Direct boundary integral equation for exterior and interior sound fields
Assuming the system is linear, the exterior and interior sound fields of the model problem shown in Figure 1 can be expressed as the superposition of the boundary sound fields generated by the scattering/reflection and radiation effects of the flexible wall and the direct sound fields generated by acoustic sources located respectively outside and inside the cavity. Both terms can be derived starting from the following inhomogeneous acoustic wave equation (Dowling and Ffowcs Williams 1983)
that takes into account the effects produced by acoustic monopole and dipole sources. In fact, the excitation terms pdq(x, t)/dt and V – f(x, t) represent a kinematic (volumetric) monopole source and a kinetic (force) dipole source respectively. More precisely q(x, t ) is the rate of volume flow per unit volume produced by the monopole source and f(x, t) is a vector with the fluctuating forces produced by the dipole source. In this equation c0 is the speed of sound, p is the mass density of the fluid and the vector x identifies the position in the
exterior or interior fluid domains. Also V2p(x, t) is the Laplacian of p(x, t) and V ■ f (x, t) is the divergence of f(x, t). Assuming time-harmonic dependence, the wave equation can be rewritten in the following form (Morse and Ingard 1968; Nelson and Elliott 1992)
V2p(x, m)+k2p(x, m)=-Q(x, m) , (2)
where k – a/c0 is the acoustic wavenumber and the volumetric monopole and force dipole sources are merged in the term Q(x, m) = jprnq(x,®)-V-f(x,«). Also p(x,«), Q(x,«), q(x,«), V – f(x,®) are the frequency dependent complex amplitudes of the co-respective time-domain functions assuming the time – harmonic dependence is given in the exponential form exp( jat), where со is the circular frequency. The remaining part of this formulation will be expressed in the frequency domain and, for brevity, the frequency dependence of the complex amplitudes will not be displayed. The solution of Eq. (2) is derived in terms of acoustic Green functions G(x | x’, ®), which are chosen according to the problem under study and satisfy the following inhomogeneous differential equation (Morse and Ingard 1968; Nelson and Elliott 1992)
V2G(x | x’)+k2G(x| x’)=-£(x-x’) , (3)
where s(x – x’) is the three-dimensional Dirac delta function, which defines a point monopole source in x’ (Nelson and Elliott 1992). Thus, the Green function describes the spatial dependence of the complex pressure field at x produced by a harmonic point monopole source at x. Eq. (2) is then solved by multiplying it by G(x | x’) and subtracting to the resulting equation Eq. (3) multiplied by p(x) . The resulting equation is then integrated in the acoustic volume V, which yields
p(x’) = JVG(x | x’)V2p(x) – p(x)V2G(x | x’)]dV +jvQ(x)G(x | x’)dV. (4)
The Green’s theorem given in the form ^ (jV2g – ) dV = £ (jVg – gVf )• n dS
and the reciprocity property G(x | x’) = G(x’ | x) are then used to transform the first volume integral into an integral over the boundary surface S of the acoustic volume V. As a result, the following “direct boundary integral equation” is derived
c(x,) p(x’) = £ (G(x’| x)Vp(x) •n – p(x)VG(x’| x) • n)dS+JVG(x’| x)Q(x)dV, (5)
x’ outside V
and n is a unit vector with direction orthogonal to the boundary surface S (note that, as shown in Figure 1, n has opposite directions for the exterior and interior domains).
In order to solve Eq. (5) it is necessary to define the pressure and pressure gradient on the boundary surface, i. e. for xeS. For a well-posed boundary – value problem, only one of the two sets of boundary conditions can be defined a priori. However, also the other set can be derived from the direct boundary integral equation by co-locating the point x’ on the boundary surface itself. Thus the derivation of the scattered/reflected or radiated sound fields by a flexible object or an enclosure is carried out in two steps based on the same integral equation. Although at first sight this may appear as a mere repetition of the same integration, the implementation of the first step is not trivial since the surface integral becomes singular when the point x’ is co-located on the boundary surface. Nevertheless this singularity is weak, and the surface integral converges in the regular sense (Wu 2000). There is also a second difficulty to be considered, that is, for exterior problems, the surface integral in Eq. (5) may not have a unique solution at certain characteristic frequencies. The reader is referred to specialised monographs that show how this problem is normally overcome with the so called CHIEF method (Wu 2000).
Besides the boundary conditions, a suitable Green function g(x’ | x) must be defined to implement the direct boundary integral Eq. (5). In principle there is a vast selection of functions that can be used to solve a given problem, since the only requirement is that they satisfy both Eq. (3) and the principle of reciprocity (Nelson and Elliott 1992). Nevertheless exterior and interior sound fields are normally handled with two specific types of functions that are described below.