Finite-beam modeling of non-collinear wave mixing in bounded media

Wave mixing is a nonlinear interaction process in which two or more waves exchange energy to generate new frequency components and is widely used in acoustics, ultrasonics, and materials characterization. The efficiency and directionality of the generated waves depend sensitively on material properties, making wave mixing a powerful probe of nonlinear responses in complex media. Although the optimal conditions for maximizing mixing amplitude are well established, predicting the actual amplitude under real experimental conditions remains challenging due to several factors, including the lack of a well-defined interaction volume and the resulting oscillatory integrals. This work addresses this limitation by extending existing models to incorporate finite beam width and the physical boundaries of the medium as constraints on the interaction region, yielding a well-defined interaction volume with good agreement with experimental results. The refined model improves understanding of wave mixing in bounded media and enables more accurate prediction of sensor placement, alignment, and material-dependent responses.