Keywords: Poisson Helmholtz Le Jean de Bessel
According to the mathematical theorem: a function about time and space can always be decomposed into the product of a function only related to time and two other functions only related to space, that is. Variable separation can be used to decompose a complex function into two unary functions, thus simplifying the problem, so variable separation is an important method to solve problems in mathematics and physics.
Variable separation method is used many times in electrodynamics: when calculating potential, a Poisson equation is given; Helmholtz equation in resonator and waveguide: Gaussian beam; Optical spatial solitons, etc. One of the same methods to solve these problems is to separate variables. The following are two concrete ways to solve these problems.
Whether it is Gaussian beam, optical soliton or Poisson equation, its essence is Helmholtz equation. Let's deduce the general solution from Laplace.
(1). In rectangular coordinate system
The unified equation of Poisson equation is, in which, so as long as the general solution of the solution is added with the special solution of Poisson, the general solution of Poisson equation can be obtained. Next, the Laplace equation is solved by separating variables.
, replace, get:
In other words, both sides of this formula are replaced by,,
. The left side of this formula is about the function of, and the right side is only about the function of. If the two sides are equal, there is only one possibility. Both sides are equal to a constant or zero. Let's set this constant to, and then make the same statement. , finishing:
So the Laplace equation is transformed into the above six formulas in the form of take, take: and Noring, so the solution obviously satisfies the above equations, so
(2) Separation of variables in column coordinates
,
(3).
Below, about pole symmetry.
The propagation of electromagnetic wave in waveguide and resonator satisfies Helmholtz equation. According to the above solution, it is not difficult to know that the expression of its solution is similar to (4), so its solution has the same form as (5).
From the above process, we can know that when solving a problem, we should choose the appropriate coordinate system for analysis according to the specific form of the problem, so as to make the calculation as simple as possible. Generally, when the problem has spherical symmetry, the solution in spherical coordinate system is simpler. When the waveguide is not rectangular, it is very complicated to analyze it in rectangular coordinates. The Laplace equation of the spherical potential is based on the spherical coordinate system, the polar axis is selected, and the solution is made simple by using the conclusion of Formula (7). Practical problems often have boundary conditions, and then the coefficients in each equation are determined according to the boundary. Obviously, there are six undetermined coefficients in the general equation, which need six boundaries to determine theoretically.
Reference: beam. Methods of Mathematical Physics, Third Edition, 229.236.