Green function for the interior sound field

In general, the Green function for enclosed sound fields can be expressed with a series expansion in terms of n = 1, 2, … acoustic natural modes of the cavity y/n(xc) and complex modal amplitudes an (x’) due to a point monopole source of unit amplitude (Morse and Ingard 1968; Pierce 1989; Nelson and Elliott 1992; Koopmann 1997), that is:

Gc (xc |x C ) = !>n (x c ) an (xC )

n=1

Here the vectors xc and x’c identify the positions of the sound pressure and point monopole source in the enclosure. The natural modes are chosen to form a complete set of functions so that any pressure field in the cavity can be derived from their linear combination. As seen for the exterior sound field, in order to avoid the two steps numerical solution of the boundary integral Eq. (5), the series expansion for the Green function in Eq. (14) is chosen to satisfy Neumann’s boundary condition such that – nGc (xc | xJ )| = 0 over the boundary surface of

П lxc=xs

the enclosure (Morse and Ingard 1968; Pierce 1989; Nelson and Elliott 1992; Koopmann 1997). From the physical point of view, this condition corresponds to rigidly walled boundary conditions. Thus the natural mode shapes used in Eq. (14) are chosen assuming the cavity is rigidly walled. The complex modal amplitudes an are derived by substituting Eq. (14) into Eq. (3), which is then

multiplied by the n – th mode and integrated over the volume of the cylindrical cavity. As a result, considering the orthonormality property of the natural modes, the following set of uncoupled ordinary equations in the unknown modal amplitudes an (x’c) are derived

V(®a2,„ -®2К = Chn П = 1 2 •••, ® ■ (15)

These equations are derived assuming the natural modes of the cavity yn (x c) are normalised in such a way as _[,^„2(xc) = V, where V is the volume of the cavity. Also <яал is the n-th natural frequency for the rigidly walled cavity and qn are

the modal excitation terms, which, using the “sifting” property of the three­dimensional Dirac delta function, are derived as follows

qn =v¥n(xcЖ^-xC)dv = ¥n(xC) . (16)

Thus, the Neumann Green function for the interior sound field is given by:

The sound absorption effects produced by internal fittings in transportation vehicles (floor, seats, wall finishing/trim layers, etc.) generate a damping action, which, for light damping, is normally taken into account in terms of modal damping so that Eq. (17) becomes (Morse and Ingard 1968; Nelson and Elliott 1992):

(18)

where ^a, n is the acoustic modal damping. The details for the derivation of this

expression can be found in Morse and Ingard (1968) and Nelson and Elliott (1992). In conclusion, assuming there are no interior acoustic sources Q (x c) and

assuming the Neumann Green function (18) is employed, also for the interior sound field the boundary integral equation (5) reduces to:

c (x c ) p(x c) = jPa>Sb(G c(x c 1 xs) vn(xs) dSb, (19)

where vn (xs) is the sound particle velocity in direction normal to the boundary surface.

When the enclosing wall has regular geometry it is possible to derive the natural frequencies ma, n and natural modes y/n (xc) in terms of simple analytic

expressions. For instance, for the cylindrical enclosure shown in Figure 1, the natural frequencies and natural modes, assuming rigid wall boundary conditions, are given by (Blevins 2001):

and

¥Sn (2Ж Г) = cos(i¥L)cos(n2^)Jn2 (Лп2Л R)

¥a„ (z,^, r) = cosinr )sinMK2 (^n2.n3 :r)

These equations are given in cylindrical coordinates z, 9 , r so that щ – 0, 1, 2,… n2 = 0, 1, 2, _ n3 = 0, 1, 2, … are the modal indices in axial, circumferential and radial directions for the n-th acoustic natural frequency and mode. Also, Jni (•••)

is the first kind Bessel function of order n2 and the term ЛП1^І is derived from the equation J’n1(K1,n3) = 0 . Finally R and L are the radius and the length of the

cylindrical cavity respectively. Since the mode shapes show a periodicity in circumferential direction, in order to represent a generic sound pressure distribution, two mode components oriented orthogonally in circumferential direction must be employed in Eqs. (17) and (18). For instance, in this formulation the symmetric cosine and anti-symmetric sine functions given in

Eqs. (21a, b) are used, which are denoted by the superscripts 5 and a respectively. Thus, to take into account the contributions of both symmetric and anti­symmetric mode shapes, the Green function of Eq. (18) must be modified as follows:

where for each mode index n, the summation is carried out considering symmetric (a = s) and antisymmetric (a = a) mode shapes. In this case, since the natural modes given by Eqs. (21a) and (21b) are not volume-normalised, the denominator includes the term V = VSnlJl1-l(^n1,ni), where £щ = 12 when n1 = 0 and є =14 when n1 > 0 . As discussed for the exterior sound problem, for

cavities with complex geometries, either FEM or BEM numerical methods can be used to formulate an eigenvalue-eigenvector problem from which the approximate natural frequencies and natural modes are derived (Wu 2000; Fahy and Gardonio 2007).