Approximations to
the properties of the acoustic medium can be found through
applying numerical integration technique to (10).
In order that the resulting computational method is applicable
to a class or arbitrary plates with arbitrary vibratory information on the plate,
a method for discretising the plate must be included. In three-dimensional
problems this will generally mean that the true plate P is replaced
by an approximate plate. For example a set of triangles can be used to
approximate a plate of arbitrary shape. Thus we may write
P »
~
P
=
m å
1
Dj
~
P
,
(13)
where each Dj [(P)\tilde] is a triangle.
The velocity potential j can be evaluated by using a numerical
approximation of the integral in (10). Perhaps the most
straightforward approach, and the one most often adopted, is to use the
mid-point numerical integration rule. This method is otherwise known
as the simple source method. The method is explained in Schenck
[2],
for example. If the mid-points of each element Dj P is
qj for j=1,2,..,m , then the application of the mid-point rule to (10)
gives
j(p) » -
1
2 p
å
j
\sf area (Dj
~
P
)
eikrj
rj
vj ,
(14)
where area(Dj [(P)\tilde]) is the area of the element Dj[(P)\tilde]j, rj = r(p,qj) is the distance between p
and qj and vj = v(qj). Clearly, since the integrand in
(10) tends to become more oscillatory with increasing k,
the accuracy of this method tends to deteriorate as k increases.
Furthermore, if p is near or on the plate then the numerical
approximation (14) is likely to be poor.
A more natural approach is by the use of product integration in which the
potentially badly behave weighting function [(eikr)/(2 pr)]
is incorporated into the quadrature rule. The normal velocity on
[(G)\tilde] is expressed in the form
~
v
(q) »
n å
j=1
v(pj)
~
c
j
(q) (q Î
~
P
)
(15)
where
[(c)\tilde]1, [(c)\tilde]2, ..., [(c)\tilde]n are basis
functions with the usual properties:
~
c
i
(pj) = dij ,
n å
j=1
~
c
j
(q) = 1 (q Î
~
P
) .
The replacement of the true plate P by the approximate plate [(P)\tilde]
and the substitution of the approximation (15) allows us to write
{ Lk m}P » { Lk
~
m
}[(P)\tilde] = { Lk
n å
j=1
m(pj)
~
c
j
}[(P)\tilde] (p) =
n å
j=1
mj { Lk
~
c
j
}[(P)\tilde] (p) .
(16)
For a particular wavenumber and a particular value of p, having
calculated the
{ Lk [(c)\tilde]j }[(P)\tilde], an approximation
to the velocity potential j(p) can be obtained by the
summation (16).
The { Lk [(c)\tilde]j }[(P)\tilde] can be evaluated by mapping the
elements of the approximate plate onto standard elements. In the case when
the integrands are bounded then a standard numerical integration
technique can be used to evaluate the
{ Lk [(c)\tilde]j }[(P)\tilde].
For the case when the Dj [(P)\tilde] are triangles, the most
efficient methods are obtained through the use of the formulae in
Laursen and Gellert [8].
When p Î [(P)\tilde] the integrand in (10) is
singular and hence at least one of the
{ Lk [(c)\tilde]j }[(P)\tilde] is singular. In this case
special techniques need to be employed. One of the most effective
way of treating with these integrals is to change the variables to
polar coordinates, transforming them into regular integrals.
This technique is described in Banerjee and Butterfield
[9],
for example.