The acoustic radiation model consists of a vibrating plate G
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. In general, the
problems that this paper addresses are both scattering and radiation
problems with a perfectly reflecting, perfectly absorbtive or
an impedance boundary condition at the plate. The problem
is that of determining the properties of the semi-infinite acoustic
medium surrounding the vibrating plate.
The development of the Rayleigh integral method that we consider
in this paper is applicable to all these cases. However,
in this paper we directly address the particular case of a perfectly
reflecting plate radiating into a three-dimensional acoustic medium.
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 G)
where v(p) = [(¶j)/(¶np)]
with f(p) given for p Î G
and np is the unit normal to the plate at p. Though in this paper
we will be mainly concerned with the Neumann problem ( a(p)=0,
b(p)=1 for p Î G).
For the problems we consider in this paper, j must also satisfy the
Sommerfeld radiation condition
lim
r ® ¥
r (
¶j(p)
¶r
- i k j(p) ) = 0 .
(5)
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 GÈE) .
(6)
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 G),
(7)
where the asterix denoted the complex conjugate. The acoustic intensity on
the baffle is zero since v is zero there.
The sound power is given by
