υn(y) is also written in a similar form, with x replaced by y/b in the right-hand sides of (12) and (13).
Summing up the above, the right-hand side of (9) is given as ζｊ(x,y) = uｍ(x)υｎ(y) from (11). Depending on the combination of odd and even numbers of m and n, these are categorized into four types, as follows:
1) ζｊ = u2m + 1(x)υ2n(y); which is odd in x and even in y, and referred to as F(X) type.
2) ζｊ = u2m(x)υ2n + 1(y); which is even in x and odd in y, and referred to as F(Y) type.
3) ζj = u2m(x)υ2n(y); which is even in both x and y, and referred to as F(Z) type.
4) ζj = u2m + 1(x)υ2n+l(71); which is odd in both x and y, and referred to as F(N) type.
In numerical computations, the mode numbers in these types are defined as in Table 1.
In fact, eq.(10) for the diffraction problem can also be categorized into the same four types as stated above.
Therefore, taking advantage of the fact that the pressure has the same symmetries as those of mode shapes, the unknowns in the integral equation (9) can be diminished to one quarter of the original.
3. NUMERICAL CALCULATION METHOD
The problem here is how accurately we solve the integral equation (9) with fewer unknowns even for very short wavelengths.
To reduce the number of unknowns, it may be indispensable to adopt a higher-order panel method. In this paper, the unknown pressure is represented by use of bi-
cubic B-spline functions, in the form
453-2.gif
Here Bk(x) and Bl(y) are normalized cubic B-spline functions, which can be obtained by Boor-Cox's recursion formula6).
NX and NY are the number of panel division in the x- and y-directions. Since one cubic spline function extends its influence over four panels, the number of total unknowns, αkl in (16), is (NX + 3) * (NY + 3).
To determine these unknowns with good accuracy, a Galerkin scheme is employed. That is, after substituting (16) in (9), both sides of (9) are multiplied by Bp(X)Bq(y), where p = 0 NX + 2 and q = 0〜NY + 2, and integrated over the plate, SH.
This procedure gives a linear system of simultaneous equations, in the form
In general, the above integrals may be evaluated using a numerical quadrature. Among these, (18) and (20) are relatively easy to evaluate, because these integrands are simply products of independent functions with respect to x and 71, respectively. For these integrals a Clenshaw-Curtis quadrature is used, with absolute error less than 10-7 required.
The most time-consuming part is evaluating (19). However, when the structure is discretized into panels of equal size, the amount of computations can be drastically diminished by taking advantage of relative similarity relations in the integrals.
For example, let us consider the case where the field point (x,y) is located at P1 and the panel number to be integrated is j = 1 (see Fig. 2). This integral with respect to (ξ,η) is equivalent to the case where the field point (x,y) is at P3 and the panel to be integrated is j = 3. (Of course, some care must be paid to the order of B-spline functions.)
Therefore, it is enough to consider only one panel for the integral with respect to (ζ,η) but for all the field
