Harmonic balance

Harmonic balance substitutes a finite Fourier series into a differential equation and projects the residual onto the retained harmonics. It determines their amplitudes and phases. A nonlinear equation generally generates additional harmonics, so vanishing of the projected residual alone does not make the finite series an exact solution. In a weakly nonlinear expansion, nonresonant generated harmonics can be found at the next order by solving their linear equations.