亥姆霍兹方程[Helmholtz Equation] - 达人(其实～我是卧底，来自外星)

2005/02/14 | 亥姆霍兹方程[Helmholtz Equation]
类别(Ω〖物理〗) | 评论(0) | 阅读(1040) | 发表于 13:45: The scalar equation is
(1)

and the vector equation is
(2)

The electromagnetic Helmholtz equation arises in the manipulation of the Maxwell equations to obtain propagating waves,
(3)

(4)

where is the angular velocity, c is the speed of light,A is magnetic vector potential,J is the current density, is the electric potential, is the charge density, and the subscript refers to the Fourier component at angular frequency . The corresponding Green's function is
(5)

where is the delta function.But the Green's function for the Helmholtz equation is given by
(6)

with A and B constants. Although, there are two solutions, if the negative exponent is chosen, the potential changes before the charge moves. This is unphysical, so causality is artificially enforced by discarding the advanced solution, and retaining only the retarded one
(7)

The solution is therefore given by
(8); 0