Green function for the exterior sound field
As shown in Figure 1, a particular boundary surface is chosen for the exterior sound field, which is composed by the external surface Sb of the flexible wall and by the surface S„ of an imaginary sphere centred in the radiating object and with radius that tends to infinity. In this way the Sommerfeld radiation condition at infinity (Sommerfeld 1949) can be assumed, which imposes that only waves travelling outwards from a source are allowed and that the pressure tends to zero at infinite distance from the source. Assuming the system of reference is located
at the centre of the radiating object, this physical condition can be expressed mathematically with the following equation:
where the vector xe identifies a point in the exterior sound field. In this case, it can be shown that the surface integral over S in Eq. (5) goes to zero. Also, for freefield sound radiation, the Green function takes the simple form
which is known as the free space Green function (Morse and Ingard 1968; Nelson and Elliott 1992). For the specific problem considered in this chapter where there is no external acoustic source distribution, i. e. Q(Xe), and the velocity distribution normal to the boundary surface is prescribed as compatible with the transverse vibration velocity of the flexible wall, the direct boundary integral equation derived above can be reformulated as follows
where the vector Xs identifies a point on the surface of the enclosure wall and dg(Xe  xs)/dn and dp(Xe)/dn are the directional derivatives[26] of g(Xe  xs) and p(Xe) along the normal n to the boundary surface Sb (Junger and Feit 1986). Equation (9) gives the sound pressure radiated by a vibrating body, provided the sound pressure in Sb and its derivative along the normal to the surface Sb are known. According to the fluid momentum equation, the sound pressure derivative dp(xs )/dn and the sound particle velocity vn (xs) along the normal to the boundary surface are related by the momentum equation cp(xs)/dn + japvn(xs) = 0 (Morse and Ingard 1968; Pierce 1989; Fahy and Gardonio 2007). Thus Eq. (9) can be rewritten in the following form
which, in literature, is known as the Helmholtz integral equation (Junger and Feit 1986; Wu 2000). Since the sound particle velocity vn (xs) is compatible with the transverse velocity of the structure w(xs), the sound pressure over the boundary surface p(xs) can be readily derived by coupling the structural wave equation with the Helmholtz integral Eq. (10) calculated in xe = xs assuming c(xs) = 1/2 and setting2 vn (xs)= —w(xs). The radiated sound field can then be derived yet again from the Helmholtz integral equation setting c(xe) = 1. Nevertheless, for
the noise problem considered in this section, this first step is sufficient to provide the external sound pressure fluid loading effect on the flexible wall. A phenomenological analysis of Eq. (10) indicates that the sound pressure p(x’s) at a given point x’s of the boundary surface of a flexible body is given by the surface integral of the superposition of the sound pressure generated by the vibration of the body, via the term ja>pg(x’s  xs) w(xs), and the sound pressure
generated by the scattering effect of the body, via the term Sg(x*’’) p(xs).
In general, for arbitrary geometries of the flexible body, the surface pressure distribution necessary to solve the Helmholtz integral equation must be derived by solving Eq. (10) numerically (Wu 2000). Analytical approximate solutions can be derived in the short – and longwavelength limits (Junger and Feit 1986; Koopmann 1997). For instance, analytical solutions can be derived either for very small sound radiating objects compared to the acoustic wavelength or when the acoustic wavelength is smaller than both the radius of curvature of the sound radiating object and the portions of the sound radiating object that vibrate in phase. Also, analytical exact solution can be derived for specific geometries of the sound radiating object so that a special class of acoustic Green functions,
Ge (xe  xs), can be defined. These functions satisfy the Neumann boundary condition on the surface of the sound radiating object, i. e. jfGe (xe  xs ) = 0
n lxe=xs
(Junger and Feit 1986; Koopmann 1997). As a result, only the velocity normal to the boundary surface is required to derive Eq. (10). According to the nomenclature in (Kellogg 1953), these functions are referred to as the Green functions of the second kind. Alternatively other authors identify them as Neumann functions (Garabedian 1964). The – jf Ge (xe  xs ) = 0 condition can
n lxe=Xs
be straightforwardly imposed when the shape of the radiating body is such that it
2 Note that, as shown in shown in Figure 1b, the standard notation used for the Helmholtz integral equation is used where the vector n points away from the acoustic domain, thus vn (x s) = – w (x s) .
can be defined in terms of a single coordinate with reference to systems of orthogonal coordinates in which the acoustic wave equation is separable (e. g. rectangular, cylindrical and spherical coordinates) (Junger and Feit 1986). In this case the second kind Green function can be formulated analytically and the simplified form of the Helmholtz integral equation
c(x e ) p(x e ) = jWPsbGe (x e Iх – )vn (x – )dSb (11)
can be used to derive the radiated sound field directly from the boundary particle velocity vn (xs), that is the transverse velocity of the structure ii(xJ). From a physical point of view, in contrast to the free space Green function, this function includes the scattering effect that would have the wall if assumed to be rigid (Junger and Feit 1986; Koopmann 1997).
It is interesting to note that in the special case where the flexible body is an infinitely extended flat plate, the scattering of sound is such that the first term equals the second term in Eq. (10). Thus, using the expression for the free space Green function given in Eq. (8) and recalling that vn (x,) = – it (x,), the radiated
sound pressure can be readily derived with the following integral
This integral expression in known as the Rayleigh integral for the radiated sound pressure by an infinitely extended flat surface with transverse velocity it (x,) (Fahy and Gardonio 2007). Thus the free space Green function can be considered as a second kind Green function for the sound radiation problem of an infinitely extended flat surface.
Since this section is focused on the coupled structuralacoustic response of a cylindrical enclosure, the second kind Green function for the baffled cylinder vibrating surface shown in Figure 1 is briefly recalled. In the literature this Green function is derived by applying spaceFourier transforms to the homogeneous counterpart of Eq. (3) formulated in cylindrical coordinates z, в, r (Morse and Ingard 1968; Junger and Feit 1986). More specifically, since the cylindrical geometry imposes a periodicity along the circumferential direction, a Fourier series is applied along the circumferential direction, such that
p(z,0, r) = ^pm2(z, r)eimi23, and a spaceFourier transform3 is applied along the
m2=—<&
3 SpaceFourier transforms are normally referred to as wavenumber transforms (Junger and Feit 1986; Fahy Gardonio 2007).
axial direction, which is given by pm2 (kz, r) =J pm2 (z, r)eJtzZdz, where the acoustic wave number is expressed as k2 = k2 + k2 and m2 = 0, 1, 2, … is the index for the Fourier series expansion in circumferential direction. In this way, the homogeneous counterpart of the partial differential Eq. (3) is transformed into a series expansion of Bessel’s differential equations in the unknown pm2 (kz, r)
functions. This series expansion is satisfied when each Bessel’s differential equation is set to zero. The solution of this set of equations can be found analytically in terms of Hankel functions. This formulation leads to a series expression whose terms are function of the axial wavenumber. Thus to obtain the Green function in the spatial coordinates, an inverse spaceFourier transform is implemented with reference to the axial wavenumber, which leads to the following expression (Morse and Ingard 1968; Stephanishen 1981; Lesueur 1988; Millard 1997)




where the position vectors x e and x s are defined in cylindrical coordinates z, в, r, and m2 = 0, 1, 2, … Here R is the radius of the cylinder, H^(…) is the first kind Hankel function of order m and smi=0 = 1, smi >0 = 2 is the Neumann factor.
A comprehensive introduction to the wave number transform approach for the solution of wave equations can be found in the monographs by (Morse and Ingard 1968; Junger and Feit 1986; Millard 1997). Since the Green function in Eq. (13) satisfies Neumann’s boundary condition such that ^Ge(xe  xJ) = 0 ,
П Ixe=xs
the integral expression in Eq. (10) reduces to Eq. (11). From the physical point of view this result follows from the fact that the Green function of Eq. (13) already includes the scattering effects that are produced by the cylinder.
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