We study sound-soft time-harmonic acoustic scattering by general scatterers, including fractal scatterers, in 2D and 3D space. For an arbitrary compact scatterer $\Gamma$ we reformulate the Dirichlet boundary value problem for the Helmholtz equation as a first kind integral equation (IE) on $\Gamma$ involving the Newton potential. The IE is well-posed, except possibly at a countable set of frequencies, and reduces to existing single-layer boundary IEs when $\Gamma$ is the boundary of a bounded Lipschitz open set, a screen, or a multi-screen. When $\Gamma$ is uniformly of $d$-dimensional Hausdorff dimension in a sense we make precise (a $d$-set), the operator in our equation is an integral operator on $\Gamma$ with respect to $d$-dimensional Hausdorff measure, with kernel the Helmholtz fundamental solution, and we propose a piecewise-constant Galerkin discretization of the IE, which converges in the limit of vanishing mesh width. When $\Gamma$ is the fractal attractor of an iterated function system of contracting similarities we prove convergence rates under assumptions on $\Gamma$ and the IE solution, and describe a fully discrete implementation using recently proposed quadrature rules for singular integrals on fractals. We present numerical results for a range of examples and make our software available as a Julia code.

翻译：我们研究一般散射体（包括分形散射体）在二维和三维空间中的软声场时谐声散射问题。对于任意紧散射体 $\Gamma$，我们将亥姆霍兹方程的狄利克雷边值问题重新表述为 $\Gamma$ 上涉及牛顿势的第一类积分方程。该积分方程在可数频率集之外是适定的，当 $\Gamma$ 为有界利普希茨开集边界、屏幕或多屏幕时，可退化为现有的单层边界积分方程。当 $\Gamma$ 在我们精确意义下一致具有 $d$ 维豪斯多夫维数（即 $d$-集）时，方程中的算子是定义在 $\Gamma$ 上关于 $d$ 维豪斯多夫测度的积分算子，其核函数为亥姆霍兹基本解。我们提出该积分方程的分片常数伽辽金离散方案，该方案在网格宽度趋于零时收敛。当 $\Gamma$ 为压缩相似迭代函数系统的分形吸引子时，我们在对 $\Gamma$ 和积分方程解的假设下证明了收敛速率，并描述了采用最近提出的分形奇异积分求积规则的全离散实现方法。最后给出多组数值算例结果，并以 Julia 代码形式开源我们的软件。