ABSTRACT
In Section 5.5, we had utilized the scalar Green function G(R,R′,ω) for homogeneous infinite space to represent acoustic source fields and, in Section 5.6, to represent acoustic scattered fields in terms of a point source synthesis. There, we emphasized the appropriateness to define additional Green functions via gradient operations-∇G(R,R′,ω), ∇∇G(R,R′,ω)—because they explicitly appear in the integral representations of acoustic fields. For electromagnetism, the constructions (I + k−2∇∇)G(R,R′,ω) and ∇G(R,R′,ω) × I played a fundamental role, and we will see in Section 13.2.3 that elastodynamics enforces additional derivatives. Despite the mathematical complexity of Green functions differential equations and their respective solutions, the physical concept of Green functions as elementary waves superimposing to wave fields in terms of the point source synthesis is very intuitive, why we start again from the beginning with G(R,R′,ω) to clarify some aspects that we ignored in Section 5.5.
