ABSTRACT
It can be easily seen that in the case of the considered triangle the rst six terms i= 1, 2, . . . , 6 in the sum (4.109) and the other six terms i = 6, 7, . . . , 12 cancel each other if n + m is odd. Indeed, the second crowd of the six virtual sources is obtained from the rst six corresponding sources if we change (ξi, ηi) by (ξi + a, ηi + b). Expression Tmn(x, y) = cos(pimx/a) cosh [√(pim)2 – (ak)2 (b – y)/a] in Eq. (4.108), under constraints (4.112), is (–1)n+m Tmn(x+a, y+ b). Therefore, the contributions of the mentioned six pairs of virtual sources cancel each other if n +m is odd, so that there is no singularity in the denominator of expression (4.108), and consequently no eigenvalues. From this consideration we can conclude that formula (4.113) should be applied only for such combinations of m and n, when n +m is even. This is equivalent to
kmn = 2pi 3c √ m2 + 3n2 = 2pi3c
√ (n + 2l)2 + 3n2 = 4pi3c
√ n2 + n l + l2
∼ kmn = 4pi3c √ m2 +mn + n2, ∀m,n = 0, 1, 2, . . . .
