The acoustic radiation model consists of a set of vibrating bodies and forced shells,
all of which are separate, surrounded by an infinite acoustic field,
and the shells assume linear elastic behaviour. Let E be the infinite
domain exterior to a set of closed surfaces S (enclosing regions D)
and shells G.
The shell(s) is assumed to have two surfaces: an upper surface G+
and a lower surface G-.
The normal to the upper surface G+ at the point
p ( Î G) is denoted np.
The model is illustrated in figure 1.
The equation governing the acoustic field is the wave equation
Ñ2 Y(p, t) =
1
c2
¶2
¶t2
Y(p,t)
(1)
where Y(p,t) is the scalar time-dependent velocity potential related to the
time-dependent particle velocity by
V(p,t) = Ñ Y(p,t)
(2)
and c is the propagation velocity (p and t are the spacial and
time variables). The time-dependent sound pressure Q(p,t) is given
in terms of the velocity potential by
Q(p,t) = - r
¶Y
¶t
(p,t)
(3)
where r is the density of the acoustic medium.
Only periodic solutions to the wave equation are considered, thus
it is sufficient to consider time-dependent velocity potentials of the form
y(p,t) = j(p) e-i wt
(4)
where w is the angular frequency (w = 2 ph,
where h is the
frequency in hertz) and j(p) is the (time-independent) velocity
potential. The substitution of expression (4) into (1) reduces
it to the Helmholtz (reduced wave) equation
Ñ2 j(p) + k2 j(p) = 0
where k2 = [(w2)/(c2)] and k is the wavenumber.
The Sommerfeld radiation condition is also satisfied.
Note that for the purpose of the acoustic modelling,
the shells are assumed to be infinitesimally thin, so that if
p Î G then p Î G+ and p Î G-.
Let the function v(p) for p Î S be defined as follows:
v(p) =
lim
e® 0
¶j
¶np
(p + enp ) .
(5)
The potential j and its derivatives are generally discontinuous at
the shell. However, j(p) (p Î G) and its
derivatives take limiting values on G+ and G-.
Let the functions j+ (p), j- (p),
v+ (p) and v- (p) (p Î G)
be defined as follows:
j+(p) =
lim
e® 0
j(p + enp ) ,
j-(p) =
lim
e® 0
j(p - enp ) ,
v+(p) =
lim
e® 0
¶j
¶np
(p + enp ) ,
v-(p) =
lim
e® 0
¶j
¶np
(p - enp ) .
Let the functions d(p), n(p), F(p) and
V(p) for p Î G be defined as follows:
Let u1(p), u2(p), ... (p Î G)
be the (orthogonal) in vacuo mode shapes of the shell(s) listed in order of
most fundamental to least. Let a unit distribution of force uj(p)
(p Î G) at wavenumber k produce a velocity of
uj (p) / lj (k) (p Î G)
for j=1,2,... for the shell lying in a vacuum.
The functions lj (k) may include the effect of structural damping.
The mode shapes uj(p) and the functions lj(k)
(j=1,2,...) can generally be computed straightforwardly via the finite element
method up to some maximum value of j, depending on the discretization
of the shell [12].
We may represent a general harmonic forcing distribution over the shell
in the form
g(p) =
å
j
Gjuj (p) (p Î G) .
(6)
The harmonic displacement w(p) and velocity v(p)
of the shell may then be written in the form
w(p) =
å
j
Wjuj (p) , v(p) =
å
j
Vjuj (p) (p Î G) .
(7)
Hence
Gj = lj (k) Vj (j=1,2,...) .
(8)
Note that since
v(p) = -i ww(p) then
Vj = -i wWj (j=1,2,...) .
(9)
Let f(p) (p Î G)
represent the direct forcing on the shell. Thus the
actual forcing distribution in the direction of np is