Integral Relations

→ See also Pierce  (1981) and others

Consider two different sound fields p${}_{{1}}$, p${}_{{2}}$ in a volume V with a bounding surface S (with outwards directed surface element $d\vec{s}$ ). Green's integral is then

$$\iiint\limits _{V}{\left({p_{1}\cdot\Delta p_{2}-p_{2}\cdot\Delta p_{1}}\right)\; d{\rm{\bf r}}}=\mathop{{\int\!\!\!\!\!\int}\bigcirc}\limits _{S}{\left({p_{1}\cdot\vec{\nabla}p_{2}-p_{2}\cdot\vec{\nabla}p_{1}}\right)\cdot d\vec{s}}.$$

(1)

The fields may differ either by different source strengths and/or locations, and/or by different boundary conditions on S, and/or are different forms (modes) for the same sources and boundaries. The surface S is either soft (p(S)=0) or hard ($\partial$p/$\partial$n=0) on parts S${}_{{0}}$ or locally reacting on parts S${}_{{a}}$ with surface admittance G, or parts S${}_{{\infty}}$ are at infinity, where the fields obey Sommerfeld's condition.

With the fundamental relations of → Sect.  B.1 it follows that

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