Numerical Modeling of Nonlinear Scalar Wave Propagation in a One-Dimensional Elastic Media: Energy Spectrum and Waveform Analysis

This study numerically investigated nonlinear scalar wave propagation in a one-dimensional elastic medium using a discrete chain of coupled oscillators. Linear elastic models proved inadequate for capturing amplitude-dependent effects seen in seismic waves, nonlinear acoustics, and heterogeneous materials. To overcome this limitation, two nonlinear extensions of the classical wave equation were developed: an asymmetric quadratic strain-gradient model and a symmetric formulation intended to maintain waveform symmetry and improve numerical stability. The governing equations were solved with an explicit finite-difference time-domain scheme employing high-order five-point spatial discretization, while stability was preserved through the Courant–Friedrichs–Lewy condition. Simulations with weak nonlinearity revealed clear departures from linear behavior, such as waveform distortion, vertical asymmetry, spectral broadening, and partial reflection. Spectral analysis detected secondary frequency components at approximately 3.8 Hz, 6.5 Hz, and 8.8 Hz—absent in linear cases—indicating nonlinear energy transfer to higher harmonics. Comparative evaluation demonstrated that the symmetric model offered superior numerical stability and preserved waveform symmetry. The findings confirmed that even mild elastic nonlinearity substantially modified wave evolution and energy distribution. The proposed framework established a reliable foundation for future work, including extensions to higher-dimensional models and incorporation of more realistic material properties. This research addressed a key gap by providing detailed insights into nonlinear mechanisms affecting waveform symmetry and energy spectra, with potential benefits for enhanced seismic hazard prediction and acoustic signal processing in engineering.