This paper presents an original approach to design perfectly matched layers (PML) for anisotropic scalar wave equations. This approach is based, first, on the introduction of a modified wave equation and, second, on the formulation of general "perfectly matched" transmission conditions for this equation. The stability of the transmission problem is discussed by way of the adaptation of a high frequency stability (necessary) condition, and we apply our approach to construct PML suited for any convex domain with straight boundaries. A new variational formulation of the problem, including a Lagrange multiplier at the interface between the physical and the absorbing domains, is then set and numerical results are presented in 2D and 3D. These results show the efficiency of our approach when using constant damping coefficients combined with high order elements.