The underlying method of interpolating the matrix (with respect to the
wavenumber k) is applicable to a wide variety of physical problems -
the structure(s) and the acoustic domain(s) may take many forms. In this
paper we will consider one general problem from this wide class,
that of an acoustic fluid occupying the infinite
domain exterior to the structure.
In figure 1, D is the domain of the structure, S is the boundary of
D that is in contact with the acoustic medium and E is the domain of the acoustic
fluid. The vector np (p Î S) is the unit outward
normal vector to the boundary S.
The acoustic field is governed by the wave equation
Ñ2 Y(p, t) =
1
c2
¶2
¶t2
Y(p,t)
(2)
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)
(3)
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)
(4)
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
(5)
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 (5) into (2) 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
sound pressure is related to the velocity potential as follows:
p(p) = i rwj(p) .
(6)
For
exterior problems we demand that the Sommerfeld radiation condition is
satisfied.
Let u1(p), u2(p), ... (p Î D)
be the (orthogonal) in vacuo mode shapes of the structure listed in order of
most fundamental to least. Let a unit distribution of force uj(p)
(p Î D) at wavenumber k produce a velocity of
uj (p) / lj (k) (p Î D) for j=1,2,... .
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 structural discretisation.
We may represent a general harmonic forcing distribution over the structure
in the form
g(p) =
å
j
Gjuj (p) (p Î D) .
(7)
The harmonic displacement w(p) and velocity of the
structure may then be written in the form
w(p) =
å
j
Wjuj (p), v(p) =
å
j
Vjuj (p) (p Î D).
(8)
Hence
Gj = lj (k) Vj (j=1,2,...) .
(9)
Note that since
v(p) = -i ww(p) then
Vj = -i wWj (j=1,2,...) .
(10)
Let f(p) = åj Fjuj (p) (p Î D)
represent the direct forcing on the structure. Thus the
actual forcing in the direction of np at the boundary S is
g(p) = f(p) - i rwj(p) = f(p) - i rk c j(p) (p Î S).