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The 2.5D acoustic finite and boundary element methods efficiently solve wave propagation problems for longitudinally invariant geometries by employing a wavenumber decomposition in the invariant direction.However, evaluating the improper Fourier integral across the wavenumber spectrum remains a significant computational challenge, as it relies on the numerical integration of a continuous spectrum resolved via discrete sampling strategies. While adaptive sampling schemes reduce the number of required 2D calculations, existing methods relying on purely local relative errors experience numerical instability when the spectral amplitudes approach zero. Furthermore, standard numerical quadrature struggles with the highly oscillatory inverse Fourier kernel, introducing additional integration errors. This paper proposes a robust adaptive wavenumber sampling approach that approximates the spectrum using strictly piecewise linear segments. This deliberate geometric restriction enables the exact analytical evaluation of the oscillatory integral via Filon-type quadrature, effectively eliminating numerical integration errors. To drive the sampling scheme, a hybrid local error norm is introduced to ensure stability near zero, combined with a global convergence check on the physical spatial pressure. Numerical examples demonstrate that the proposed approach drastically reduces computational effort while maintaining the strict accuracy of dense equidistant sampling.