The class of Trefftz methods has been extensively employed for solving wave problems (see, e.g, [1]-[2], among other references). In addition to their main feature of using basis functions that are solutions of the underlying considered PDE, these methods exhibit interesting abilities for limiting numerical pollution while preventing -to some extent- prohibitive computational cost increases, particularly when equipped with the GMRES method. However, the Trefftz methods present the disadvantage of ignoring the propagative nature of the model due to the double integration that occurs in the variational formulation that yields in replacing the trace of the numerical solution at the element interfaces with Discontinuous Galerkin numerical fluxes. Given that, we propose a new approach consisting of deriving the Trefftz variational formulation in the context of a general hyperbolic framework. This is accomplished by introducing suitable intermediate variables and performing a spectral decomposition of the Flux operators. This new approach allows a physical characterization of the propagation of the field, which happens to be crucial for ensuring the stability of the formulation. Numerical results will be presented to highlight the salient features of the proposed solution methodology and illustrate its potential for efficiently solving high-frequency three-dimensional wave problems. In addition, a procedure for reducing roundoff errors based on a principal component analysis will be presented with a performance assessment analysis [3]. References [1] O. Cessenat, and B. Despres. "Application of an ultraweak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problem." SIAM journal on numerical analysis 35.1 (1998): 255-299. [2] R. Hiptmair, A. Moiola, and I. Perugia. "A survey of Trefftz methods for the Helmholtz equation." Building bridges: connections and challenges in modern approaches to numerical partial differential equations (2016): 237-279. [3] H. Barucq, A. Bendali, J. Diaz, & S. Tordeux. (2021). Local strategies for improving the conditioning of the plane-wave Ultra-Weak Variational Formulation. Journal of Computational Physics, 441, 110449.