© 2018 authors. Published by the American Physical Society. Published by the American Physical Society under the terms of the »https://creativecommons.org/licenses/by/4.0/» Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. We introduce a perturbation-theory, mode-analysis method for longitudinal multibunch instabilities driven by the higher harmonic cavity (HHC) fundamental mode. The method, based on the exact solution of the unperturbed particle motion in the rf bucket and suitable for modeling the effect of cavities with general settings, is applied to study the feasibility of reutilizing the existing Advanced Light Source (ALS) HHCs in the ALS Upgrade (ALS-U). We find that with ALS cavities the ALS-U would be susceptible to a fast =1 mode instability. Interestingly, the instability is driven by the imaginary rather than the real part of the cavity fundamental-mode impedance.