The acoustic field is governed by the damped wave equation.
However, in this paper we are concerned only with the modal
solution and thus it is sufficient (and also the convention) to model
the acoustic field by the undamped wave equation
Ñ2 Y(p, t) =
1
c2
¶2
¶t2
Y(p,t) (p Î D)
(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) (p Î D ÈS)
(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) (p Î D ÈS)
(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 (p Î D)
(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 (p Î D)
(5)
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) (p Î D ÈS) .
(6)
It is necessary to specify a
homogeneous boundary condition which generally takes the form
a(p) j(p) + b(p)
¶j(p)
¶np
= 0 (p Î S)
(7)
where a(p) and b(p) are known complex-valued functions of
p ( Î S) and np is the unit outward normal to the
boundary at p. The non-trivial solutions k=k* and j(p) = j*(p) (p Î D ÈS) are termed the eigenfrequencies
and eigenfunctions. They are dependent on the
boundary S and the boundary functions
a(p) and b(p), and the eigenfrequencies are all
real numbers. In this paper only the
Neumann eigenproblem (a(p)=0 and b(p)=1) is
considered.