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We consider the classical coupled, combined-field integral equationformulations for time-harmonic acoustic scattering by a sound soft boundedobstacle. In recent work, we have proved lower and upper bounds on the $L^2$condition numbers for these formulations, and also on the norms of theclassical acoustic single- and double-layer potential operators. These boundsto some extent make explicit the dependence of condition numbers on the wavenumber $k$, the geometry of the scatterer, and the coupling parameter. Forexample, with the usual choice of coupling parameter they show that, while thecondition number grows like $k^{1/3}$ as $k\to\infty$, when the scatterer is acircle or sphere, it can grow as fast as $k^{7/5}$ for a class of `trapping'obstacles. In this paper we prove further bounds, sharpening and extending ourprevious results. In particular we show that there exist trapping obstacles forwhich the condition numbers grow as fast as $\exp(\gamma k)$, for some$\gamma>0$, as $k\to\infty$ through some sequence. This result depends onexponential localisation bounds on Laplace eigenfunctions in an ellipse that weprove in the appendix. We also clarify the correct choice of coupling parameterin 2D for low $k$. In the second part of the paper we focus on the boundaryelement discretisation of these operators.  We discuss the extent to which the bounds on the continuous operators arealso satisfied by their discrete counterparts and, via numerical experiments,we provide supporting evidence for some of the theoretical results, bothquantitative and asymptotic, indicating further which of the upper and lowerbounds may be sharper.

Condition number estimates for combined potential integral operators in acoustics and their boundary element discretisation