This work compares two Nitsche‐type approaches to treat non‐conforming triangulations for a high‐order discontinuous Galerkin (DG) solver for the acoustic conservation equations. The first approach (point‐to‐point interpolation) uses inconsistent integration with quadrature points prescribed by a primary element. The second approach uses consistent integration by choosing quadratures depending on the intersection between non‐conforming elements. In literature, some excellent properties regarding performance and ease of implementation are reported for point‐to‐point interpolation. However, we show that this approach can not safely be used for DG discretizations of the acoustic conservation equations since, in our setting, it yields spurious oscillations that lead to instabilities. This work presents a test case in that we can observe the instabilities and shows that consistent integration is required to maintain a stable method. Additionally, we provide a detailed analysis of the method with consistent integration. We show optimal spatial convergence rates globally and in each mesh region separately. The method is constructed such that it can natively treat overlaps between elements. Finally, we highlight the benefits of non‐conforming discretizations in acoustic computations by a numerical test case with different fluids.