Generalized reaction principles and reciprocity theorems for the wave equations, and the relationship between the time-advanced and time-retarded fields

Generalized reaction principles and generalized reciprocity theorems for the scalar and vector wave equations are derived. It is shown that the reaction principles of Rumsey [Phys. Rev. 94, 1483–1491 (1954)] and the reciprocity theorems of Welsh [IRE Trans. Antennas Propag. 8(1), 68–73 (1960)] are special cases of these generalized reaction principles and generalized reciprocity theorems. The generalized reciprocity theorems are cast in terms of the time-retarded fields only, as well as in terms of the time-retarded and time-advanced fields. It is shown that the advanced fields are an improper solution of the wave equations subject to the proper Sommerfeld radiation boundary conditions at infinity. A relationship between the time-advanced and time-retarded fields is derived.