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This paper presents a novel boundary integral equation (BIE) formulation forthe two-dimensional time-harmonic water-waves problem. It utilizes acomplex-scaled Laplace's free-space Green's function, resulting in a BIE posedon the infinite boundaries of the domain. The perfectly matched layer (PML)coordinate stretching that is used to render propagating waves exponentiallydecaying, allows for the effective truncation and discretization of the BIEunbounded domain. We show through a variety of numerical examples that, despitethe logarithmic growth of the complex-scaled Laplace's free-space Green'sfunction, the truncation errors are exponentially small with respect to thetruncation length. Our formulation uses only simple function evaluations (e.g.complex logarithms and square roots), hence avoiding the need to compute theinvolved water-wave Green's function. Finally, we show that the proposedapproach can also be used to find complex resonances through a \emph{linear}eigenvalue problem since the Green's function is frequency-independent.

A complex-scaled boundary integral equation for time-harmonic water waves