Wave propagation in finite discrete chains unravelled by virtual measurement of dispersion properties

Travelling waves in circuit chains are studied to measure continuous dispersion. A lock-in frequency meter (LIF) is suitable for precisely determining k for each set $\omega$ of waves in finite alternate LC chains, where LIF has been proven to be more accurate than the fast Fourier transform. In addition to the $\omega$–k measurement, the wave impedance spectrum of the travelling wave can be measured simultaneously, for investigating the dispersion and splitting of pulse propagation. The measured dispersion is validated to be consistent with the derived theoretical equations. The result provides an independent way to precisely obtain dynamical system properties for chains composed of non-ideal components, such as resistors for researching non-Hermitian behaviour under dissipation. Systematical mapping of relative deviation dependence of wave dispersion measurement with LIF on different chain length and component variation is studied, indicating boundaries of 1%, 0.1%, and 0.01% precision for guidance of experiments.