FIG. 13.

(Color online) As used for field projection calculations, two surfaces ${\Sigma }_1$ and ${\Sigma }_2$ are depicted schematically relative to a coordinate system with origin O. Each surface corresponds to the locus of points represented by position vectors ${\mathbf{r}}_1$ or ${\mathbf{r}}_2$, respectively. ${\Sigma }_1$ represents a boundary surface with an acoustic source that radiates waves into the half-space region that includes ${\Sigma }_2$. This surface is further described by the outward facing unit normal ${\mathbf{n}}_1$ and differential area elements $d{\Sigma }_1$. ${\Sigma }_2$ is described by analogous notation and represents a non-physical surface at which field measurements are made.