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13.3. The Kirchhoff–Helmholtz boundary integral

In the previous section we introduced the Green's function

g (r | r_{0}) = \frac{e^{- j k (r - r_{0})}}{4 π (r - r_{0})},

$g (r | r_{0}) = \frac{e^{- j k (r - r_{0})}}{4 π (r - r_{0})},$

(13.16)

which turns out to be a solution of the following inhomogeneous wave equation:

(\nabla^{2} + k^{2}) g (r | r_{0}) = - δ (r - r_{0}) .

$(\nabla^{2} + k^{2}) g (r | r_{0}) = - δ (r - r_{0}) .$

(13.17)

An important principle in acoustics is that of reciprocity whereby the locations of the sound source r ₀ and its observation point r are interchangeable. It can be seen that Eqs. (13.16) and (13.17) are unaffected by interchanging r and r ₀. Hence

g (r | r_{0}) = g (r_{0} | r)

$g (r | r_{0}) = g (r_{0} | r)$

(13.18)

and

δ (r - r_{0}) = δ (r_{0} - r) .

$δ (r - r_{0}) = δ (r_{0} - r) .$

(13.19)

Eq. (13.17) differs from the homogeneous wave Eq. (13.1) in that the Dirac delta function δ on the right hand side represents the excitation at the point r ₀. Eq. (13.17) describes the normalized pressure field (that is, divided by

i k ρ c {\tilde{U}}_{0}

$i k ρ c {\tilde{U}}_{0}$

) of a point source. However, it is desirable to solve the following inhomogeneous wave equation for any source distribution:

(\nabla^{2} + k^{2}) \tilde{p} (r) = - \tilde{f} (r),

$(\nabla^{2} + k^{2}) \tilde{p} (r) = - \tilde{f} (r),$

(13.20)

where

\tilde{f} (r)

$\tilde{f} (r)$

is a source pressure distribution in Pa/m². This can be achieved [1,2] by multiplying Eq. (13.17) by

\tilde{p} (r)

$\tilde{p} (r)$

and then subtracting it from Eq. (13.20) multiplied by

g (r | r_{0})

$g (r | r_{0})$

, which leads to

g (r | r_{0}) \nabla^{2} \tilde{p} (r) - \tilde{p} (r) \nabla^{2} g (r | r_{0}) = \tilde{p} (r) δ (r - r_{0}) - g (r | r_{0}) \tilde{f} (r) .

$g (r | r_{0}) \nabla^{2} \tilde{p} (r) - \tilde{p} (r) \nabla^{2} g (r | r_{0}) = \tilde{p} (r) δ (r - r_{0}) - g (r | r_{0}) \tilde{f} (r) .$

(13.21)

Using the reciprocity relationships of Eqs. (13.18) and (13.19), we can exchange r and r ₀ in Eq. (13.21) and integrate over an arbitrary volume containing all the sources to obtain

\begin{matrix} ∭ (g (r | r_{0}) \nabla_{0}^{2} \tilde{p} (r_{0}) - \tilde{p} (r_{0}) \nabla_{0}^{2} g (r | r_{0})) d V_{0} \\ = ∭ \tilde{p} (r_{0}) δ (r - r_{0}) d V_{0} - ∭ g (r | r_{0}) \tilde{f} (r_{0}) d V_{0}, \end{matrix}

$\begin{matrix} ∭ (g (r | r_{0}) \nabla_{0}^{2} \tilde{p} (r_{0}) - \tilde{p} (r_{0}) \nabla_{0}^{2} g (r | r_{0})) d V_{0} \\ = ∭ \tilde{p} (r_{0}) δ (r - r_{0}) d V_{0} - ∭ g (r | r_{0}) \tilde{f} (r_{0}) d V_{0}, \end{matrix}$

(13.22)

where the zero subscripts indicate differentiation with respect to the r ₀ coordinates. Using Green's theorem [3], which essentially states that anything created within a diverging volume passes through its outer surface, the volume integral of the term in parentheses can be replaced with a surface integral:

\begin{matrix} ∭ (g (r | r_{0}) \nabla_{0}^{2} \tilde{p} (r_{0}) - \tilde{p} (r_{0}) \nabla_{0}^{2} g (r | r_{0})) d V_{0} \\ = ∭ \nabla_{0} (g (r | r_{0}) \nabla_{0} \tilde{p} (r_{0}) - \tilde{p} (r_{0}) \nabla_{0} g (r | r_{0})) d V_{0} \\ = \iint (g (r | r_{0}) \frac{\partial}{\partial n_{0}} \tilde{p} (r_{0}) - \tilde{p} (r_{0}) \frac{\partial}{\partial n_{0}} g (r | r_{0})) d S_{0}, \end{matrix}

$\begin{matrix} ∭ (g (r | r_{0}) \nabla_{0}^{2} \tilde{p} (r_{0}) - \tilde{p} (r_{0}) \nabla_{0}^{2} g (r | r_{0})) d V_{0} \\ = ∭ \nabla_{0} (g (r | r_{0}) \nabla_{0} \tilde{p} (r_{0}) - \tilde{p} (r_{0}) \nabla_{0} g (r | r_{0})) d V_{0} \\ = \iint (g (r | r_{0}) \frac{\partial}{\partial n_{0}} \tilde{p} (r_{0}) - \tilde{p} (r_{0}) \frac{\partial}{\partial n_{0}} g (r | r_{0})) d S_{0}, \end{matrix}$

(13.23)

where the surface of integration bounds the volume of the original volume integral, and the Laplace operator is replaced with a first-order derivative normal to the surface, pointing away from the space enclosed by the surface integral. We can verify the first step of Eq. (13.23) by working backwards. Although taking the derivative of the two products in the second line leads to four terms, two of them cancel to leave the remaining two terms in the first line. The third line is obtained from the second by the divergence theorem of Gauss. Inserting Eq. (13.23) into Eq. (13.22) and using the property of the Dirac delta function to solve the volume integral

\tilde{p} (r_{0}) δ (r - r_{0})

$\tilde{p} (r_{0}) δ (r - r_{0})$

yields

\tilde{p} (r) = {\tilde{p}}_{V} (r) + {\tilde{p}}_{S} (r),

$\tilde{p} (r) = {\tilde{p}}_{V} (r) + {\tilde{p}}_{S} (r),$

(13.24)

where

{\tilde{p}}_{V} (r)

${\tilde{p}}_{V} (r)$

is a volume integral given by

{\tilde{p}}_{V} (r) = ∭ g (r | r_{0}) \tilde{f} (r_{0}) d V_{0}

${\tilde{p}}_{V} (r) = ∭ g (r | r_{0}) \tilde{f} (r_{0}) d V_{0}$

(13.25)

and

{\tilde{p}}_{S} (r)

${\tilde{p}}_{S} (r)$

is the Kirchhoff–Helmholtz surface integral given by

{\tilde{p}}_{S} (r) = {\tilde{p}}_{M} (r) + {\tilde{p}}_{D} (r)

${\tilde{p}}_{S} (r) = {\tilde{p}}_{M} (r) + {\tilde{p}}_{D} (r)$

(13.26)

where

{\tilde{p}}_{M} (r)

${\tilde{p}}_{M} (r)$

is the monopole integral given by

{\tilde{p}}_{M} (r) = \iint g (r | r_{0}) \frac{\partial}{\partial n_{0}} \tilde{p} (r_{0}) d S_{0}

${\tilde{p}}_{M} (r) = \iint g (r | r_{0}) \frac{\partial}{\partial n_{0}} \tilde{p} (r_{0}) d S_{0}$

(13.27)

and

{\tilde{p}}_{D} (r)

${\tilde{p}}_{D} (r)$

is the dipole integral given by

{\tilde{p}}_{D} (r) = - \iint \tilde{p} (r_{0}) \frac{\partial}{\partial n_{0}} g (r | r_{0}) d S_{0} .

${\tilde{p}}_{D} (r) = - \iint \tilde{p} (r_{0}) \frac{\partial}{\partial n_{0}} g (r | r_{0}) d S_{0} .$

(13.28)

What is remarkable about Eq. (13.24) is that, merely given a solution

g (r | r_{0})

$g (r | r_{0})$

to the wave equation for a point source, it provides a solution for the pressure field

\tilde{p} (r)

$\tilde{p} (r)$

everywhere in the presence of an arbitrary source distribution

\tilde{f} (r_{0})

$\tilde{f} (r_{0})$

within the volume of integration.

It should be noted that in this instance, the integrals

{\tilde{p}}_{M} (r)

${\tilde{p}}_{M} (r)$

and

{\tilde{p}}_{D} (r)

${\tilde{p}}_{D} (r)$

have nothing to do with reflections, although they can be applied to problems of scattering surfaces when appropriate boundary conditions are applied. The volume of integration does not have a physical reflecting boundary surface but a transparent notional one. Inside the volume,

{\tilde{p}}_{M} (r)

${\tilde{p}}_{M} (r)$

and

{\tilde{p}}_{D} (r)

${\tilde{p}}_{D} (r)$

cancel each other so that there is no net contribution from

{\tilde{p}}_{S} (r)

${\tilde{p}}_{S} (r)$

, and the field is entirely given by the volume integral, or

{\tilde{p} (r) |}_{r \in V_{0}} = {\tilde{p}}_{V} (r), {\tilde{p}}_{S} (r) = 0, {\tilde{p}}_{M} (r) = - {\tilde{p}}_{D} (r) .

${\tilde{p} (r) |}_{r \in V_{0}} = {\tilde{p}}_{V} (r), {\tilde{p}}_{S} (r) = 0, {\tilde{p}}_{M} (r) = - {\tilde{p}}_{D} (r) .$

(13.29)

However, outside the boundary, the field due to the surface integral cancels the field due to the volume integral:

{\tilde{p} (r) |}_{r \notin V_{0}} = 0, {\tilde{p}}_{S} (r) = - {\tilde{p}}_{V} (r), (\begin{matrix} {\tilde{p}}_{M} (r) = {\tilde{p}}_{D} (r) = - {\tilde{p}}_{V} (r) / 2 \\ if planar infinite surface \end{matrix}) .

${\tilde{p} (r) |}_{r \notin V_{0}} = 0, {\tilde{p}}_{S} (r) = - {\tilde{p}}_{V} (r), (\begin{matrix} {\tilde{p}}_{M} (r) = {\tilde{p}}_{D} (r) = - {\tilde{p}}_{V} (r) / 2 \\ if planar infinite surface \end{matrix}) .$

(13.30)

Hence,

{\tilde{p}}_{S} (r)

${\tilde{p}}_{S} (r)$

is a discontinuous solution to Eq. (13.20), which is only valid outside the volume containing the sources (provided that the sign is reversed). If the volume is infinitely large, the Sommerfeld condition applies and the boundary integrals vanish so that

\tilde{p} (r) = {\tilde{p}}_{V} (r)

$\tilde{p} (r) = {\tilde{p}}_{V} (r)$

is a solution to Eq. (13.20) everywhere. In practice, however, the volume of integration only has to include all the sources under consideration, but not necessarily all the observation points. The usefulness of the boundary surface integral of Eq. (13.26) for solving acoustical problems cannot be overstated; it forms the basis for many numerical methods such as Boundary Element Modeling (or BEM). It is an embodiment of the Huygens–Fresnel principle discussed in Section 13.1. The surface of integration must be a closed one, which fully encloses all the sources, although they may form part or all of the surface. By a closed surface, we could also mean an infinite plane that isolates the sources on one side of the plane (or within the plane itself) from the observation field on the other. Although

{\tilde{p}}_{M} (r)

${\tilde{p}}_{M} (r)$

and

{\tilde{p}}_{D} (r)

${\tilde{p}}_{D} (r)$

are both needed in the case of general surfaces, such as the spherical cap in a sphere in Section 13.5, we shall see that in the case of planar sources, one of the integrals can often be eliminated because of the symmetry of the problem. Before we apply the boundary integral to some problems of practical importance, we shall take a further look at the Green's function.

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13.3. The Kirchhoff–Helmholtz boundary integral

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Acoustics