Rayleigh Quotient Eigenvalue-Based Array Beamforming Optimization for Targeted Angular Energy Concentration in Underwater Acoustic Energy Transfer

Underwater acoustic energy transmission (UAET) is critical for sustaining long-term operations of underwater platforms, but its efficiency is constrained by the limited aperture of underwater receivers—requiring acoustic energy to be concentrated within a pre-defined target angular domain. Existing array-weighting methods face inherent limitations: traditional window-based techniques optimize mainlobe–sidelobe trade-offs rather than target-specific energy concentration, while intelligent algorithms suffer from high computational cost, quasi-optimality, and poor reproducibility. To address these gaps, this study proposes an array beam energy aggregation optimization method based on Rayleigh quotient eigenvalues for UAET. First, a rigorous mathematical model of the acoustic energy concentration problem was established: by defining a target-domain energy operator matrix RΘ with a Toeplitz–sinc structure (Hermitian positive definite), the energy-focusing problem was transformed into a tractable linear algebra problem. Second, the optimization objective of maximizing target-domain energy was formulated as a generalized Rayleigh quotient maximization problem, where the optimal amplitude weights correspond to the eigenvector of the maximum eigenvalue of RΘ—solved via Cholesky whitening and eigenvalue decomposition to ensure theoretical optimality and low computational complexity. Comprehensive validations were conducted via simulations and underwater physical experiments. Simulations on 1D uniform linear arrays and 2D 4-layer circular ring arrays showed that the proposed method outperformed traditional weighting methods and PSO in target angular energy concentration: for the 16-element linear array, its energy radiation efficiency in the 30° domain was 14% higher than classical methods (Blackman weighting). Underwater physical tests further confirmed its superiority: for the 4-layer circular ring array at 1 m, the acoustic energy efficiency in the 30° target domain reached 21.5% higher than Blackman weighting. Additionally, the method exhibited strong adaptivity (dynamic weight adjustment with target angular width) and scalability (performance improvement with array size), meeting UAET’s real-time and reliability requirements. This work provides a theoretically optimal and engineering-feasible solution for directional acoustic energy transfer in underwater environments, offering valuable insights for UAET system design.