We present a novel approach for simulating acoustic (pressure) wave propagation across different media separated by a diffuse interface through the use of a weak compressibility formulation. Our method builds on our previous work on an entropy-stable discontinuous Galerkin spectral element method for the incompressible Navier-Stokes/Cahn-Hilliard system %\cite{manzanero2020entropyNSCH}% (Manzanero et al. (2020)), and incorporates a modified weak compressibility formulation that allows different sound speeds in each phase. We validate our method through numerical experiments, demonstrating spectral convergence for acoustic transmission and reflection coefficients in one dimension and for the angle defined by Snell's law in two dimensions. Special attention is given to quantifying the modeling errors introduced by the width of the diffuse interface. Our results show that the method successfully captures the behavior of acoustic waves across interfaces, allowing exponential convergence in transmitted waves. The transmitted angles in two dimensions are accurately captured for air-water conditions, up to the critical angle of $13^\circ$. In a final example, we show a three-dimensional wave transmission from air into water to demonstrate the potential of this methodology for addressing general multiphase acoustic problems. This work represents a step forward in modeling acoustic propagation in incompressible multiphase systems, with potential applications to marine aeroacoustics.

翻译：我们提出了一种新颖的方法，通过弱可压缩性公式模拟声压波在不同介质间通过扩散界面的传播。该方法基于我们先前关于不可压缩Navier-Stokes/Cahn-Hilliard系统的熵稳定间断伽辽金谱元方法的工作（Manzanero等人，2020），并引入了一种改进的弱可压缩性公式，允许各相中具有不同的声速。我们通过数值实验验证了该方法，展示了一维声学透射与反射系数的谱收敛性，以及二维中由Snell定律定义角度的收敛性。特别关注了量化由扩散界面宽度引入的建模误差。结果表明，该方法成功捕捉了声波跨越界面的行为，实现了透射波的指数收敛。在空气-水条件下，二维透射角度可准确捕捉至临界角$13^\circ$。在最后一个示例中，我们展示了三维声波从空气到水的透射过程，以证明该方法在处理广义多相声学问题中的潜力。本工作代表了在不可压缩多相系统中建模声学传播的重要进展，在海洋航空声学等领域具有潜在应用价值。