The negative-mass instability (NMI), previously found in ion traps, appears as a distinct regime of the sideband instability in nonlinear plasma waves with trapped particles. As the bounce frequency of these particles decreases with the bounce action, bunching can occur if the action distribution is inverted in trapping islands. In contrast to existing theories that also infer instabilities from the anharmonicity of bounce oscillations, spatial periodicity of the islands turns out to be unimportant, and the particle distribution can be unstable even if it is flat at the resonance. An analytical model is proposed which describes both single traps and periodic nonlinear waves and concisely generalizes the conventional description of the sideband instability in plasma waves. The theoretical results are supported by particle-in-cell simulations carried out for a regime accentuating the NMI effect. Introduction. -It is well known that bounce oscillations of particles autoresonantly trapped in a wave can couple to wave sidebands, rendering them unstable [1][2][3]. The sideband instability (SI) was extensively studied in the past [4][5][6][7][8], more recently in application to free electron lasers [9] and storage rings [10], and now is attracting renewed attention [11,12] in the context of intense laserplasma interactions (LPI) and the associated trappedparticle modulational instability (TPMI) [13], which is the SI's geometrical-optics limit [14]. Yet little effort was paid to unifying SI theories that appeared after the original Kruer-Dawson-Sudan work [1], further termed KDS. As a consequence, their results are often neglected today, and that, in turn, leads to misapplications [15]. Thus, even though quantitative predictions may be better left to simulations in any case, a transparent theory is needed (particularly as a practical tool for interpreting LPI-related numerical data) that would both comprehensively capture and elucidate the SI paradigmatic physics.