Approximations to
the properties of the acoustic medium can be found through
applying numerical integration technique to (11).
This requires us to represent the plate by a set of panels and
go on to approximate the integral (11); rewriting it
in discrete form.
In order that the resulting computational method is applicable
to a class or arbitrary plates there must be a facility
for representing the plate as a set of panels.
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
,
(14)
where each Dj [(P)\tilde] is a triangle.
Triangulations of circular and square plates are illustrated in
sections 7 and 8 of this document.
The velocity potential j can be evaluated by using a numerical
approximation of the integral in (11). 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
[11],
for example. If the mid-points of each element Dj [(P)\tilde] is
qj for j=1,2,..,m , then the application of the mid-point rule to (11)
gives
j(p) » -
1
2 p
å
j
\sf area (Dj
~
P
)
eikrj
rj
vj ,
(15)
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
(11) 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 (15) is likely to be poor.
However, the main difficulty with this approach is that the
approximation is only directly applicable
in the case of the Neumann condition.
A more natural approach is by the use of collocation in which the
potentially badly behaved weighting function [(eikr)/(2 pr)]
is incorporated into the quadrature rule. The normal velocity on
[(P)\tilde] is expressed in the form
~
v
(q) »
n å
j=1
v(pj)
~
c
j
(q) =
n å
j=1
vj
~
c
j
(q) (q Î
~
P
)
(16)
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
)
and vj=v(pj), the velocity at the jth collocation point.
The replacement of the true plate P by the approximate plate [(P)\tilde]
and the substitution of the approximation (16) 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) .
(17)
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 (17).
The { Lk [(c)\tilde]j }[(P)\tilde] can be evaluated by mapping the
elements of the approximate plate onto the standard triangle with
vertices (0,0), (0,1) and (1,0).
When
the integrands are bounded then a standard numerical integration
technique can be used to evaluate the
{ Lk [(c)\tilde]j }[(P)\tilde].
Efficient methods may be obtained through the use of the
Gaussian Quadrature formulae in
Laursen and Gellert [8].
When p Î [(P)\tilde] the integrand in (11) 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 these integrals is to change the variables to
polar coordinates, transforming them into regular integrals.
The method employed for discretising the Lk
operator is described in Kirkup [2,5,6].