coordinate system to obtain the following magnetic field intensity H(r): 
                     H   ⁡     (   r   )       =         1   μ     ⁢     ∇     ×     A   ⁡     (   r   )             =         -     φ   ^       ⁢     1   μ     ⁢       ∂       A   z     ⁡     (     ρ   ,   z     )           ∂   ρ         =       φ   ^     ⁢           ⁢     H   φ     ⁢     ;                   (   4   )               
in the formula, {circumflex over (φ)} is the unit vector along the co direction.
 
step 4.2, the integral formula (3) of the vector magnetic potential A(r) containing the source point position vector is put into the formula (4) of the magnetic field intensity H(r), to obtain:
 
                     H   φ     =       -       I   0       4   ⁢           ⁢   π         ⁢       ∂               ∂   ρ       ⁢         ∫     l   ⁢     /     ⁢   2           -   l     ⁢     /     ⁢   2       ⁢       cos   ⁡     (     kz   ′     )       ⁢       e     -   jkR       R     ⁢     dz   ′     ⁢     ;                   (     5   ⁢   a     )               
step 4.3, the exact solution is obtained by using Euler formula and homogeneous Maxwell equation: according to the Euler formula:
 
                     cos   ⁡     (     kz   ′     )       =         e     j   ⁡     (     kz   ′     )         +     e     -     j   ⁡     (     kz   ′     )             2             (     5   ⁢   b     )               
formula (5b) is put into the formula (5a) for expansion to obtain
 
                           H   φ     =       ⁢       -       I   0       8   ⁢           ⁢   π         ⁢         ∫       -   l     ⁢     /     ⁢   2         l   ⁢     /     ⁢   2       ⁢       [           ∂               ∂   ρ       ⁢       e     -     jk   ⁡     (     R   -     z   ′       )           R       +         ∂               ∂   ρ       ⁢       e     -     jk   ⁡     (     R   +     z   ′       )           R         ]     ⁢     dz   ′                       =       ⁢       -       I   0       8   ⁢           ⁢   π         ⁢     {           ∫     l   ⁢     /     ⁢   2           -   l     ⁢     /     ⁢   2       ⁢       [         1   R     ⁢       ∂               ∂   ρ       ⁢     e     -     jk   ⁡     (     R   -     z   ′       )             +       e     -     jk   ⁡     (     R   -     z   ′       )           ⁢       ∂               ∂   ρ       ⁢     1   R         ]     ⁢     dz   ′         +                         ⁢         ∫     l   ⁢     /     ⁢   2           -   l     ⁢     /     ⁢   2       ⁢       [         1   R     ⁢       ∂               ∂   ρ       ⁢     e     -     jk   ⁡     (     R   +     z   ′       )             +       e     -     jk   ⁡     (     R   +     z   ′       )           ⁢       ∂               ∂   ρ       ⁢     1   R         ]     ⁢     dz   ′         }               =       ⁢         I   0       8   ⁢           ⁢   π       ⁢     {       ρ   ⁢         ∫     l   ⁢     /     ⁢   2           -   l     ⁢     /     ⁢   2       ⁢       [         -   jk     ⁢           ⁢       e     -     jk   ⁡     (     R   -     z   ′       )             R   2         -       e     -     jk   ⁡     (     R   -     z   ′       )             R   3         ]     ⁢     dz   ′           +                         ⁢     ρ   ⁢           ⁢         ∫     l   ⁢     /     ⁢   2           -   l     ⁢     /     ⁢   2       ⁢       [         -   jk     ⁢           ⁢       e     -     jk   ⁡     (     R   +     z   ′       )             R   2         -       e     -     jk   ⁡     (     R   +     z   ′       )             R   3         ]     ⁢     dz   ′           }               =       ⁢       -       I   0       8   ⁢           ⁢   π         ⁢     {       ρ   ⁢           ⁢         ∫     l   ⁢     /     ⁢   2           -   l     ⁢     /     ⁢   2       ⁢         e     -     jk   ⁡     (     R   -     z   ′       )           ⁡     (         -   jk     ⁢           ⁢     1     R   2         -     1     R   3         )       ⁢     dz   ′           +                         ⁢     ρ   ⁢           ⁢         ∫     l   ⁢     /     ⁢   2           -   l     ⁢     /     ⁢   2       ⁢         e     -     jk   ⁡     (     R   +     z   ′       )           ⁡     (         -   jk     ⁢           ⁢     1     R   2         -     1     R   3         )       ⁢     dz   ′           }               =       ⁢       -       I   0       8   ⁢           ⁢   π         ⁢     {     ρ   ⁢           ⁢         ∫     l   ⁢     /     ⁢   2           -   l     ⁢     /     ⁢   2       ⁢       e     -     jk   ⁡     (     R   -     z   ′       )           [         -   jk     ⁢           ⁢     1     R   2       ⁢     dz   ′       +           z   ′     -   z         R   3     ⁡     (     R   -     z   ′     +   z     )         ⁢     dz   ′       -                                 ⁢       1       R   2     ⁡     (     R   -     z   ′     +   z     )         ⁢     dz   ′       ]     +     ρ   ⁢         ∫     l   ⁢     /     ⁢   2           -   l     ⁢     /     ⁢   2       ⁢       e     -     jk   ⁡     (     R   +     z   ′       )           [         -   jk     ⁢           ⁢     1     R   2       ⁢     dz   ′       -           z   ′     -   z         R   3     ⁡     (     R   +     z   ′     -   z     )         ⁢     dz   ′       -                             ⁢       1       R   2     ⁡     (     R   +     z   ′     -   z     )         ⁢     dz   ′       ]     }               =       ⁢       -       I   0       8   ⁢           ⁢   π         ⁢     {     ρ   ⁢         ∫     l   ⁢     /     ⁢   2           -   l     ⁢     /     ⁢   2       ⁢       e     -     jk   ⁡     (     R   -     z   ′       )           [       jk   ⁢           ⁢       d   ⁡     (     R   -     z   ′       )         R   ⁡     (     R   -     z   ′     +   z     )           -         dR     -   1         (     R   -     z   ′     +   z     )       ⁢     dz   ′       -                                 ⁢         d   ⁡     (     R   -     z   ′     +   z     )         -   1       R     ]     +                   ⁢     ρ   ⁢         ∫     l   ⁢     /     ⁢   2           -   l     ⁢     /     ⁢   2       ⁢       e     -     jk   ⁡     (     R   +     z   ′       )           ⁡     [         -   jk     ⁢           ⁢       d   ⁡     (     R   +     z   ′       )         R   ⁡     (     R   +     z   ′     -   z     )           +       dR     -   1         (     R   +     z   ′     -   z     )       +         d   ⁡     (     R   +     z   ′     -   z     )         -   1       R       ]           }               =       ⁢         I   0       8   ⁢           ⁢   π       ⁢     {       ρ   ⁢           ⁢         ∫     l   ⁢     /     ⁢   2           -   l     ⁢     /     ⁢   2       ⁢     d   ⁡     [       e     -     jk   ⁡     (     R   -     z   ′       )             R   ⁡     (     R   -     z   ′     +   z     )         ]           -     ρ   ⁢         ∫     l   ⁢     /     ⁢   2           -   l     ⁢     /     ⁢   2       ⁢     d   ⁡     [       e     -     jk   ⁡     (     R   +     z   ′       )             R   ⁡     (     R   +     z   ′     -   z     )         ]             }                   =       ⁢             I   0     ⁢   ρ       8   ⁢           ⁢   π       ⁡     [         e     -     jk   ⁡     (     R   -     z   ′       )             R   ⁡     (     R   -     z   ′     +   z     )         +       e     -     jk   ⁡     (     R   +     z   ′       )             R   ⁡     (     R   +     z   ′     -   z     )           ]       ⁢     |       -   l     ⁢     /     ⁢   2       l   ⁢     /     ⁢   2                     =       ⁢           I   0     ⁢   ρ       8   ⁢           ⁢   π       [         e     -     jk   ⁡     (       R   1     -     l   ⁢     /     ⁢   2       )               R   1     ⁡     (       R   1     -     l   ⁢     /     ⁢   2     +   z     )         -       e     -     jk   ⁡     (       R   2     +     l   ⁢     /     ⁢   2       )               R   2     ⁡     (       R   2     +     l   ⁢     /     ⁢   2     -   z     )         -       e     -     jk   ⁡     (       R   1     +     l   ⁢     /     ⁢   2       )               R   1     ⁡     (       R   1     +     l   ⁢     /     ⁢   2     -   z     )         +                       ⁢       e     -     jk   ⁡     (       R   2     -     l   ⁢     /     ⁢   2       )               R   2     ⁡     (       R   2     -     l   ⁢     /     ⁢   2     -   z     )         ]               =       ⁢           I   0     ⁢   ρ       8   ⁢           ⁢   π       ⁢     {           e     -     jkR   1           R   1       ⁡     [         e     jk   ⁢           ⁢   l   ⁢     /     ⁢   2         (       R   1     -     l   ⁢     /     ⁢   2     +   z     )       -       e       -   jk     ⁢           ⁢   l   ⁢     /     ⁢   2         (       R   1     +     l   ⁢     /     ⁢   2     -   z     )         ]       +         e     -     jkR   2           R   2       [         e     jk   ⁢           ⁢   l   ⁢     /     ⁢   2         (       R   2     -     l   ⁢     /     ⁢   2     -   z     )       -                               ⁢       e       -   jk     ⁢           ⁢   l   ⁢     /     ⁢   2         (       R   2     +     l   ⁢     /     ⁢   2     -   z     )       ]     }               =       ⁢           I   0     ⁢   ρ       8   ⁢           ⁢   π       ⁢     {           e     -     jkR   1           R   1       ⁡     [           cos   ⁡     (     kl   ⁢     /     ⁢   2     )       +     j   ⁢           ⁢     sin   ⁡     (     kl   ⁢     /     ⁢   2     )             (       R   1     -     l   ⁢     /     ⁢   2     +   z     )       -         cos   ⁢     (     kl   ⁢     /     ⁢   2     )       -     j   ⁢           ⁢     sin   ⁡     (     kl   ⁢     /     ⁢   2     )             (       R   1     +     l   ⁢     /     ⁢   2     -   z     )         ]       -                         ⁢         e     -     jkR   2           R   2       ⁡     [           cos   ⁡     (     kl   ⁢     /     ⁢   2     )       +     j   ⁢           ⁢     sin   ⁡     (     kl   ⁢     /     ⁢   2     )             (       R   2     -     l   ⁢     /     ⁢   2     -   z     )       -         cos   ⁡     (     kl   ⁢     /     ⁢   2     )       -     j   ⁢           ⁢     sin   ⁡     (     kl   ⁢     /     ⁢   2     )             (       R   2     +     l   ⁢     /     ⁢   2     -   z     )         ]       }               =       ⁢         I   0       4   ⁢           ⁢   π   ⁢           ⁢   ρ       [         (             l   ⁢     /     ⁢   2     -   z       R   1       ⁢     e     -     jkR   1           +           l   ⁢     /     ⁢   2     +   z       R   2       ⁢     e     -     jkR   2             )     ⁢     cos   ⁡     (     kl   2     )         +                         ⁢       j   ⁡     (       e     -     jkR   1         +     e     -     jkR   2           )       ⁢     sin   ⁡     (     kl   2     )         ]     ⁢     ;                   (   6   )               
according to the homogeneous Maxwell equation:
 
                     E   ⁡     (   r   )       =       1       j   ⁢           ⁢   ω   ⁢           ⁢   ɛ     +   σ       ⁢     ∇     ×     H   ⁡     (   r   )                     (   7   )               
an electric field expression is obtained:
 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           E 
                           ρ 
                         
                         = 
                           
                         ⁢ 
                         
                           
                             - 
                             
                               1 
                               
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   ω 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   ɛ 
                                 
                                 + 
                                 σ 
                               
                             
                           
                           ⁢ 
                           
                             
                               ∂ 
                               
                                 H 
                                 φ 
                               
                             
                             
                               ∂ 
                               z 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               j 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 I 
                                 0 
                               
                             
                             
                               4 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ρ 
                             
                           
                           ⁢ 
                           
                             
                               μ 
                               
                                 ɛ 
                                 - 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     σ 
                                     ω 
                                   
                                 
                               
                             
                           
                           ⁢ 
                           
                             { 
                             
                               [ 
                               
                                 ( 
                                 
                                   
                                     1 
                                     
                                       kR 
                                       1 
                                       3 
                                     
                                   
                                   + 
                                   
                                     
                                       j 
                                       
                                         R 
                                         1 
                                         2 
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         ( 
                                         
                                           
                                             l 
                                             2 
                                           
                                           - 
                                           z 
                                         
                                         ) 
                                       
                                       2 
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         - 
                                         
                                           jkR 
                                           1 
                                         
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       1 
                                       
                                         kR 
                                         1 
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         - 
                                         
                                           jkR 
                                           1 
                                         
                                       
                                     
                                   
                                   - 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                               
                             ⁢ 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       1 
                                       
                                         kR 
                                         2 
                                         3 
                                       
                                     
                                     + 
                                     
                                       j 
                                       
                                         R 
                                         2 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   
                                     ( 
                                     
                                       
                                         l 
                                         2 
                                       
                                       + 
                                       z 
                                     
                                     ) 
                                   
                                   2 
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       2 
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                   1 
                                   
                                     kR 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       2 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 kl 
                                 2 
                               
                               ) 
                             
                           
                         
                         - 
                       
                     
                   
                   
                     
                       
                         
                             
                           ⁢ 
                           
                             
                               [ 
                               
                                 
                                   
                                     1 
                                     
                                       R 
                                       1 
                                     
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         l 
                                         2 
                                       
                                       - 
                                       z 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       - 
                                       
                                         jkR 
                                         1 
                                       
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     1 
                                     
                                       R 
                                       2 
                                     
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         l 
                                         2 
                                       
                                       + 
                                       z 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       - 
                                       
                                         jkR 
                                         2 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 
                                   kl 
                                   2 
                                 
                                 ) 
                               
                             
                           
                           } 
                         
                         ⁢ 
                         
                           ; 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             E 
                             z 
                           
                           = 
                             
                           ⁢ 
                           
                             
                               
                                 1 
                                 
                                   
                                     j 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     ωɛ 
                                   
                                   + 
                                   σ 
                                 
                               
                               ⁢ 
                               
                                 1 
                                 ρ 
                               
                               ⁢ 
                               
                                 
                                   ∂ 
                                   
                                     ( 
                                     
                                       ρ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         H 
                                         φ 
                                       
                                     
                                     ) 
                                   
                                 
                                 
                                   ∂ 
                                   ρ 
                                 
                               
                             
                             = 
                             
                               
                                 1 
                                 
                                   
                                     j 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     ω 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     ɛ 
                                   
                                   + 
                                   σ 
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     
                                       ∂ 
                                       
                                         H 
                                         φ 
                                       
                                     
                                     
                                       ∂ 
                                       ρ 
                                     
                                   
                                   + 
                                   
                                     
                                       1 
                                       ρ 
                                     
                                     ⁢ 
                                     
                                       H 
                                       φ 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   I 
                                   0 
                                 
                               
                               
                                 4 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 π 
                               
                             
                             ⁢ 
                             
                               
                                 μ 
                                 
                                   ɛ 
                                   - 
                                   
                                     j 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       σ 
                                       ω 
                                     
                                   
                                 
                               
                             
                             ⁢ 
                             
                               { 
                               
                                 [ 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           j 
                                           
                                             R 
                                             1 
                                             2 
                                           
                                         
                                         + 
                                         
                                           1 
                                           
                                             kR 
                                             1 
                                             3 
                                           
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         
                                           l 
                                           2 
                                         
                                         - 
                                         z 
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         - 
                                         
                                           jkR 
                                           1 
                                         
                                       
                                     
                                   
                                   + 
                                   
                                     ( 
                                     
                                       
                                         j 
                                         
                                           R 
                                           2 
                                           2 
                                         
                                       
                                       + 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 
                                     
                                   ⁢ 
                                   
                                     1 
                                     
                                       kR 
                                       2 
                                       3 
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       l 
                                       2 
                                     
                                     + 
                                     z 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       2 
                                     
                                   
                                 
                               
                               ] 
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 
                                   kl 
                                   2 
                                 
                                 ) 
                               
                             
                           
                           - 
                           
                             [ 
                             
                               
                                 
                                   1 
                                   
                                     R 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       1 
                                     
                                   
                                 
                               
                               + 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                   
                                 ⁢ 
                                 
                                   
                                     1 
                                     
                                       R 
                                       2 
                                     
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       - 
                                       
                                         jkR 
                                         2 
                                       
                                     
                                   
                                 
                                 ] 
                               
                               ⁢ 
                               
                                 sin 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     kl 
                                     2 
                                   
                                   ) 
                                 
                               
                             
                             } 
                           
                           ⁢ 
                           
                             ; 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   where 
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         R 
                         1 
                       
                       = 
                       
                         
                           
                             ρ 
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 z 
                                 - 
                                 
                                   l 
                                   ⁢ 
                                   
                                     / 
                                   
                                   ⁢ 
                                   2 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                     , 
                   
                 
               
               
                 
                   ( 
                   
                     10 
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       R 
                       2 
                     
                     = 
                     
                       
                         
                           
                             ρ 
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 z 
                                 + 
                                 
                                   l 
                                   ⁢ 
                                   
                                     / 
                                   
                                   ⁢ 
                                   2 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                       ⁢ 
                       
                         ; 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     10 
                     ⁢ 
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
     The present invention has the following beneficial effects: 
     Compared with uniform approximate expression, the cosine expression of the linear current source of the present invention can reflect the fundamental attributes of the harmonic current better. In the environment of geophysical electromagnetic exploration, it truly reflects the volatility of current along the line (weak or significant) whatever the ratio between the wavelength λ and the line length l is (an electric dipole that may be considered as the point source in the far zone, or a finite-length linear current source); and compared with the original uniform current expression, the cosine current expression of the present invention has wider applicability for changes of the working frequency and the conductivity of the space medium. 
     The closed form exact solution for the field generated by a finite-length harmonic linear current source in the whole space obtained through the cosine expression of the finite-length linear current source is applicable in the all zone. The closed form solution composed of elementary function can intuitively reveal the change rule of the electromagnetic field, intuitively exhibit the relation between the field quantity and the parameters, and play a key role in theoretical study on antenna and radio wave propagation. 
     Furthermore, the cosine expression of the harmonic linear current source of the present invention can better reflect the assumption of the uniform distribution of the electric dipole current in a free space; and the obtained closed form exact solution for the field generated by a finite-length harmonic linear current source in the whole space has the same beneficial effects for underwater communication. 
    
    
     
       IV. BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  shows a linear current source and a coordinate system; 
         FIG. 2  shows the current distribution of a finite-length harmonic linear current source in a free space; 
         FIG. 3  shows the current distribution of a finite-length harmonic linear current source in the wet soil environment; 
         FIG. 4  shows the current distribution of a finite-length harmonic linear current source in the seawater environment; and 
         FIG. 5  shows a flow diagram of a method of solving closed form exact solution for the field generated by a finite-length harmonic linear current source in a whole space. 
     
    
    
     V. DETAILED DESCRIPTION OF THE INVENTION 
     To clarify the purpose, technical solutions and advantages of the present invention, the present invention is further described below in conjunction with the drawings and examples. It should be understood that the examples described herein are only used to explain the present invention, not to limit the present invention. 
     As shown in  FIG. 5 , a method of solving closed form exact solution for the field generated by a finite-length harmonic linear current source in a whole space includes: 
     step 1, listing integral formula for vector magnetic potential containing a source point position vector of a finite-length harmonic linear current source; 
     step 2, performing a cosine processing for the uniform current; 
     step 3, expressing current in the integral formula for vector magnetic potential containing the source point position vector by a cosine function; and 
     step 4, solving the integral containing the source point position vector to obtain the closed form exact solution for the field generated by the finite-length harmonic linear current source in the whole space. 
     Further, taking a cylindrical-coordinate system, a middle point of the linear current source coincides with coordinates origin O and is placed along the z axis; as shown in  FIG. 1 , in a uniform, linear, isotropic and time-invariant unbounded medium, a vector magnetic potential formula A(r) of the harmonic linear current source containing the source point position vector is listed: 
                     A   ⁡     (   r   )       =         z   ^     ⁢       A   z     ⁡     (     ρ   ,   z     )         =         z   ^     ⁢           ⁢     μ     4   ⁢           ⁢   π       ⁢       ∫       -   l     ⁢     /     ⁢   2       l   ⁢     /     ⁢   2       ⁢           I   ⁡     (     z   ′     )       ⁢     e       -   jk     ⁢          r   -     r   ′                         r   -     r   ′              ⁢     dz   ′           =         z   ^     ⁢           ⁢     μ     4   ⁢           ⁢   π       ⁢         ∫       -   l     ⁢     /     ⁢   2         l   ⁢     /     ⁢   2       ⁢           I   ⁡     (     z   ′     )       ⁢     e       -   jk     ⁢         ρ   2     +       (     z   -     z   ′       )     2                     ρ   2     +       (     z   -     z   ′       )     2           ⁢     dz   ′           =       z   ^     ⁢     μ     4   ⁢           ⁢   π       ⁢         ∫     l   ⁢     /     ⁢   2           -   l     ⁢     /     ⁢   2       ⁢           I   ⁡     (     z   ′     )       ⁢     e     -   jkR         R     ⁢     dz   ′                         (     1   ⁢   a     )               
where A z (ρ,z) is the z component of the vector magnetic potential A(r); due to the symmetry, A z  is only the function of (ρ,z) among three coordinate variables (ρ,φ,z) of the cylindrical-coordinate system, {circumflex over (z)} is the unit vector along the z-axis direction, r is the field point position vector, r′ is the source point position vector, z′ is the source point position coordinate, μ is the permeability, l is the length of the linear current source, I(z) is the current distribution function, j is the imaginary unit, R is the distance from the source point to the field point, k is the wave number, and co is the radial distance in the cylindrical-coordinate system.
 
 k =√{square root over (ω2με− j ωμσ=)}=√{square root over (2π f μ(2π fε−j σ))}  (1b)
 
where ω is the angular frequency (in the unit of rad/s) of the source current, f is the frequency (in the unit of Hz) of the source current, ε is the permittivity (in the unit of F/m), and σ is the conductivity (in the unit of S/m).
 
     Further, the uniform current is subjected to cosine processing as following: setting the current at the middle point of the linear current source to I 0 ; 
                       I   ⁡     (     z   ′     )       =       I   0     ⁢     cos   ⁡     (     kz   ′     )           ,       -     l   2       ≤     z   ′     ≤       l   2     ⁢     ;                 (   2   )               
where I 0  is the peak value of harmonic current.
 
     Further, the formula (2) is put into the formula (1a) to obtain the integral expression containing the source point position vector: 
     
       
         
           
             
               
                 
                   
                     A 
                     ⁡ 
                     
                       ( 
                       r 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         z 
                         ^ 
                       
                       ⁢ 
                       
                         
                           μ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             I 
                             0 
                           
                         
                         
                           4 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                         
                       
                       ⁢ 
                       
                         
                           
                             ∫ 
                             
                               l 
                               ⁢ 
                               
                                 / 
                               
                               ⁢ 
                               2 
                             
                           
                           
                             
                               - 
                               l 
                             
                             ⁢ 
                             
                               / 
                             
                             ⁢ 
                             2 
                           
                         
                         ⁢ 
                         
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 kz 
                                 ′ 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               e 
                               
                                 
                                   - 
                                   jk 
                                 
                                 ⁢ 
                                 
                                    
                                   
                                     r 
                                     - 
                                     
                                       r 
                                       ′ 
                                     
                                   
                                    
                                 
                               
                             
                             
                                
                               
                                 r 
                                 - 
                                 
                                   r 
                                   ′ 
                                 
                               
                                
                             
                           
                           ⁢ 
                           
                             dz 
                             ′ 
                           
                         
                       
                     
                     = 
                     
                       
                         z 
                         ^ 
                       
                       ⁢ 
                       
                         
                           μ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             I 
                             0 
                           
                         
                         
                           4 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                         
                       
                       ⁢ 
                       
                         
                           
                             ∫ 
                             
                               l 
                               ⁢ 
                               
                                 / 
                               
                               ⁢ 
                               2 
                             
                           
                           
                             
                               - 
                               l 
                             
                             ⁢ 
                             
                               / 
                             
                             ⁢ 
                             2 
                           
                         
                         ⁢ 
                         
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 kz 
                                 ′ 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               e 
                               
                                 - 
                                 jkR 
                               
                             
                             R 
                           
                           ⁢ 
                           
                             dz 
                             ′ 
                           
                           ⁢ 
                           
                             ; 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     Further, a process for obtaining the exact solution includes: 
     step 4.1, the relation formula between the vector magnetic potential A(r) containing the source point position vector and the magnetic field intensity H(r) is expanded in the cylindrical-coordinate system to obtain the following magnetic field intensity H(r): 
                     H   ⁡     (   r   )       =         1   μ     ⁢     ∇     ×     A   ⁡     (   r   )             =         -     φ   ^       ⁢     1   μ     ⁢       ∂       A   z     ⁡     (     ρ   ,   z     )           ∂   ρ         =       φ   ^     ⁢     H   φ     ⁢     ;                   (   4   )               
in the formula, {circumflex over (φ)} is the unit vector along the co direction.
 
step 4.2, the integral formula (3) of the vector magnetic potential A(r) containing the source point position vector is put into the formula (4) of the magnetic field intensity H(r), to obtain:
 
                     H   φ     =       -       I   0       4   ⁢           ⁢   π         ⁢       ∂               ∂   ρ       ⁢         ∫     l   ⁢     /     ⁢   2           -   l     ⁢     /     ⁢   2       ⁢       cos   ⁡     (     kz   ′     )       ⁢       e     -   jkR       R     ⁢     dz   ′     ⁢     ;                   (     5   ⁢   a     )               
step 4.3, the exact solution is obtained by using Euler formula and homogeneous Maxwell equation: according to the Euler formula:
 
                     cos   ⁡     (     kz   ′     )       =         e     j   ⁡     (     kz   ′     )         +     e     -     j   ⁡     (     kz   ′     )             2             (     5   ⁢   b     )               
formula (5b) is put into the formula (5a) for expansion to obtain
 
                           H   φ     =       ⁢       -       I   0       8   ⁢           ⁢   π         ⁢         ∫       -   l     ⁢     /     ⁢   2         l   ⁢     /     ⁢   2       ⁢       [           ∂               ∂   ρ       ⁢       e     -     jk   ⁡     (     R   -     z   ′       )           R       +         ∂               ∂   ρ       ⁢       e     -     jk   ⁡     (     R   +     z   ′       )           R         ]     ⁢     dz   ′                       =       ⁢       -       I   0       8   ⁢           ⁢   π         ⁢     {         ∫     l   ⁢     /     ⁢   2           -   l     ⁢     /     ⁢   2       ⁢     [         1   R     ⁢       ∂               ∂   ρ       ⁢     e     -     jk   ⁡     (     R   -     z   ′       )             +       e     -     jk   ⁡     (     R   -     z   ′       )           ⁢       ∂               ∂   ρ       ⁢     1   R     ⁢     dz   ′       +                             ⁢         ∫     l   ⁢     /     ⁢   2           -   l     ⁢     /     ⁢   2       ⁢       [         1   R     ⁢       ∂               ∂   ρ       ⁢     e     -     jk   ⁡     (     R   +     z   ′       )             +       e     -     jk   ⁡     (     R   +     z   ′       )           ⁢       ∂               ∂   ρ       ⁢     1   R         ]     ⁢     dz   ′         }               =       ⁢       -       I   0       8   ⁢           ⁢   π         ⁢     {       ρ   ⁢           ⁢         ∫     l   ⁢     /     ⁢   2           -   l     ⁢     /     ⁢   2       ⁢       [         -   jk     ⁢           ⁢       e     -     jk   ⁡     (     R   -     z   ′       )             R   2         -       e     -     jk   ⁡     (     R   -     z   ′       )             R   3         ]     ⁢     dz   ′           +     ρ   ⁢         ∫     l   ⁢     /     ⁢   2           -   l     ⁢     /     ⁢   2       ⁢     [         -   jk     ⁢           ⁢       e     -     jk   (     R   +     z   ′               R   2         -                                     ⁢       e     -     jk   ⁡     (     R   +     z   ′       )             R   3       ]     ⁢     dz   ′       }               =       ⁢       -       I   0       8   ⁢           ⁢   π         ⁢     {       ρ   ⁢         ∫     l   ⁢     /     ⁢   2           -   l     ⁢     /     ⁢   2       ⁢         e     -     jk   ⁡     (     R   -     z   ′       )           ⁡     (         -   jk     ⁢           ⁢     1     R   2         -     1     R   3         )       ⁢     dz   ′           +     ρ   ⁢         ∫     l   ⁢     /     ⁢   2           -   l     ⁢     /     ⁢   2       ⁢       e     -     jk   ⁡     (     R   +     z   ′       )           (         -   jk     ⁢           ⁢     1     R   2         -                                     ⁢     1     R   3       )     ⁢     dz   ′       }               =       ⁢       -       I   0       8   ⁢           ⁢   π         ⁢     {     ρ   ⁢         ∫     l   ⁢     /     ⁢   2           -   l     ⁢     /     ⁢   2       ⁢       e     -     jk   ⁡     (     R   -     z   ′       )           [         -   jk     ⁢           ⁢     1     R   2       ⁢     dz   ′       +           z   ′     -   z         R   3     ⁡     (     R   -     z   ′     +   z     )         ⁢     dz   ′       -                                 ⁢       1       R   2     ⁡     (     R   -     z   ′     +   z     )         ⁢     dz   ′       ]     +     ρ   ⁢         ∫     l   ⁢     /     ⁢   2           -   l     ⁢     /     ⁢   2       ⁢       e     -     jk   ⁡     (     R   +     z   ′       )           [         -   jk     ⁢           ⁢     1     R   2       ⁢     dz   ′       -                             ⁢             z   ′     -   z         R   3     ⁡     (     R   +     z   ′     -   z     )         ⁢     dz   ′       -       1       R   2     ⁡     (     R   +     z   ′     -   z     )         ⁢     dz   ′         ]     }               =       ⁢       -       I   0       8   ⁢           ⁢   π         ⁢     {     ρ   ⁢         ∫     l   ⁢     /     ⁢   2           -   l     ⁢     /     ⁢   2       ⁢       e     -     jk   ⁡     (     R   -     z   ′       )           [       jk   ⁢           ⁢       d   ⁡     (     R   -     z   ′       )         R   ⁡     (     R   -     z   ′     +   z     )           -         dR     -   1         (     R   -     z   ′     +   z     )       ⁢     dz   ′       -                                 ⁢         d   ⁡     (     R   -     z   ′     +   z     )         -   1       R     ]     +     ρ   ⁢         ∫     l   ⁢     /     ⁢   2           -   l     ⁢     /     ⁢   2       ⁢       e     -     jk   ⁡     (     R   +     z   ′       )           [         -   jk     ⁢           ⁢       d   ⁡     (     R   +     z   ′       )         R   ⁡     (     R   +     z   ′     -   z     )           +                             ⁢         dR     -   1         (     R   +     z   ′     -   z     )       +         d   ⁡     (     R   +     z   ′     -   z     )         -   1       R       ]     }               =       ⁢         I   0       8   ⁢           ⁢   π       ⁢     {       ρ   ⁢           ⁢         ∫     l   ⁢     /     ⁢   2           -   l     ⁢     /     ⁢   2       ⁢     d   ⁡     [       e     -     jk   ⁡     (     R   -     z   ′       )             R   ⁡     (     R   -     z   ′     +   z     )         ]           -     ρ   ⁢         ∫     l   ⁢     /     ⁢   2           -   l     ⁢     /     ⁢   2       ⁢     d   ⁡     [       e     -     jk   ⁡     (     R   +     z   ′       )             R   ⁡     (     R   +     z   ′     -   z     )         ]             }                   =       ⁢             I   0     ⁢   ρ       8   ⁢           ⁢   π       ⁡     [         e     -     jk   ⁡     (     R   -     z   ′       )             R   ⁡     (     R   -     z   ′     +   z     )         +       e     -     jk   ⁡     (     R   +     z   ′       )             R   ⁡     (     R   +     z   ′     -   z     )           ]       ⁢     |       -   l     ⁢     /     ⁢   2       l   ⁢     /     ⁢   2                     =       ⁢           I   0     ⁢   ρ       8   ⁢           ⁢   π       [         e     -     jk   ⁡     (       R   1     -     l   ⁢     /     ⁢   2       )               R   1     ⁡     (       R   1     -     l   ⁢     /     ⁢   2     +   z     )         -       e     -     jk   ⁡     (       R   2     +     l   ⁢     /     ⁢   2       )               R   2     ⁡     (       R   2     +     l   ⁢     /     ⁢   2     -   z     )         -       e     -     jk   ⁡     (       R   1     +     l   ⁢     /     ⁢   2       )               R   1     ⁡     (       R   1     +     l   ⁢     /     ⁢   2     -   z     )         +                       ⁢       e     -     jk   ⁡     (       R   2     -     l   ⁢     /     ⁢   2       )               R   2     ⁡     (       R   2     -     l   ⁢     /     ⁢   2     -   z     )         ]               =       ⁢           I   0     ⁢   ρ       8   ⁢           ⁢   π       ⁢     {           e     -     jkR   1           R   1       ⁡     [         e     jk   ⁢           ⁢   l   ⁢     /     ⁢   2         (       R   1     -     l   ⁢     /     ⁢   2     +   z     )       -       e       -   jk     ⁢           ⁢   l   ⁢     /     ⁢   2         (       R   1     +     l   ⁢     /     ⁢   2     -   z     )         ]       +                         ⁢         e     -     jkR   2           R   2       ⁡     [         e     jk   ⁢           ⁢   l   ⁢     /     ⁢   2         (       R   2     -     l   ⁢     /     ⁢   2     -   z     )       -       e       -   jk     ⁢           ⁢   l   ⁢     /     ⁢   2         (       R   2     +     l   ⁢     /     ⁢   2     -   z     )         ]       }               =       ⁢           I   0     ⁢   ρ       8   ⁢           ⁢   π       ⁢     {           e     -     jkR   1           R   1       ⁡     [           cos   ⁡     (     kl   ⁢     /     ⁢   2     )       +     j   ⁢           ⁢     sin   ⁡     (     kl   ⁢     /     ⁢   2     )             (       R   1     +     l   ⁢     /     ⁢   2     +   z     )       -         cos   ⁡     (     kl   ⁢     /     ⁢   2     )       -     j   ⁢           ⁢     sin   ⁡     (     kl   ⁢     /     ⁢   2     )             (       R   1     +     l   ⁢     /     ⁢   2     -   z     )         ]       -                         ⁢         e     -     jkR   2           R   2       ⁡     [           cos   ⁡     (     kl   ⁢     /     ⁢   2     )       +     j   ⁢           ⁢     sin   ⁡     (     kl   ⁢     /     ⁢   2     )             (       R   2     -     l   ⁢     /     ⁢   2     -   z     )       -         cos   ⁡     (     kl   ⁢     /     ⁢   2     )       -     j   ⁢           ⁢     sin   ⁡     (     kl   ⁢     /     ⁢   2     )             (       R   2     +     l   ⁢     /     ⁢   2     -   z     )         ]       }               =       ⁢         I   0       4   ⁢           ⁢   π   ⁢           ⁢   ρ       [         (             l   ⁢     /     ⁢   2     -   z       R   1       ⁢     e     -     jkR   1           +           l   ⁢     /     ⁢   2     +   z       R   2       ⁢     e     -     jkR   2             )     ⁢     cos   ⁡     (     kl   2     )         +                         ⁢       j   ⁡     (       e     -     jkR   1         +     e     -     jkR   2           )       ⁢     sin   ⁡     (     kl   2     )         ]     ⁢     ;                   (   6   )               
according to the homogeneous Maxwell equation:
 
                     E   ⁡     (   r   )       =       1       j   ⁢           ⁢   ω   ⁢           ⁢   ɛ     +   σ       ⁢     ∇     ×     H   ⁡     (   r   )                     (   7   )               
an electric field expression is obtained:
 
                           E   ρ     =       ⁢       -     1       j   ⁢           ⁢   ω   ⁢           ⁢   ɛ     +   σ         ⁢       ∂     H   φ         ∂   z                     =       ⁢         j   ⁢           ⁢     I   0         4   ⁢           ⁢   π   ⁢           ⁢   ρ       ⁢       μ     ɛ   -     j   ⁢           ⁢     σ   ω             ⁢     {     [     (       1     kR   1   3       +       j     R   1   2       ⁢       (       l   2     -   z     )     2     ⁢     e     -     jkR   1           -       1     kR   1       ⁢     e     -     jkR   1           -                                 ⁢         (       1     kR   2   3       +     j     R   2   2         )     ⁢       (       l   2     +   z     )     2     ⁢     e     -     jkR   2           +       1     kR   2       ⁢     e     -     jkR   2             ]     ⁢     cos   ⁡     (     kl   2     )         -                     ⁢       [         1     R   1       ⁢     (       l   2     -   z     )     ⁢     e     -     jkR   1           -       1     R   2       ⁢     (       l   2     +   z     )     ⁢     e     -     jkR   2             ]     ⁢     sin   ⁡     (     kl   2     )         }     ⁢     ;                   (   8   )                       E   z     =       ⁢         1       j   ⁢           ⁢   ωɛ     +   σ       ⁢     1   ρ     ⁢       ∂     (     ρ   ⁢           ⁢     H   φ       )         ∂   ρ         =       1       j   ⁢           ⁢   ω   ⁢           ⁢   ɛ     +   σ       ⁢     (         ∂     H   φ         ∂   ρ       +       1   ρ     ⁢     H   φ         )                     =       ⁢         j   ⁢           ⁢     I   0         4   ⁢           ⁢   π       ⁢       μ     ɛ   -     j   ⁢           ⁢     σ   ω             ⁢     {     [         (       j     R   1   2       +     1     kR   1   3         )     ⁢     (       l   2     -   z     )     ⁢     e     -     jkR   1           +       (       j     R   2   2       +     1     kR   2   3         )     ⁢     (       l   2     +                                             ⁢   z   )     ⁢     e     -     jkR   2           ]     ⁢     cos   ⁡     (     kl   2     )         -       [         1     R   1       ⁢     e     -     jkR   1           +       1     R   2       ⁢     e     -     jkR   2             ]     ⁢     sin   ⁡     (     kl   2     )           }     ⁢     ;                   (   9   )               
where
 
 R   1 =√{square root over (ρ 2 +( z−l/ 2) 2 )},  (10a)
 
 R   2 =√{square root over (ρ 2 ( z+l/ 2) 2 )};  (10b)
 
     Example 1, in the free space I 0 =1 A, l=2 m, f=10 6  Hz, μ=μ 0 =4π×10 −7  H/m, 
             ɛ   =       ɛ   0     ≈       10     -   9         36   ⁢           ⁢   π               
F/m and σ=0 S/m are put into the formula (2) to obtain current distribution of a finite-length harmonic linear current source in a free space, as shown in  FIG. 2 ;
 
according to the formula (6), formula (8) and formula (9) of the exact solution, in the free space, the field generated by the harmonic linear current source is (wherein l=2 m, I 0 =1 A and f=10 6  Hz):
 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           H 
                           φ 
                         
                         = 
                           
                         ⁢ 
                         
                           
                             
                               I 
                               0 
                             
                             
                               4 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ρ 
                             
                           
                           [ 
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         
                                           l 
                                           ⁢ 
                                           
                                             / 
                                           
                                           ⁢ 
                                           2 
                                         
                                         - 
                                         z 
                                       
                                       
                                         R 
                                         1 
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         - 
                                         
                                           jkR 
                                           1 
                                         
                                       
                                     
                                   
                                   + 
                                   
                                     
                                       
                                         
                                           l 
                                           ⁢ 
                                           
                                             / 
                                           
                                           ⁢ 
                                           2 
                                         
                                         + 
                                         z 
                                       
                                       
                                         R 
                                         2 
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         - 
                                         
                                           jkR 
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     kl 
                                     2 
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               j 
                               ( 
                               
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       1 
                                     
                                   
                                 
                                 + 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                               
                             ⁢ 
                             
                               e 
                               
                                 - 
                                 
                                   jkR 
                                   2 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 kl 
                                 2 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             1 
                             
                               4 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ρ 
                             
                           
                           [ 
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         1 
                                         - 
                                         z 
                                       
                                       
                                         R 
                                         1 
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         - 
                                         
                                           jkR 
                                           1 
                                         
                                       
                                     
                                   
                                   + 
                                   
                                     
                                       
                                         1 
                                         + 
                                         z 
                                       
                                       
                                         R 
                                         2 
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         - 
                                         
                                           jkR 
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                             + 
                             
                               j 
                               ( 
                               
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       1 
                                     
                                   
                                 
                                 + 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                               
                             ⁢ 
                             
                               e 
                               
                                 - 
                                 
                                   jkR 
                                   2 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           E 
                           ρ 
                         
                         = 
                           
                         ⁢ 
                         
                           
                             
                               jI 
                               0 
                             
                             
                               4 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ρ 
                             
                           
                           ⁢ 
                           
                             
                               μ 
                               
                                 ɛ 
                                 - 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     σ 
                                     ω 
                                   
                                 
                               
                             
                           
                           ⁢ 
                           
                             { 
                             
                               [ 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         1 
                                         
                                           kR 
                                           1 
                                           3 
                                         
                                       
                                       + 
                                       
                                         j 
                                         
                                           R 
                                           1 
                                           2 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     
                                       ( 
                                       
                                         
                                           l 
                                           2 
                                         
                                         - 
                                         z 
                                       
                                       ) 
                                     
                                     2 
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       - 
                                       
                                         jkR 
                                         1 
                                       
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     1 
                                     
                                       kR 
                                       1 
                                     
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       - 
                                       
                                         jkR 
                                         1 
                                       
                                     
                                   
                                 
                                 - 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                               
                             ⁢ 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       1 
                                       
                                         kR 
                                         2 
                                         3 
                                       
                                     
                                     + 
                                     
                                       j 
                                       
                                         R 
                                         2 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   
                                     ( 
                                     
                                       
                                         l 
                                         2 
                                       
                                       + 
                                       z 
                                     
                                     ) 
                                   
                                   2 
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       2 
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                   1 
                                   
                                     kR 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       2 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 kl 
                                 2 
                               
                               ) 
                             
                           
                         
                         - 
                         
                           [ 
                           
                             
                               1 
                               
                                 R 
                                 1 
                               
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   l 
                                   2 
                                 
                                 - 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             
                               
                                 
                                     
                                   ⁢ 
                                   z 
                                   ) 
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       1 
                                     
                                   
                                 
                               
                               - 
                               
                                 
                                   1 
                                   
                                     R 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       l 
                                       2 
                                     
                                     + 
                                     z 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       2 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 kl 
                                 2 
                               
                               ) 
                             
                           
                         
                         } 
                       
                     
                   
                   
                     
                       
                         ≈ 
                           
                         ⁢ 
                         
                           
                             
                               j 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               30 
                             
                             ρ 
                           
                           ⁢ 
                           
                             { 
                             
                               [ 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         1 
                                         
                                           kR 
                                           1 
                                           3 
                                         
                                       
                                       + 
                                       
                                         j 
                                         
                                           R 
                                           1 
                                           2 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     
                                       ( 
                                       
                                         1 
                                         - 
                                         z 
                                       
                                       ) 
                                     
                                     2 
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       - 
                                       
                                         jkR 
                                         1 
                                       
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     1 
                                     
                                       kR 
                                       1 
                                     
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       - 
                                       
                                         jkR 
                                         1 
                                       
                                     
                                   
                                 
                                 - 
                                 
                                   ( 
                                   
                                     
                                       1 
                                       
                                         kR 
                                         2 
                                         3 
                                       
                                     
                                     + 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             
                               
                                 
                                     
                                   ⁢ 
                                   
                                     j 
                                     
                                       R 
                                       2 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   
                                     ( 
                                     
                                       1 
                                       + 
                                       z 
                                     
                                     ) 
                                   
                                   2 
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       2 
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                   1 
                                   
                                     kR 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       2 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                         - 
                         
                           [ 
                           
                             
                               
                                 1 
                                 
                                   R 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   z 
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   - 
                                   
                                     jkR 
                                     1 
                                   
                                 
                               
                             
                             - 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                               
                             ⁢ 
                             
                               
                                 1 
                                 
                                   R 
                                   2 
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   z 
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   - 
                                   
                                     jkR 
                                     2 
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           E 
                           z 
                         
                         = 
                           
                         ⁢ 
                         
                           
                             
                               jI 
                               0 
                             
                             
                               4 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               π 
                             
                           
                           ⁢ 
                           
                             
                               μ 
                               
                                 ɛ 
                                 - 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     σ 
                                     ω 
                                   
                                 
                               
                             
                           
                           ⁢ 
                           
                             { 
                             
                               [ 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         j 
                                         
                                           R 
                                           1 
                                           2 
                                         
                                       
                                       + 
                                       
                                         1 
                                         
                                           kR 
                                           1 
                                           3 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         l 
                                         2 
                                       
                                       - 
                                       z 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       - 
                                       
                                         jkR 
                                         1 
                                       
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       
                                         j 
                                         
                                           R 
                                           2 
                                           2 
                                         
                                       
                                       + 
                                       
                                         1 
                                         
                                           kR 
                                           2 
                                           3 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         l 
                                         2 
                                       
                                       + 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             
                               
                                 
                                     
                                   ⁢ 
                                   z 
                                   ) 
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       2 
                                     
                                   
                                 
                               
                               ] 
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 
                                   kl 
                                   2 
                                 
                                 ) 
                               
                             
                           
                           - 
                           
                             
                               [ 
                               
                                 
                                   
                                     1 
                                     
                                       R 
                                       1 
                                     
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       - 
                                       
                                         jkR 
                                         1 
                                       
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     1 
                                     
                                       R 
                                       2 
                                     
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       - 
                                       
                                         jkR 
                                         2 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 
                                   kl 
                                   2 
                                 
                                 ) 
                               
                             
                           
                         
                         } 
                       
                     
                   
                   
                     
                       
                         ≈ 
                           
                         ⁢ 
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           30 
                           ⁢ 
                           
                             { 
                             
                               [ 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         j 
                                         
                                           R 
                                           1 
                                           2 
                                         
                                       
                                       + 
                                       
                                         1 
                                         
                                           kR 
                                           1 
                                           3 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       1 
                                       - 
                                       z 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       - 
                                       
                                         jkR 
                                         1 
                                       
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       
                                         j 
                                         
                                           R 
                                           2 
                                           2 
                                         
                                       
                                       + 
                                       
                                         1 
                                         
                                           kR 
                                           2 
                                           3 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       1 
                                       + 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             
                               
                                 
                                     
                                   ⁢ 
                                   z 
                                   ) 
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       2 
                                     
                                   
                                 
                               
                               ] 
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           - 
                           
                             
                               [ 
                               
                                 
                                   
                                     1 
                                     
                                       R 
                                       1 
                                     
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       - 
                                       
                                         jkR 
                                         1 
                                       
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     1 
                                     
                                       R 
                                       2 
                                     
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       - 
                                       
                                         jkR 
                                         2 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     According to the formula (10), it can be known that in the above formula (11), formula (12) and formula (13):
 
 R   1 =√{square root over (ρ 2 ( z−l/ 2) 2 )}=√{square root over (ρ 2 +( z− 1) 2 )}
 
 R   2 =√{square root over (ρ 2 +( z+l/ 2) 2 )}=√{square root over (ρ 2 ( z+ 1) 2 )}
 
     According to the formula (1b), it can be known that the wave number k in the above formula (11), formula (12) and formula (13) are: 
     
       
         
           
             k 
             = 
             
               
                 
                   
                     2 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     π 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     f 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       μ 
                       ⁡ 
                       
                         ( 
                         
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             f 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ɛ 
                           
                           - 
                           
                             j 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             σ 
                           
                         
                         ) 
                       
                     
                   
                 
                 ≈ 
                 
                   
                     2 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     π 
                     × 
                     
                       10 
                       6 
                     
                     × 
                     4 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     π 
                     × 
                     
                       10 
                       
                         - 
                         7 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                           × 
                           
                             10 
                             6 
                           
                           × 
                           
                             
                               10 
                               
                                 - 
                                 9 
                               
                             
                             
                               36 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               π 
                             
                           
                         
                         - 
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           0 
                         
                       
                       ) 
                     
                   
                 
               
               = 
               
                 
                   
                     8 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       π 
                       2 
                     
                     × 
                     
                       10 
                       
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           10 
                           
                             - 
                             3 
                           
                         
                         18 
                       
                       ) 
                     
                   
                 
                 = 
                 
                   
                     
                       2 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       π 
                     
                     3 
                   
                   × 
                   
                     10 
                     
                       - 
                       2 
                     
                   
                 
               
             
           
         
       
     
     The field point coordinate P(ρ,φ,z) is put into the formula (11), formula (12) and formula (13) to obtain the spatial distribution of the field generated by the finite-length harmonic linear current source in the free space, wherein l=2 m, I 0 =1 A and f=10 6  Hz. 
     Example 2, in the wet soil environment 
     I 0 =1 A, l=2000 m, f=100 Hz, μ=μ 0 =4π×10 −7  H/m, 
             ɛ   =       10   ×     ɛ   0       ≈     10   ⁢           ⁢       10     -   9         36   ⁢           ⁢   π       ⁢   F   ⁢     /     ⁢   m             
and σ=0.01 S/m are put into the formula (2) to obtain current distribution of a finite-length harmonic linear current source in the wet soil environment, as shown in  FIG. 3 ;
 
according to the formula (6), formula (8) and formula (9) of the exact solution, in the wet soil environment, the field generated by the harmonic line current source is (wherein l=2000 m, I 0 =1 A and f=100 Hz):
 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           H 
                           φ 
                         
                         = 
                           
                         ⁢ 
                         
                           
                             
                               I 
                               0 
                             
                             
                               4 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ρ 
                             
                           
                           [ 
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         
                                           l 
                                           ⁢ 
                                           
                                             / 
                                           
                                           ⁢ 
                                           2 
                                         
                                         - 
                                         z 
                                       
                                       
                                         R 
                                         1 
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         - 
                                         
                                           jkR 
                                           1 
                                         
                                       
                                     
                                   
                                   + 
                                   
                                     
                                       
                                         
                                           l 
                                           ⁢ 
                                           
                                             / 
                                           
                                           ⁢ 
                                           2 
                                         
                                         + 
                                         z 
                                       
                                       
                                         R 
                                         2 
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         - 
                                         
                                           jkR 
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     kl 
                                     2 
                                   
                                   ) 
                                 
                               
                             
                             + 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             j 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       1 
                                     
                                   
                                 
                                 + 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       2 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 kl 
                                 2 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             1 
                             
                               4 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ρ 
                             
                           
                           [ 
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         1000 
                                         - 
                                         z 
                                       
                                       
                                         R 
                                         1 
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         - 
                                         
                                           jkR 
                                           1 
                                         
                                       
                                     
                                   
                                   + 
                                   
                                     
                                       
                                         1000 
                                         + 
                                         z 
                                       
                                       
                                         R 
                                         2 
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         - 
                                         
                                           jkR 
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     1000 
                                     ⁢ 
                                     k 
                                   
                                   ) 
                                 
                               
                             
                             + 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             j 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       1 
                                     
                                   
                                 
                                 + 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       2 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 1000 
                                 ⁢ 
                                 k 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                   
                     
                       
                         
                           E 
                           ρ 
                         
                         = 
                           
                         ⁢ 
                         
                           
                             
                               jI 
                               0 
                             
                             
                               4 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ρ 
                             
                           
                           ⁢ 
                           
                             
                               μ 
                               
                                 ɛ 
                                 - 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     σ 
                                     ω 
                                   
                                 
                               
                             
                           
                           ⁢ 
                           
                             { 
                             
                               [ 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         1 
                                         
                                           kR 
                                           1 
                                           3 
                                         
                                       
                                       + 
                                       
                                         j 
                                         
                                           R 
                                           1 
                                           2 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     
                                       ( 
                                       
                                         
                                           l 
                                           2 
                                         
                                         - 
                                         z 
                                       
                                       ) 
                                     
                                     2 
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       - 
                                       
                                         jkR 
                                         1 
                                       
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     1 
                                     
                                       kR 
                                       1 
                                     
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       - 
                                       
                                         jkR 
                                         1 
                                       
                                     
                                   
                                 
                                 - 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                               
                             ⁢ 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       1 
                                       
                                         kR 
                                         2 
                                         3 
                                       
                                     
                                     + 
                                     
                                       j 
                                       
                                         R 
                                         2 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   
                                     ( 
                                     
                                       
                                         l 
                                         2 
                                       
                                       + 
                                       z 
                                     
                                     ) 
                                   
                                   2 
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       2 
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                   1 
                                   
                                     kR 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       2 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 kl 
                                 2 
                               
                               ) 
                             
                           
                         
                         - 
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             [ 
                             
                               
                                 
                                   1 
                                   
                                     R 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       l 
                                       2 
                                     
                                     - 
                                     z 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       1 
                                     
                                   
                                 
                               
                               - 
                               
                                 
                                   1 
                                   
                                     R 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       l 
                                       2 
                                     
                                     + 
                                     z 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       2 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 kl 
                                 2 
                               
                               ) 
                             
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         ≈ 
                           
                         ⁢ 
                         
                           
                             0.05 
                             
                               ρ 
                               ⁢ 
                               
                                 5 
                               
                             
                           
                           ⁢ 
                           
                             { 
                             
                               [ 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         1 
                                         
                                           kR 
                                           1 
                                           3 
                                         
                                       
                                       + 
                                       
                                         j 
                                         
                                           R 
                                           1 
                                           2 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     
                                       ( 
                                       
                                         1000 
                                         - 
                                         z 
                                       
                                       ) 
                                     
                                     2 
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       - 
                                       
                                         jkR 
                                         1 
                                       
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     1 
                                     
                                       kR 
                                       1 
                                     
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       - 
                                       
                                         jkR 
                                         1 
                                       
                                     
                                   
                                 
                                 - 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                               
                             ⁢ 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       1 
                                       
                                         kR 
                                         2 
                                         3 
                                       
                                     
                                     + 
                                     
                                       j 
                                       
                                         R 
                                         2 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   
                                     ( 
                                     
                                       1000 
                                       + 
                                       z 
                                     
                                     ) 
                                   
                                   2 
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       2 
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                   1 
                                   
                                     kR 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       2 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 1000 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 k 
                               
                               ) 
                             
                           
                         
                         - 
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             [ 
                             
                               
                                 
                                   1 
                                   
                                     R 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     ( 
                                     
                                       1000 
                                       - 
                                       z 
                                     
                                     ) 
                                   
                                   
                                     - 
                                     
                                       jkR 
                                       1 
                                     
                                   
                                 
                               
                               - 
                               
                                 
                                   1 
                                   
                                     R 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     1000 
                                     + 
                                     z 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       2 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 1000 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 k 
                               
                               ) 
                             
                           
                         
                         } 
                       
                     
                   
                   
                     
                       
                         
                           E 
                           z 
                         
                         = 
                           
                         ⁢ 
                         
                           
                             
                               jI 
                               0 
                             
                             
                               4 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               π 
                             
                           
                           ⁢ 
                           
                             
                               μ 
                               
                                 ɛ 
                                 - 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     σ 
                                     ω 
                                   
                                 
                               
                             
                           
                           ⁢ 
                           
                             { 
                             
                               [ 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         j 
                                         
                                           R 
                                           1 
                                           2 
                                         
                                       
                                       + 
                                       
                                         1 
                                         
                                           kR 
                                           1 
                                           3 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         l 
                                         2 
                                       
                                       - 
                                       z 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       - 
                                       
                                         jkR 
                                         1 
                                       
                                     
                                   
                                 
                                 + 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                               
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   
                                     j 
                                     
                                       R 
                                       2 
                                       2 
                                     
                                   
                                   + 
                                   
                                     1 
                                     
                                       kR 
                                       2 
                                       3 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     l 
                                     2 
                                   
                                   + 
                                   z 
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   - 
                                   
                                     jkR 
                                     2 
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 kl 
                                 2 
                               
                               ) 
                             
                           
                         
                         - 
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             [ 
                             
                               
                                 
                                   1 
                                   
                                     R 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       1 
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                   1 
                                   
                                     R 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       2 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 kl 
                                 2 
                               
                               ) 
                             
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                             
                         
                         ⁢ 
                         
                           ≈ 
                             
                           ⁢ 
                           
                             
                               0.05 
                               
                                 5 
                               
                             
                             ⁢ 
                             
                               { 
                               
                                 [ 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           j 
                                           
                                             R 
                                             1 
                                             2 
                                           
                                         
                                         + 
                                         
                                           1 
                                           
                                             kR 
                                             1 
                                             3 
                                           
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         1000 
                                         - 
                                         z 
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         - 
                                         
                                           jkR 
                                           1 
                                         
                                       
                                     
                                   
                                   + 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                               
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   
                                     j 
                                     
                                       R 
                                       2 
                                       2 
                                     
                                   
                                   + 
                                   
                                     1 
                                     
                                       kR 
                                       2 
                                       3 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   1000 
                                   + 
                                   z 
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   - 
                                   
                                     jkR 
                                     2 
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 1000 
                                 ⁢ 
                                 k 
                               
                               ) 
                             
                           
                         
                         - 
                         
                           [ 
                           
                             
                               
                                 1 
                                 
                                   R 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   - 
                                   
                                     jkR 
                                     1 
                                   
                                 
                               
                             
                             + 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                               
                             ⁢ 
                             
                               
                                 1 
                                 
                                   R 
                                   2 
                                 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   - 
                                   
                                     jkR 
                                     2 
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 1000 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 k 
                               
                               ) 
                             
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     According to the formula (10), it can be known that in the above formula (14), formula (15) and formula (16):
 
 R   1 =√{square root over (ρ 2 +( z−l/ 2) 2 )}=√{square root over (ρ 2 +( z− 1000) 2 )}
 
 R   2 =√{square root over (ρ 2 +( z+l/ 2) 2 )}=√{square root over (ρ 2 ( z+ 1000) 2 )}
 
     According to the formula (1b), it can be known that the wave number k in the above formula (14), formula (15) and formula (16) are: 
     
       
         
           
             k 
             = 
             
               
                 
                   
                     2 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     π 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     f 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       μ 
                       ⁡ 
                       
                         ( 
                         
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             f 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ɛ 
                           
                           - 
                           
                             j 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             σ 
                           
                         
                         ) 
                       
                     
                   
                 
                 ≈ 
                 
                   
                     2 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     π 
                     × 
                     100 
                     × 
                     4 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     π 
                     × 
                     
                       10 
                       
                         - 
                         7 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                           × 
                           100 
                           × 
                           10 
                           ⁢ 
                           
                             
                               
                                   
                               
                               ⁢ 
                               
                                 10 
                                 
                                   - 
                                   9 
                                 
                               
                             
                             
                               36 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               π 
                             
                           
                         
                         - 
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           0.01 
                         
                       
                       ) 
                     
                   
                 
               
               = 
               
                 
                   
                     8 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       π 
                       2 
                     
                     × 
                     
                       10 
                       
                         - 
                         5 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           10 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               10 
                               
                                 - 
                                 7 
                               
                             
                             18 
                           
                         
                         - 
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             10 
                             
                               - 
                               2 
                             
                           
                         
                       
                       ) 
                     
                   
                 
                 = 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         π 
                       
                       3 
                     
                     × 
                     
                       10 
                       
                         - 
                         6 
                       
                     
                     ⁢ 
                     
                       
                         10 
                         - 
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           18 
                           × 
                           
                             10 
                             5 
                           
                         
                       
                     
                   
                   ≈ 
                   
                     j 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     4 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     π 
                     × 
                     
                       10 
                       
                         - 
                         4 
                       
                     
                     ⁢ 
                     
                       5 
                     
                   
                 
               
             
           
         
       
     
     The field point coordinate P(ρ,φ,z) is put into the formula (14), formula (15) and formula (16) to obtain the spatial distribution of the field generated by the finite-length harmonic linear current source the wet soil environment, wherein l=2000 m, I 0 =1 A and f=100 Hz. 
     Example 3, in the seawater environment 
     I 0 =1 A, l=500 m, f=100 Hz, μ=μ 0 =4π×10 −7  H/m, 
             ɛ   =       80   ×     ɛ   0       ≈     80   ⁢           ⁢       10     -   9         36   ⁢           ⁢   π       ⁢   F   ⁢     /     ⁢   m             
and σ=4 S/m are put into the formula (2) to obtain current distribution of a finite-length harmonic linear current source in the seawater environment, as shown in  FIG. 4 .
 
according to the formula (6), formula (8) and formula (9), in the seawater environment, the field generated by the harmonic line current source is (l=500 m, I 0 =1 A and f=100 Hz):
 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           H 
                           φ 
                         
                         = 
                           
                         ⁢ 
                         
                           
                             
                               I 
                               0 
                             
                             
                               4 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ρ 
                             
                           
                           [ 
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         
                                           l 
                                           ⁢ 
                                           
                                             / 
                                           
                                           ⁢ 
                                           2 
                                         
                                         - 
                                         z 
                                       
                                       
                                         R 
                                         1 
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         - 
                                         
                                           jkR 
                                           1 
                                         
                                       
                                     
                                   
                                   + 
                                   
                                     
                                       
                                         
                                           l 
                                           ⁢ 
                                           
                                             / 
                                           
                                           ⁢ 
                                           2 
                                         
                                         + 
                                         z 
                                       
                                       
                                         R 
                                         2 
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         - 
                                         
                                           jkR 
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     kl 
                                     2 
                                   
                                   ) 
                                 
                               
                             
                             + 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             j 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       1 
                                     
                                   
                                 
                                 + 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       2 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 kl 
                                 2 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             1 
                             
                               4 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ρ 
                             
                           
                           [ 
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         250 
                                         - 
                                         z 
                                       
                                       
                                         R 
                                         1 
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         - 
                                         
                                           jkR 
                                           1 
                                         
                                       
                                     
                                   
                                   + 
                                   
                                     
                                       
                                         250 
                                         + 
                                         z 
                                       
                                       
                                         R 
                                         2 
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         - 
                                         
                                           jkR 
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     250 
                                     ⁢ 
                                     k 
                                   
                                   ) 
                                 
                               
                             
                             + 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             j 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       1 
                                     
                                   
                                 
                                 + 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       2 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 250 
                                 ⁢ 
                                 k 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                   
                     
                       
                         
                           E 
                           ρ 
                         
                         = 
                           
                         ⁢ 
                         
                           
                             
                               jI 
                               0 
                             
                             
                               4 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               ρ 
                             
                           
                           ⁢ 
                           
                             
                               μ 
                               
                                 ɛ 
                                 - 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     σ 
                                     ω 
                                   
                                 
                               
                             
                           
                           ⁢ 
                           
                             { 
                             
                               [ 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         1 
                                         
                                           kR 
                                           1 
                                           3 
                                         
                                       
                                       + 
                                       
                                         j 
                                         
                                           R 
                                           1 
                                           2 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     
                                       ( 
                                       
                                         
                                           l 
                                           2 
                                         
                                         - 
                                         z 
                                       
                                       ) 
                                     
                                     2 
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       - 
                                       
                                         jkR 
                                         1 
                                       
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     1 
                                     
                                       kR 
                                       1 
                                     
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       - 
                                       
                                         jkR 
                                         1 
                                       
                                     
                                   
                                 
                                 - 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                               
                             ⁢ 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       1 
                                       
                                         kR 
                                         2 
                                         3 
                                       
                                     
                                     + 
                                     
                                       j 
                                       
                                         R 
                                         2 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   
                                     ( 
                                     
                                       
                                         l 
                                         2 
                                       
                                       + 
                                       z 
                                     
                                     ) 
                                   
                                   2 
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       2 
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                   1 
                                   
                                     kR 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       2 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 kl 
                                 2 
                               
                               ) 
                             
                           
                         
                         - 
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             [ 
                             
                               
                                 
                                   1 
                                   
                                     R 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       l 
                                       2 
                                     
                                     - 
                                     z 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       1 
                                     
                                   
                                 
                               
                               - 
                               
                                 
                                   1 
                                   
                                     R 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       l 
                                       2 
                                     
                                     + 
                                     z 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       2 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 kl 
                                 2 
                               
                               ) 
                             
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         ≈ 
                           
                         ⁢ 
                         
                           
                             0.0025 
                             
                               ρ 
                               ⁢ 
                               
                                 5 
                               
                             
                           
                           ⁢ 
                           
                             { 
                             
                               [ 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         1 
                                         
                                           kR 
                                           1 
                                           3 
                                         
                                       
                                       + 
                                       
                                         j 
                                         
                                           R 
                                           1 
                                           2 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     
                                       ( 
                                       
                                         250 
                                         - 
                                         z 
                                       
                                       ) 
                                     
                                     2 
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       - 
                                       
                                         jkR 
                                         1 
                                       
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     1 
                                     
                                       kR 
                                       1 
                                     
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       - 
                                       
                                         jkR 
                                         1 
                                       
                                     
                                   
                                 
                                 - 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                               
                             ⁢ 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       1 
                                       
                                         kR 
                                         2 
                                         3 
                                       
                                     
                                     + 
                                     
                                       j 
                                       
                                         R 
                                         2 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   
                                     ( 
                                     
                                       250 
                                       + 
                                       z 
                                     
                                     ) 
                                   
                                   2 
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       2 
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                   1 
                                   
                                     kR 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       2 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 250 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 k 
                               
                               ) 
                             
                           
                         
                         - 
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             [ 
                             
                               
                                 
                                   1 
                                   
                                     R 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     250 
                                     - 
                                     z 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       1 
                                     
                                   
                                 
                               
                               - 
                               
                                 
                                   1 
                                   
                                     R 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     250 
                                     + 
                                     z 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       2 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 250 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 k 
                               
                               ) 
                             
                           
                         
                         } 
                       
                     
                   
                   
                     
                       
                         
                           E 
                           z 
                         
                         = 
                           
                         ⁢ 
                         
                           
                             
                               jI 
                               0 
                             
                             
                               4 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               π 
                             
                           
                           ⁢ 
                           
                             
                               μ 
                               
                                 ɛ 
                                 - 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     σ 
                                     ω 
                                   
                                 
                               
                             
                           
                           ⁢ 
                           
                             { 
                             
                               [ 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         j 
                                         
                                           R 
                                           1 
                                           2 
                                         
                                       
                                       + 
                                       
                                         1 
                                         
                                           kR 
                                           1 
                                           3 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         l 
                                         2 
                                       
                                       - 
                                       z 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       - 
                                       
                                         jkR 
                                         1 
                                       
                                     
                                   
                                 
                                 + 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                               
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   
                                     j 
                                     
                                       R 
                                       2 
                                       2 
                                     
                                   
                                   + 
                                   
                                     1 
                                     
                                       kR 
                                       2 
                                       3 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     l 
                                     2 
                                   
                                   + 
                                   z 
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   - 
                                   
                                     jkR 
                                     2 
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 kl 
                                 2 
                               
                               ) 
                             
                           
                         
                         - 
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             [ 
                             
                               
                                 
                                   1 
                                   
                                     R 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       1 
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                   1 
                                   
                                     R 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     - 
                                     
                                       jkR 
                                       2 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 kl 
                                 2 
                               
                               ) 
                             
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                             
                         
                         ⁢ 
                         
                           ≈ 
                             
                           ⁢ 
                           
                             
                               0.0025 
                               
                                 ρ 
                                 ⁢ 
                                 
                                   5 
                                 
                               
                             
                             ⁢ 
                             
                               { 
                               
                                 [ 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           j 
                                           
                                             R 
                                             1 
                                             2 
                                           
                                         
                                         + 
                                         
                                           1 
                                           
                                             kR 
                                             1 
                                             3 
                                           
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         250 
                                         - 
                                         z 
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       e 
                                       
                                         - 
                                         
                                           jkR 
                                           1 
                                         
                                       
                                     
                                   
                                   + 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                               
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   
                                     j 
                                     
                                       R 
                                       2 
                                       2 
                                     
                                   
                                   + 
                                   
                                     1 
                                     
                                       kR 
                                       2 
                                       3 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   250 
                                   + 
                                   z 
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   - 
                                   
                                     jkR 
                                     2 
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 250 
                                 ⁢ 
                                 k 
                               
                               ) 
                             
                           
                         
                         - 
                         
                           [ 
                           
                             
                               
                                 1 
                                 
                                   R 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   - 
                                   
                                     jkR 
                                     1 
                                   
                                 
                               
                             
                             + 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                               
                             ⁢ 
                             
                               
                                 1 
                                 
                                   R 
                                   2 
                                 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   - 
                                   
                                     jkR 
                                     2 
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 250 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 k 
                               
                               ) 
                             
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     According to the formula (10), it can be known that in the above formula (17), formula (18) and formula (19):
 
 R   1 =√{square root over (ρ 2 +( z−l/ 2) 2 )}=√{square root over (ρ 2 ( z −250) 2 )}
 
 R   2 =√{square root over (ρ 2 +( z+l/ 2) 2 )}=√{square root over (ρ 2 ( z+ 250) 2 )}
 
     According to the formula (1b), it can be known that the wave number k in the above formula (17), formula (18) and formula (19) are: 
     
       
         
           
             k 
             = 
             
               
                 
                   
                     2 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     π 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     f 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       μ 
                       ⁡ 
                       
                         ( 
                         
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             f 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ɛ 
                           
                           - 
                           
                             j 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             σ 
                           
                         
                         ) 
                       
                     
                   
                 
                 ≈ 
                 
                   
                     2 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     π 
                     × 
                     100 
                     × 
                     4 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     π 
                     × 
                     
                       10 
                       
                         - 
                         7 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                           × 
                           100 
                           × 
                           80 
                           ⁢ 
                           
                             
                               
                                   
                               
                               ⁢ 
                               
                                 10 
                                 
                                   - 
                                   9 
                                 
                               
                             
                             
                               36 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               π 
                             
                           
                         
                         - 
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           4 
                         
                       
                       ) 
                     
                   
                 
               
               = 
               
                 
                   
                     8 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       π 
                       2 
                     
                     × 
                     
                       10 
                       
                         - 
                         5 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           80 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               10 
                               
                                 - 
                                 7 
                               
                             
                             18 
                           
                         
                         - 
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           4 
                         
                       
                       ) 
                     
                   
                 
                 = 
                 
                   
                     
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         π 
                       
                       3 
                     
                     × 
                     
                       10 
                       
                         - 
                         6 
                       
                     
                     ⁢ 
                     
                       
                         80 
                         - 
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           72 
                           × 
                           
                             10 
                             7 
                           
                         
                       
                     
                   
                   ≈ 
                   
                     j 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     8 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     π 
                     × 
                     
                       10 
                       
                         - 
                         3 
                       
                     
                     ⁢ 
                     
                       5 
                     
                   
                 
               
             
           
         
       
     
     The field point coordinate P(ρ,φ,z) is put into the formula (17), formula (18) and formula (19) to obtain the spatial distribution of the field generated by the finite-length harmonic linear current source in the seawater environment (wherein l=500 m, I 0 =1 A and f=100 Hz). 
     In conclusion, the exact solution for the field generated by a finite-length harmonic linear current source in the whole space is obtained with the method of solving the closed form exact solution for the field generated by a finite-length harmonic linear current source in the whole space provided in the present invention; the closed form exact solution is not only applicable in the far zone, the middle zone and the near zone, but also can better reflect the situation of distribution along the linear source of the harmonic current; and compared with the original uniform current expression, the cosine current expression of the present invention has better adaptability for changes of the working frequency and the conductivity of the medium. 
     The above examples are only used for illustrating the design idea and characteristics of the present invention and the purpose thereof is to enable the person skilled in the art to understand the contents of the present invention and make implementation; and the protection scope of the present invention is not limited to the above examples. Therefore, the equivalent changes or modifications made on the basis of principles and design idea disclosed in the present invention are within the protection scope of the present invention.