18矢量波函数(8-0).doc

18. Vector Wave Functions Vector wave equation As discussed previously, in source-free region, the homogeneous Helmholtz equations (wave equations) for EM fields and potentials are given by where k is the intrinsic wave number, The Laplacian operator is defined as and Let represent , or , or in source-free region, then it will obey the following vector wave equation: B. Scalar wave equation In rectangular coordinates, is denoted by and the homogenous vector Helmholtz equation for becomes or Therefore the three rectangular components, , should satisfy the following homogeneous scalar Helmholtz equations: . However, in a curvilinear coordinate system, only the rectangular coordinate components (if any) satisfy the homogeneous scalar Helmholtz equation and the others do not. For instance, in a cylindrical coordinate system, and its coordinate components satisfy, respectively, the scalar equations: It is found that in this coordinate system, only the rectangular-coordinate component satisfies the homogeneous scalar Helmholtz equation and the other two components, and , do not. Let denote or the rectangular-coordinate components of , , and , then it must satisfy the following homogeneous scalar wave equation C. Scalar wave functions The solution to the scalar wave equation is called the scalar wave function. Although the scalar wave equation is set up only for the rectangular field components, the equation may be solved in various coordinate systems. The solution to the scalar wave equation in rectangular coordinates is called the plane wave function, which is discussed in Chapter 4 of Harrington’s book. The solution to the scalar wave equation in cylindrical coordinates is called the cylindrical wave function, which appears in Chapter 5 of Harrington’s book. The solution to the scalar wave equation in spherical coordinates is called the spherical wave function, which is studied in Chapter 6 of Harrington’s book. The homogeneous scalar wave equation is usual