The acoustic radiation model consists of a vibrating plate P
of arbitrary shape lying in a infinite rigid baffle and surrounded by a
semi-infinite acoustic medium E, as shown in figure 1
for the three-dimensional case. The baffle is perfectly rigid and reflecting.
The problems covered in this work allow a general
Robin boundary condition on the panel; modelling
radiation
problems with a perfectly reflecting, perfectly absorbtive or
an impedance boundary condition at the plate. The aim
is that of determining the properties of the semi-infinite acoustic
medium surrounding the plate.
The acoustic field is governed by the wave equation in E,
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 of interest, 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 .
In general, let it be assumed that we have a Robin condition on the
plate of the form
a(p) j(p) + b(p) v(p) = f(p) (p Î P)
(5)
where v(p) = [(¶j)/(¶np)]
with f(p) given for p Î P
and np is the unit normal to the plate at p.
Though in this report
we will be mainly concerned with the Neumann problem (a(p)=0,
b(p)=1 for p Î P). The true velocity of the
can be expressed as v(p) e-iwt for p Î P.
For the problems we consider in this document, j must also satisfy the
Sommerfeld radiation condition
lim
r ® ¥
r (
¶j(p)
¶r
- i k j(p) ) = 0 ,
(6)
where r is the distance between the point p and a fixed origin.
2.2 Properties of the time-harmonic acoustic field
The substitution of (4) into equation (1) gives the time-independent
sound pressure
P(p) = i rwj(p) (p Î PÈE) .
(7)
The sound intensity on the plate with respect to the normal to the plate is
I(p) =
1
2
Re{P*(p) v(p) } (p Î P),
(8)
where the asterix denotes the complex conjugate. The acoustic intensity on
the baffle is zero since v is zero there.
The sound power is given by
W =
ó õ
P
I(q) dSq .
(9)
The radiation ratio (often also termed the
radiation resistance of radiation efficiency) is given by