  We study the dynamics of isospectral granular chains that are highly tunable
due to the nonlinear Hertz contact law interaction between the granular
particles. The system dynamics can thus be tuned easily from being linear to
strongly nonlinear by adjusting the initial compression applied to the chain.
In particular, we introduce both discrete and continuous spectral
transformation schemes to generate a family of granular chains that are
isospectral in their linear limit. Inspired by the principle of supersymmetry
in quantum systems, we also introduce a methodology to add or remove certain
eigenfrequencies, and we demonstrate numerically that the corresponding
physical system can be constructed in the setting of one-dimensional granular
crystals. In the linear regime, we highlight the commonalities and differences
in the elastic wave transmission characteristics of such isospectral systems,
and emphasize that the presented mathematical framework allows one to suitably
tailor the wave transmission through a general class of granular chains.
Moreover, we show how the dynamic response of these structures deviates from
its linear limit as we introduce Hertzian nonlinearity in the chain and how
nonlinearity breaks the notion of linear isospectrality.
