ABSTRACT
For a long time there were numerous heated debates whether this theory is asymptotic at k → ∞ or not. Recently it was proved that the leading asymptotic term of the exact solution coincides with Kirchhoff’s prediction (see, for example, Taylor, 1981), but the proofs required too abstract mathematical instruments. We give here a simple and graphical proof. It is based upon the basic boundary integral equation (see Section 2.2). The only restriction is the established fact that the BIEs studied in this chapter are uniquely solvable for all k > 0, except a countable set of values {kn} (kn→∞, n→∞) corresponding to the eigenvalues of the corresponding interior boundary value problem. So, strictly speaking, the asymptotic property of Kirchhoff’s theory stated by the following theorem is valid only for regular high›frequency values of k. A more rened analysis should be carried out in order to prove that Kirchhoff’s solution is asymptotically valid also for irregular values of k (k → ∞), i.e., Kirchhoff’s approximation represents the leading asymptotic term of the solution for all large k. Assuming that p(x0) = o(1), k → ∞, x0 ∈ l-, we prove the following
THEOREM. Let the boundary contour l be smooth, convex, and acoustically hard, and x0 ∈ l+. Then
p(x0) = 2pinc(x0) + o(1), k →∞ (k ≠ kn). (2.114) Proof. If k ≠ kn, then the boundary integral equation in the case of the Neumann
boundary condition (∂p/∂n) = 0, x→ l, and with p(x0) ≡ 0(1), k →∞, is asymptotically equivalent to
p(y0) – 2 ∫ l+ p(y) ∂Φ(|y0 – y|)
∂ny dly = 2pinc(y0) + o(1), y0 → l+, (2.115)
being uniquely solvable. Here Φ(r) = (i/4)H (1)0 (kr), r = |y – y0|, is the Green’s function for full 2D space. Moreover, for the considered k, the integral operator of the left›hand side, I –G, has a bonded continuous operator (I –G)–1 in the Banach space C(l).
